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Izvestiya: Mathematics, 2023, Volume 87, Issue 1, Pages 154–199
DOI: https://doi.org/10.4213/im9250e
(Mi im9250)
 

This article is cited in 6 scientific papers (total in 6 papers)

Deterministic and random attractors for a wave equation with sign changing damping

Q. Changa, D. Lia, Ch. Suna, S. V. Zelikabc

a School of Mathematics and Statistics, Lanzhou University, China
b University of Surrey, Department of Mathematics, UK
c Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
References:
Abstract: The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping coefficient depends explicitly on time and may change sign. It is shown that in the case, where the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipation rate remains positive and that the conditions of this type are not sufficient in the linear case. Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rate), it is shown that the considered system possesses smooth uniform attractors as well as non-autonomous exponential attractors. In the case where the mean is not uniform (which corresponds to the random dissipation rate, for instance, when this dissipation rate is generated by the Bernoulli process), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have infinite Hausdorff and fractal dimensions. The simplified model example demonstrating infinite-dimensionality of the random attractor is also presented.
Keywords: damped wave equation, negative damping, random dynamics, asymptotic regularity, infinite-dimensional attractors.
Funding agency Grant number
Engineering and Physical Sciences Research Council EP/P024920/1
National Natural Science Foundation of China 11471148
11522109
11871169
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1623
Supported by Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2019-1623 with Ministry of Education and Science of the Russian Federation. Also supported by EPSRC EP/P024920/1 (the UK) и NSFC grant nos. 11471148, 11522109, 11871169 (China).
Received: 02.08.2021
Bibliographic databases:
Document Type: Article
UDC: 517.956.35
MSC: 35L72, 35B41, 35L20
Language: English
Original paper language: Russian

§ 1. Introduction

The paper gives a comprehensive study of the semilinear wave equation

$$ \begin{equation} \partial_t^2 u+\gamma(t)\, \partial_t u-\Delta_x u+f(u)=g \end{equation} \tag{1.1} $$
in a bounded smooth domain $\Omega$ of $\mathbb{R}^3$ endowed with Dirichlet boundary conditions. Here, $\Delta_x$ is the Laplacian with respect to the variable $x\in\Omega$, $f(u)$ and $g(x)$ are given non-linear interaction function and the external force, respectively, and $\gamma(t)$ is the dissipation rate which, in contrast to the standard situation, may change sign.

Various types of equations of the form (1.1) is of a great permanent interest. On the one hand, they model many important phenomena arising in modern science, for instance, in quantum mechanics, see [1], [2], semiconductor devices (for example, Josephson junctions, see [3] and the references there), propagation of waves in a transmission wire (the so-called telegraph equation, see [4], [5]), geophysical flows (see, for example, [6], [7]), and mathematical biology (see, for example, [8]), etc.

On the other hand, this type of equations also possesses a nice and deep mathematical theory combining very different topics such as inverse scattering, harmonic analysis, Strichartz type estimates, non-concentration estimates and Pohozhaev–Morawetz inequalities, etc., which makes the theoretical study of these equations also interesting and important, see [9]–[12] and the references there.

It is believed that the analytic properties of solutions of (1.1) depend mainly on the sign and the growth rate of the non-linearity $f$ (the dissipative term $\gamma(t)\, \partial_t u$ is subordinated and is not essential when the solutions on a finite time interval are considered). Namely, if

$$ \begin{equation} f(u)=u|u|^p+\text{lower order terms}, \end{equation} \tag{1.2} $$
which will be always assumed in this paper, these properties depend on the value of the exponent $p$ (the sign assumption is already incorporated in this condition, and we will not consider the self-focusing case $f(u)\sim-u|u|^p$ in the present paper).

The key tool for the mathematical study of these equations is the so-called energy identity which can be obtained by formal multiplication of (1.1) by $\partial_t u$ and integration over $x$,

$$ \begin{equation} \frac d{dt}\biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+(F(u),1)-(g,u)\biggr)+\gamma(t)\|\partial_t u\|^2_{L^2}=0, \end{equation} \tag{1.3} $$
where $F(u):=\int_0^uf(v)\,dv$, which determines the energy phase space
$$ \begin{equation*} E:=[H^1_0(\Omega)\cap L^{p+2}(\Omega)]\times L^2(\Omega),\qquad \xi_u:=\{u,\partial_t u\}\in E \end{equation*} \notag $$
of the problem, and gives the natural control of the energy norm of a solution.

In the energy subcritical and critical cases $p<2$ and $p=2$, respectively, the global existence and uniqueness of the energy solutions (that is, solutions with finite energy) can be obtained relatively easily since the energy control (1.3) is enough to treat the non-linear term $f(u)$ as a perturbation (see, for example, [13]–[15]). In contrast to this, in the supercritical case $p>4$, the global well-posedness of equation (1.1) remains an open problem. Indeed, similarly to the 3D Navier–Stokes problem, we have here global existence of weak energy solutions (without uniqueness) and local existence of smooth solutions (which a priori may blow up in finite time), see [16], [17] and the references there for more details.

The most interesting here is the intermediate case $2<p\leqslant 4$. In this case, the mere energy control (1.3) is not sufficient to treat the non-linearity properly, but it remains subordinated to the linear part if more delicate space-time integrability properties for the solutions of the linear equation are used. The global well-posedness of energy solutions for the case of $\Omega=\mathbb{R}^3$ and $p<4$ was obtained in [18], [19].

The quintic case ($p=4$) is much more difficult since it is not clear how to “lift” the extra space-time integrability from linear equation to the non-linear one (at least in a straightforward way). For instance, in order to get the uniqueness of the solution, one needs the so-called Strichartz estimates for the solutions of (1.1) like

$$ \begin{equation} u\in L^4_{\mathrm{loc}}(\mathbb{R},L^{12}(\Omega)) \end{equation} \tag{1.4} $$
(see [12], [20], [21] and the references there). In order to prove that these Strichartz norms do not blow up in finite time, one usually exploits the so-called non-concentration estimates and Pohozhaev–Morawetz inequalities (see [10]–[12]). Such results were first obtained for the case $\Omega=\mathbb{R}^3$, but then have been extended to the case of bounded domains based on relatively recent results for Strichartz estimates in bounded domains (see [21]–[23]). Note also that it is still an open problem whether or not the extra regularity (1.4) holds for any energy solutions, so we will refer to the energy solutions satisfying (1.4) as Shatah–Struwe (SS) solutions.

We now discuss the asymptotic behaviour of solutions of (1.1) as $t\to\infty$. Of course, the structure of the dissipation term $\gamma(t)\, \partial_t u$ plays a crucial role here. The most studied is the case of strictly positive and separated from zero dissipation rate $\gamma(t)$; for instance, $\gamma\equiv \mathrm{const}>0$. In this case, the energy identity (1.3) readily gives in an immediate way the global Lyapunov functional. This functional trivializes the long-time dynamics and guarantees the convergence of trajectories to the set of equilibria. Nevertheless, even in this case, proving the asymptotic compactness (which is necessary for the proof of the convergence in the energy space) and smoothness of the so-called global attractor may be a non-trivial task. For energy subcritical case $p<2$, this result was obtained in [24], [25] (see also [14], [15], [26]–[28] and reference therein), the energy critical case $p=2$ was treated in [14], [29] (see also [17] for treating the non-autonomous case and [30] for degenerate case), the subcritical case $p<4$ was studied in [31] and [32] for the case of the whole space $\Omega=\mathbb{R}^3$ or periodic boundary conditions, respectively. The critical case $p=4$ in general bounded domains was considered in [33]. Note that the proof given there used in a crucial way the Lyapunov functional and cannot be extended to the non-autonomous case. This drawback was overcome in [34] for the case of periodic boundary conditions using the so-called energy-to-Strichartz estimates, but derivation of such estimates in the critical case for general domains is still an open problem. This is the reason why, in the present paper, we mainly consider the subcritical case $p<4$. It is also worth mentioning of the results that can be extended to the supercritical case $p>4$ using the so-called trajectory attractors technique to overcome possible non-uniqueness (see [16], [17] for details).

The next well-studied case is when the dissipation rate $\gamma$ is still non-negative, but may be vanish on some non-trivial subset of $\Omega$. In this case, identity (1.3) does not immediately give the global Lyapunov functional, and so some new technique should come into play. In the case of degenerate $\gamma=\gamma(x)$, but which is still non-negative, the results on the existence and further regularity of attractors are usually obtained based on a combination of two types of estimates: 1) Carleman type estimates, capable of delivering the global Lyapunov function, and 2) the exponential decay estimates for the linear equation ($f(u)=0$), which require the so-called geometric control conditions on the support of $\gamma$ (see [27], [35]–[39] and the references there). The complementary case when the degenerate dissipation rate $\gamma=\gamma(t)\geqslant0$ depends only on $t$, has also been intensively studied (see [40]–[43] and the references given there), although the general case $\gamma=\gamma(t,x)\geqslant0$ has not been sufficiently studied yet.

In contrast to this, not much is known for the case of sign-changing dissipation. The key difference here is that the right-hand side of the energy equality (1.3) is no more non-negative, so the global Lyapunov functional disappears, and even proving the global boundedness or/and dissipativity of solutions becomes quite a challenge. It is also worth mentioning that in the absence of a Lyapunov functional the associated dynamics can easily be chaotic (this may be observed even in the simplest examples; see , for example, [44]).

As far as we know (at least for PDEs of the form (1.1)), only the case where the negative part of the dissipative coefficient

$$ \begin{equation*} \gamma_-(t):=-\max\{0,-\gamma(t)\} \end{equation*} \notag $$
is small with respect to the positive part and can be treated as a perturbation was studied in the literature (see [40]–[42], [45]–[48] and the references there). In addition, in all these papers, the linear equation (which corresponds to the case $f(u)=g=0$) is assumed to be stable, an so the non-linearity is, in a sense, treated as a perturbation which does not affect the dissipation mechanism. Some exception here is the paper [49], where doubly non-linear strongly damped wave equation with sign-changing dissipation rate was considered.

On the other hand, the classical model example here is the Van der Pol equation

$$ \begin{equation} y''+(y^2-1)y'+y=0, \end{equation} \tag{1.5} $$
which describes generation of auto-oscillations in various physical systems arising, for example, in radio-electronics, classical mechanics, biology, etc. (see [44] and the references given there). The PDE analogue of this equation as well as related Fitz–Hugh–Nagumo equation, describe oscillatory processes in excitable media (see, for example, [50]). Equation (1.1) can be considered as a simplified model for such problems where the non-linear damping term is replaced by $\gamma(t):=y^2(t)-1$ for some special solution $y(t)$ of an ODE, for example, of the Van der Pol equation or of its multi-dimensional analogue.

The example of the Van der Pol equation suggests that the assumption that the linear part of the equation must be stable is too restrictive and that the equation can be stabilized by the non-linear terms. As we will see later, this is exactly the case for equation (1.1). Surprisingly, the global stability analysis of the non-linear problem (1.1), with $f(u)$ satisfying (1.2) with $p>0$, is simpler than in the linear case, and reasonable conditions for $\gamma(t)$ can be stated in this case.

Thus, the ultimate goal of the present paper is to give a detailed study of equation (1.1) in the super-linear case where $f$ satisfies (1.2) with $p>0$. Our main assumption on the dissipation rate is as follows:

$$ \begin{equation} \liminf_{T\to\infty}\frac1T\int_{\tau-T}^\tau \biggl(\frac12\gamma_+(t)-\frac{p+2}{p+4}\gamma_-(t)\biggr)\, dt>0,\qquad \tau\in\mathbb{R}, \end{equation} \tag{1.6} $$
here $\gamma_+=\max\{0,\gamma\}$, $\gamma_-=\gamma_+-\gamma$. The heuristic arguments showing why this condition looks necessary are given in § 2.

We start with the discussion why the analogue of (1.6) does not work in the linear case. A simplest model example here is the following ODE:

$$ \begin{equation} y''+\gamma(t)y'+\omega^2 y=0 \end{equation} \tag{1.7} $$
with time-periodic $\gamma$. Here, the stability analysis is already an interesting and non-trivial task. Indeed, a standard periodic change of variables (see § 2) transform this equation to the classical Mathieu–Hill’s equation
$$ \begin{equation} y''+\langle\gamma\rangle y'+ (\omega^2+\psi(t))y=0, \end{equation} \tag{1.8} $$
where $\langle\gamma\rangle$ is the mean value of $\gamma$ over the period and the periodic function $\psi(t)$ is evaluated from $\gamma(t)$. This equation describes resonances in parametrically excited mechanical systems (see [44] and the references there); stability and bifurcations of limit cycles (see, for example, [51]), etc. It also can be interpreted (at least for $\langle\gamma\rangle=0$) as a Schrödinger equation with periodic potential which is central in the quantum theory of solids (see, for example, [52]). It is well-known that the stability analysis for this equation is complicated and is not described by conditions like (1.6); see [53] and references therein. In the case where $\gamma$ is not periodic (for instance, random), the situation becomes more difficult since the effects related to Anderson localization come into play (see [52]).

In the present paper, we demonstrate that assumptions like (1.6) do not work in the linear case. The following result (see Proposition 2.1) will be proved.

Proposition 1.1. Let $a,b,\omega>0$ be arbitrary. Then there exists a $\pi/\omega$-periodic function $\gamma\in L^1(0,\pi/\omega)$ such that $\langle\gamma_+\rangle=a$, $\langle\gamma_-\rangle=b$, and equation (1.7) is exponentially unstable.

However, this instability mechanism does not work in the superlinear case $p>0$, since, in contrast to the linear case, the frequency of internal oscillations grows with the energy growth and parametric resonances become impossible on higher energy levels. By this reason, under condition (1.6), the dissipation becomes prevalent at higher energy levels, which makes the equation globally dissipative (see § 2 for details). To preserve this effect in the case where $\gamma(t)$ s not periodic, we need to assume an extra regularity assumption on $\gamma$, which guarantees that the frequency of “external oscillations” of $\gamma$ does not grow with time. Namely, we assume that $\gamma$ is translation compact in the $L^1$-metric:

$$ \begin{equation} \gamma\in L^1_{\textrm{tr-c}}(\mathbb{R}). \end{equation} \tag{1.9} $$
Roughly speaking, assumption (1.9) means that $\gamma$ can be approximated in the mean uniformly with respect to time by smooth bounded functions (see § 3 for a rigorous definition). This condition is also natural, since otherwise one can construct time-growing solutions for the non-linear problem by arguing exactly as in the proof of Proposition 1.1.

As we will show in the present paper, there are two principally different cases depending on whether or not the limit in (1.6) is uniform with respect to $\tau\in\mathbb{R}$. The first case (where the limit is uniform) is more standard and is related the deterministic dissipation rate $\gamma(t)$, for example, $\gamma(t)$ is periodic or quasi/almost periodic in time, and the second one is natural for chaotic or random in time dissipation rates.

We start with the uniform case. The key result in this case is the following uniform dissipative estimate for the SS solutions of equation (1.1).

Theorem 1.1. Let the dissipation rate $\gamma(t)$ satisfy (1.9), and obey condition (1.6) uniformly with respect $\tau\in\mathbb{R}$. Assume also that $g\in L^2(\Omega)$, and the non-linearity $f$ satisfies (1.2) for some $0<p\leqslant 4$. Then, for every $\xi_\tau\in E$, there exist a unique SS solution $u(t)$, $t\geqslant\tau$, of problem (1.1) with the initial data $\xi_u\big|_{t=\tau}=\xi_\tau$, and such that

$$ \begin{equation} \|\xi_u(t)\|_E^2\leqslant Q(\|\xi_u(\tau)\|_E^2)e^{-\alpha (t-\tau)}+Q(\|g\|_{L^2}),\qquad t\geqslant\tau\in\mathbb{R}, \end{equation} \tag{1.10} $$
where the positive constant $\alpha$ and the monotone function $Q$ are independent of $t$, $\tau$, $u$ and $g$. Here and below, $\xi_u(t):=\{u(t),\partial_t u(t)\}$.

The proof of this result is given in § 4.

The dissipative estimate (1.10) makes it possible to apply the main techniques of the attractors theory to equation (1.1) in a more or less standard way. Indeed, Theorem 1.1 allows us to define the dynamical process $U(t,\tau)$, $t\geqslant\tau$, in the energy space $E=H^1_0(\Omega)\times L^2(\Omega)$ (under the condition $p\leqslant 4$, the Sobolev embedding $H^1\subset L^{p+2}$ holds, and so the term $L^{p+2}$ is not necessary in the definition of $E$), and study its attractors. Since the equation under consideration depends explicitly on time, one has to use the proper extensions of a global attractor to the non-autonomous case. One of possible extensions is the so-called uniform attractor (see [16] and the references there). By definition, a uniform attractor $\mathcal A_{\mathrm{un}}$ is a minimal compact set in $E$ which attracts all bounded subsets in $E$ uniformly with respect to $\tau\in\mathbb{R}$. Namely, for every bounded set $B$, we have the attraction

$$ \begin{equation} \lim_{s\to\infty}\sup_{\tau\in\mathbb{R}}\operatorname{dist}_E\bigl(U(\tau+s,\tau)B,\mathcal A_{\mathrm{un}}\bigr)=0, \end{equation} \tag{1.11} $$
where “$\operatorname{dist}$” stands for the Hausdorff distance in $E$ (see § 4 for more details). The following theorem will be proved in § 4.

Theorem 1.2. Under the conditions of Theorem 1.1, let, in addition, $p<4$. Then, the dynamical process $U(t,\tau)$ generated by solution operators of problem (1.1) possesses a uniform attractor $\mathcal A_{\mathrm{un}}$, which is a bounded set in the higher energy space $E^1:=[H^2(\Omega)\cap H^1_0(\Omega)]\times H^1_0(\Omega)$.

To describe the structure of a uniform attractor, we need, according to the general theory (see [16] for details) to consider not only equation (1.1), but also all its time shifts together with their limits in the proper topology. Namely, we need to consider the hull $\mathcal H(\gamma)$ of the initial dissipation rate $\gamma$,

$$ \begin{equation} \mathcal H(\gamma):=[T_h\gamma,\, h\in\mathbb{R}]_{L^1_{\mathrm{loc}}(\mathbb{R})},\qquad (T_h\gamma)(t):=\gamma(t+h), \end{equation} \tag{1.12} $$
where $[\,\cdot\,]_V$ stands for the closure in $V$. In particular, assumption (1.9) implies that $\mathcal H(\gamma)$ is compact in $L^1_{\mathrm{loc}}(\mathbb{R})$. For every $\eta\in \mathcal H(\gamma)$ we consider equation (1.1) with $\gamma$ replaced by $\eta$, and denote by $\mathcal K_\eta\subset L^\infty(\mathbb{R},E)$ the set of all solutions of this equation defined for all $t\in\mathbb{R}$ and bounded as $t\to-\infty$ (the so-called kernel of this equation in the terminology of Chepyzhov and Vishik; see [16]). Now the uniform attractor $\mathcal A_{\mathrm{un}}$ of problem (1.1) can be described as follows:
$$ \begin{equation} \mathcal A_{\mathrm{un}}=\bigcup_{\eta\in\mathcal H(\gamma)}\mathcal K_\eta\big|_{t=0}. \end{equation} \tag{1.13} $$
Moreover, following the general procedure, we may define the kernel sections by
$$ \begin{equation} \mathcal K_\eta(\tau):=\mathcal K_\eta\big|_{t=\tau},\qquad \eta\in\mathcal H(\gamma),\quad \tau\in\mathbb{R}. \end{equation} \tag{1.14} $$
It is known (see [16], [54] for details) that these sections are compact in $E$ and possess the strict invariance property:
$$ \begin{equation*} U_\eta(t,\tau)\mathcal K_\eta(\tau)=\mathcal K_\eta(t), \end{equation*} \notag $$
where $U_\eta(t,\tau)$ is the dynamical process generated by equation (1.1), with $\gamma$ replaced by $\eta\in\mathcal H(\gamma)$. Moreover, these sections enjoy the so-called pullback attraction property
$$ \begin{equation} \lim_{s\to\infty}\operatorname{dist}_E\bigl(U_\eta(\tau,\tau-s)B,\mathcal K_\eta(\tau)\bigr)=0. \end{equation} \tag{1.15} $$
By this reason, the above family of kernel sections $\mathcal K_\eta(t)$, $t\in\mathbb{R}$, are often referred as a pullback attractor associated with the dynamical process $U_\eta(t,\tau)$ (see [16], [54]–[56] for more details).

Similarly to global attractors for autonomous case, these kernel sections are usually compact and have finite fractal and Hausdorff dimensions, but in contrast to uniform attractors, the rate of attraction in (1.15) is typically not uniform with respect to $\tau\in\mathbb{R}$ (and $\eta\in\mathcal H(\gamma)$). Moreover, as elementary examples show, an exponentially repelling equilibrium may easily be a pullback “attractor” for the dynamical process considered.

One of the ways to overcome this drawback is to use the concept of an exponential attractor introduced in [57] and extended to the non-autonomous case in [58], [59] (see also the survey [60] and the references there). By definition, a non-autonomous exponential attractor $\mathcal M_\eta(t)$, $t\in\mathbb{R}$, for the dynamical process $U_\eta(t,\tau)$ is a semi-invariant family of compact sets which have finite Hausdorff and fractal dimensions and possesses the uniform exponential attraction property, namely, there exist a positive constant $\alpha$ and a monotone function $Q$ such that, for every bounded set $B$ of $E$,

$$ \begin{equation} \operatorname{dist}_E\bigl(U_\eta(\tau+s,\tau)B,\mathcal M_\eta(\tau+s)\bigr)\leqslant Q(\|B\|_E)e^{-\alpha s}, \end{equation} \tag{1.16} $$
uniformly with respect to $\tau\in\mathbb{R}$ (and also with respect to $\eta\in\mathcal H(\gamma)$). In particular, as is easily checked, $\mathcal K_\eta(t)\subset\mathcal M_\eta(t)$, if the exponential attractor exists. We also emphasize that, in contrast to the kernel sections, the non-autonomous exponential attractor $\mathcal M_\eta(t)$, $t\in\mathbb{R}$, is not only pullback attracting, but also forward in time (exponentially) attracting.

The next theorem, which will be proved in § 5, establishes the existence of a non-autonomous exponential attractor for the wave equation (1.1).

Theorem 1.3. Under the conditions of Theorem 1.2, let, in addition, the dissipation rate $\gamma$ is more regular: $\gamma\in L^{1+\varepsilon}_b(\mathbb{R})$ for some $\varepsilon>0$ (see § 4 for details). Then the dynamical process $U_\eta(t,\tau)$, $\eta\in\mathcal H(\gamma)$, associated with wave equation (1.1) possess non-autonomous exponential attractors $\mathcal M_\eta(t)$ which are bounded sets of the higher energy space $E^1$.

We now turn to the second (probably more interesting) case where the dissipativity assumption (1.6) is not uniform with respect to $\tau\in\mathbb{R}$. In this case, typically, the dissipation is not strong enough to provide boundedness of trajectories and dissipativity forward in time, and so a uniform attractor may fail to exist. Moreover, the kernels $\mathcal K_\eta(t)$, defined as above via all bounded solutions, may be either empty at all or too small to get any type of attraction, and so the theory should be properly modified.

A natural way to overcome this issue, which comes from the theory of random attractors (see [55], [56], [61] and the references there), is to replace bounded trajectories by tempered ones, and, respectively, bounded sets by tempered sets. Namely, a complete trajectory $u(t)$, $t\in\mathbb{R}$, of problem (1.1) is tempered if $\|\xi_u(t)\|_E$ grows as $t\to-\infty$ slower than any exponent and a family of bounded sets $B(t)$, $t\in\mathbb{R}$, is tempered if $\|B(t)\|_E$ grows as $t\to-\infty$ slower than any exponent. Then, the theory of kernel sections developed in [16], [54], can be naturally extended to the tempered case by considering tempered kernels (sets of all tempered complete trajectories) and tempered kernel sections (tempered pullback attractors), see [55], [56] and the references there, and this is exactly the key technical tool for dealing with the wave equation (1.1) in the non-uniform case (see § 6 for more details).

However, as in the bounded case, tempered kernel sections have an intrinsic drawback due to the absence of attraction forward in time, which disappears in random case where forward attraction usually holds in probability. Keeping also in mind that the non-uniformity with respect to $\tau\in\mathbb{R}$ in the dissipative condition (1.6) appears naturally when the dissipation rate is random (or chaotic), we introduce the required random formalism from the very beginning. Namely, we assume that there is a Borel probability measure $\mu$ on the hull $\mathcal H(\gamma)$ such that it is invariant and ergodic with respect to time shifts

$$ \begin{equation} T_h\colon\mathcal H(\gamma)\to\mathcal H(\gamma),\quad h\in\mathbb{R},\qquad (T_h\eta)(t)=\eta(t+h). \end{equation} \tag{1.17} $$
Now assumption (1.6) is replaced by
$$ \begin{equation} \int_{\eta\in\mathcal H(\gamma)} \biggl(\int_0^1\frac12\eta_+(t)-\frac{p+2}{p+4}\eta_-(t)\,dt\biggr)\, \mu(d\eta)>0, \end{equation} \tag{1.18} $$
and the initial assumption (1.6) holds for almost all $\eta\in\mathcal H(\gamma)$ due to the Birkhoff ergodic theorem.

Recall that a $\mu$-measurable set-valued function $\eta\to\mathcal A(\eta)\subset E$ is called a tempered random attractor for the family $U_\eta(t,\tau)\colon E\to E$ of dynamical processes if

1) $\mathcal A(\eta)$ are well-defined and compact in $E$ for almost all $\eta\in\mathcal H(\gamma)$;

2) the family of bounded sets $t\to \mathcal A(T_t\eta)$ is tempered for almost all $\eta\in\mathcal H$;

3) the strict invariance property holds: $U_\eta(t,0)\mathcal A(\eta)=\mathcal A(T_t\eta)$, $t\geqslant0$;

4) for any other measured tempered random set $\eta\to B(\eta)$,

$$ \begin{equation*} \lim_{s\to\infty}\operatorname{dist}_E\bigl(U_\eta(0,-s)B(T_{-s}\eta),\mathcal A(\eta)\bigr)=0 \end{equation*} \notag $$
for almost all $\eta\in\mathcal H(\gamma)$.

The next theorem, which will be proved in § 6, verifies existence of a tempered random attractor for equation (1.1).

Theorem 1.4. Let $g\in L^2(\Omega)$, the non-linearity $f$ satisfy condition (1.2) with $0<p<4$, and $\gamma$ satisfy condition (1.9). Assume also that the Borel probability measure $\mu$ on $\mathcal H(\gamma)$ is invariant and ergodic with respect to time shifts and assumption (1.18) is satisfied. Then the family of dynamical processes $U_\eta(t,\tau)$, $\eta\in\mathcal H(\gamma)$, possesses a tempered random attractor $\mathcal A(\eta)$. Moreover, this attractor is attracting forward in time in sense of convergence in measure:

$$ \begin{equation} \mu\textit{-}\lim_{t\to\infty}\operatorname{dist}_E\bigl(U_\eta(t,0)B(\eta),\mathcal A(T_t\eta)\bigr)=0 \end{equation} \tag{1.19} $$
for every tempered random set $B(\eta)$.

As usual, this random attractor is constructed by means of the tempered kernel sections $\mathcal K_\eta(t)$ for almost all $\eta\in\mathcal H(\gamma)$, after which one sets $\mathcal A(\eta):=\mathcal K_\eta(0)$.

As a key model example of a random dissipation rate, we consider the piecewise constant function

$$ \begin{equation} \eta(t):=\eta_n,\qquad t\in[n,n+1),\quad n\in\mathbb Z, \end{equation} \tag{1.20} $$
where $\{\eta_n\}_{n\in\mathbb Z}\in \Gamma:=\{a,-b\}^{\mathbb Z}$ is a Bernoulli scheme with two symbols $a>0$ and $b>0$. We assume that the value $a$ appears with probability $q$ (for some $0<q<1$) and the remaining value $-b$ appears with probability $1-q$. Let $\mu$ be a product measure on the Bernoulli scheme $\Gamma$. It is known (see, for example, [51]) that this measure is invariant and ergodic with respect to discrete shifts $T_l\colon \Gamma\to\Gamma$, $l\in\mathbb Z$. Moreover, the Bernoulli scheme $\Gamma$ endowed with the Tichonoff topology is compact and possesses a dense trajectory which we take as the initial $\gamma$, and construct $\gamma(t)$ by (1.20). In this case, the hull $\mathcal H(\gamma)$ generates the whole Bernoulli scheme $\Gamma$. Note also that the discrete shifts on $\Gamma$ are conjugated to the discrete shifts on the hull $\mathcal H(\gamma)$. Thus, the conditions of Theorem 1.4 will be satisfied if we verify (1.18). A simple algebra show that it is satisfied if and only if
$$ \begin{equation} aq-\frac{2(p+2)}{p+4}b(1-q)>0. \end{equation} \tag{1.21} $$
So, under condition (1.21), equation (1.1) with the dissipation rate generated by the Bernoulli process possesses a tempered random attractor.

Up to the moment, the application of the random attractors theory to equation (1.1) is more or less standard. However, there is a principal difference here. Namely, in contrast to the usual situation, we cannot guarantee that the constructed random attractor has a finite first moment. Moreover, we expect that

$$ \begin{equation} \int_{\eta\in\mathcal H(\gamma)}\|\mathcal A(\eta)\|_{E}\, \mu(d\eta)=\infty. \end{equation} \tag{1.22} $$
At least, we have this equality for the random absorbing set constructed in the proof of Theorem 1.4 in the case of Bernoulli process which satisfies (1.21) and does not satisfy the stronger assumption
$$ \begin{equation} \ln\bigl(e^{-a}q+e^{2(p+2)b/(p+4)}(1-q)\bigr)<0, \end{equation} \tag{1.23} $$
and we do not now how to obtain that this moment is finite for the attractor. This infiniteness has a drastic impact on the dynamics of the considered random system. Indeed, if (1.22) is infinite, then there are no reasons to expect that the random Lyapunov exponents (see [62]–[64]) will be finite and this, in turn, may lead to infinite-dimensionality of the corresponding random attractor $\mathcal A(\eta)$.

Our conjecture is that in this case the random attractor $\mathcal A(\eta)$ is indeed infinite-dimensional for almost all $\eta\in\mathcal H(\gamma)$ in this case.

Since the fact that random/stochastic perturbation of a “good” dissipative system with finite-dimensional attractor may lead to infinite-dimensional dynamics may potentially have a fundamental impact on the theory of random dynamical systems and, to the best of our knowledge, has never been considered before, we give in § 7 a simple model example demonstrating this effect. Namely, we consider the following infinite system of ODEs in a Hilbert space $H=l_2$:

$$ \begin{equation} u_1'+\eta(t)u_1=1,\qquad u_n'+n^4u_n=u_1u_n-u_n^3,\quad n=2,\dots\,. \end{equation} \tag{1.24} $$
In this case, if $\eta$ is generated by a Bernoulli process andsuch that (1.21) is satisfied and (1.23) is not satisfied (both for $p=0$), then the associated random attractor exists, but has infinite Hausdorff and fractal dimensions (see § 7 for details). Note that in this case the attractor is clearly finite-dimensional in the deterministic case, for example, if $\gamma=\mathrm{const}>0$ or satisfies condition (1.6) uniformly with respect to $\tau\in\mathbb{R}$.

The paper is organized as follows. In § 2 we will be mainly concerned with heuristic arguments demonstrating that the posed assumptions are natural and reasonable. In addition, we prove Proposition 1.1 and give a rigorous analysis of the ODE (1.1).

In § 3, we prove the key dissipative estimate (1.10). Rigorous definitions for weak energy and Shatah–Struwe solutions and a more rigorous discussion of the known result about global solvability of (1.1) are also presented there. Asymptotic compactness estimates which guarantee that a bounded ball of $E^1$ attracts exponentially all solutions of (1.1) in the uniformly dissipative case are presented in § 4 and uniform and exponential attractors for this case are constructed in § 5. The non-uniform and random cases are considered in § 6. In particular, Theorem 1.4 is proved there. Finally, the related model example where the dimension of a random attractor is infinite is studied in § 7.

§ 2. Preliminaries and heuristics

In this section, we show that the conditions on the mean value of the dissipation rate $\gamma(t)$ are not relevant for the case of linear equations and give some evidence that they are natural and, in a sense, necessary in the super-linear case.

2.1. Linear ODE

We start with the simplest, but already non-trivial case of a scalar equation

$$ \begin{equation} y''(t)+\gamma(t) y'(t)+ y(t)=0; \end{equation} \tag{2.1} $$
here we assume for simplicity that $\gamma(t)$ is smooth and $T$-periodic. Let $\gamma_0:=(1/T)\int_0^T\gamma(t)\,dt$ be the mean value of $\gamma$. Then the standard time-periodic change of variables
$$ \begin{equation*} y(t)=\exp\biggl\{-\frac12\int_0^t(\gamma(s)-\gamma_0)\,ds\biggr\}z(t) \end{equation*} \notag $$
reduces the equation to the damped version of the classical Mathieu–Hill’s equation
$$ \begin{equation} z''+\gamma_0z'+(1+\psi(t))z=0,\qquad \psi(t):=\frac14\bigl(\gamma_0^2-\gamma^2(t)-2\gamma'(t)\bigr). \end{equation} \tag{2.2} $$
The most studied here is the non-dissipative case $\gamma_0=0$ and $\psi(t)=\varepsilon\sin(\omega t)$, which corresponds to the original Mathieu equation. The instability in this equation is caused by the so-called parametric resonances and the instability zone (where the exponentially growing/decaying solutions of (2.2)) on the $(\omega,\varepsilon)$-plane touches the $\varepsilon=0$ line at infinitely many points $\omega=n/2$, $n\in\mathbb Z$, and forms the famous Arnold tongues (see [44], [51], [53] for more details). For non-zero dissipation rate $\gamma_0>0$, the number of tongues crossing $\varepsilon=\varepsilon_0>0$, becomes finite, but it grows as $\gamma_0\to 0$.

The above picture remain similar for a general periodic function $\psi$, but becomes much more complicated if the periodicity assumption is broken (see [52]).

Thus, the stability of equation (2.1) is an interesting and delicate problem and it is unlikely that more or less sharp conditions for it can be formulated in a simple way. The next proposition gives an alternative way to generate instability directly in equation (2.1), and has an independent interest.

Consider the class of functions

$$ \begin{equation*} \Gamma_{a,b}:=\biggl\{\gamma\in L^1_{\mathrm{per}}(0,\pi),\ \int_0^{\pi}\gamma_+(t)\,dt=a,\ \int_0^{\pi}\gamma_-(t)\,dt=b\biggr\}, \end{equation*} \notag $$
where $a,b\geqslant0$ are two given numbers, $\gamma_+=\max\{\gamma,0\}$ and $\gamma_-=\gamma_+-\gamma$. The following result holds.

Proposition 2.1. Let $\gamma\in \Gamma_{a,b}$, and let $\mu_+(\gamma)$ and $\mu_-(\gamma)$ be the maximal and minimal Lyapunov exponents for equation (2.1), respectively. Then

$$ \begin{equation} \sup_{\gamma\in\Gamma_{a,b}}\mu_+(\gamma)=\frac b{\pi},\qquad \inf_{\gamma\in\Gamma_{a,b}}\mu_-(\gamma)=-\frac a{\pi}. \end{equation} \tag{2.3} $$

Proof. We first mention that by the Liouville theorem,
$$ \begin{equation*} \mu_+(\gamma)+\mu_-(\gamma)=\frac{b-a}{\pi}, \end{equation*} \notag $$
and so we only need to prove the first equality in (2.3). Let us start with the estimate from above. To this end, multiplying equation (2.1) by $y'(t)$, we get
$$ \begin{equation} \frac12\frac d{dt}\bigl(y'(t)^2+y(t)^2)\bigr)=-\gamma(t)y'(t)^2\leqslant \gamma_-(t)y'(t)^2\leqslant\gamma_-(t)\bigl(y'(t)^2+y(t)^2\bigr). \end{equation} \tag{2.4} $$
Integrating this estimate, we arrive at
$$ \begin{equation*} y'(2\pi)^2+y(2\pi)^2\leqslant e^{2b}\bigl(y'(0)^2+y(0)^2\bigr) \end{equation*} \notag $$
which implies that $\mu_+(\gamma)\leqslant b/\pi$.

For the lower bound, we use an explicit construction of the function $\gamma_h(t)\in\Gamma_{a,b}$, depending on a small parameter $h$:

$$ \begin{equation} \gamma_h(t)=\begin{cases} \dfrac{a}{h}, &t\in[0,h], \\ 0, &t\in\biggl(h,\dfrac{\pi}2\biggr)\cup\biggl(\dfrac{\pi}2+h,\pi\biggr), \\ -\dfrac{b}{h}, &t\in\biggl[\dfrac{\pi}2,\dfrac{\pi}2+h\biggr]. \end{cases} \end{equation} \tag{2.5} $$
To find the Lyapunov exponents, we need to compute the eigenvalues of the period map related with this choice of the function $\gamma_h$. Let us denote by $U_h(t,s)$ the solution matrix related to equation (2.1), that is,
$$ \begin{equation*} \begin{pmatrix} y(t)\\y'(t)\end{pmatrix}:=U_h(t,s)\begin{pmatrix} y(s)\\y'(s) \end{pmatrix}. \end{equation*} \notag $$
Next, we decompose the required period map as
$$ \begin{equation*} P(h)=U_h\biggl(\pi,\frac{\pi}2+h\biggr)U_h\biggl(\frac{\pi}2+h,\frac{\pi}2\biggr) U_h\biggl(\frac{\pi}2,h\biggr)U_h(h,0), \end{equation*} \notag $$
and find the limit $P(h)$ as $h\to0$. Obviously,
$$ \begin{equation*} U_h\biggl(\frac{\pi}2,h\biggr)=U_h\biggl(\pi,\frac{\pi}2+h\biggr) =\begin{pmatrix}0&-1\\1&0\end{pmatrix}+O(h), \end{equation*} \notag $$
and the matrices $U_h(h,0)$ and $U_h(\pi/2+h,\pi/2)$ also coincide up to changing $a$ to $-b$. Finally, the straightforward computations involving the explicit formula for the solution give
$$ \begin{equation*} U_h(h,0)=\begin{pmatrix} 1&0\\0&-e^{-a}\end{pmatrix}+O(h), \end{equation*} \notag $$
and therefore,
$$ \begin{equation*} P(h)=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix} 1&0\\0&-e^{b}\end{pmatrix} \begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix} 1&0\\0&-e^{-a}\end{pmatrix} +O(h)=\begin{pmatrix}e^{b}&0\\0&e^{-a}\end{pmatrix}+O(h). \end{equation*} \notag $$
Thus, $\mu_+(\gamma_h)=b/{\pi}+O(h)$, and the proposition is proved.

Remark 2.1. Using the energy arguments as in the proof of the upper bound together with the fact that the solution $y(t)$ cannot be identically zero on any interval, it can be shown that neither the supremum and nor the infimum in (2.3) can be attained if $ab\ne0$. We also note that the proved result shows that in any class $\Gamma_{a,b}$ with $ab\ne0$ there is an element $\gamma$, with positive Lyapunov exponent.

2.2. Super-linear ODE

As we have seen, the conditions on the mean value of $\gamma_+$ or $\gamma_-$ are not sufficient for establishing the absence of growing solutions in the linear case. Surprisingly, the situation is essentially simpler in the case of non-linear equations with super-linear non-linearities. As already mentioned in the introduction, the reason for this is that, in contrast to the linear case, the frequency of internal oscillations grows with energy, and so adding the energy to the system destroys the conditions for parametric resonances. We start with a heuristic derivation for the dissipative estimate, which will be rigorously justified later.

Consider the equation

$$ \begin{equation} y''(t)+\gamma(t)y'(t)+y(t)|y(t)|^p=0. \end{equation} \tag{2.6} $$
If $\gamma=0$, then the equation is invariant with respect to scaling $t\to tE^{-p/(2(p+2))}$, $u\to uE^{1/(p+2)}$, and so the energy $E$ and the frequency $\omega$ of internal oscillations are related by
$$ \begin{equation*} \omega\sim E^{p/(2(p+2))}, \end{equation*} \notag $$
and the dissipative term $\gamma y'$ will be of order $E^{-p/(2(p+2))}$ in the scaled time, and so it cannot change the oscillatory nature of the solutions (at least if $\gamma$ is bounded). By this reason, all solutions of equation (2.6) will oscillate rapidly in time on high energy levels.

We now write the energy equality

$$ \begin{equation} \frac d{dt} E(t):=\frac d{dt}\biggl(\frac12 y'(t)^2+\frac1{p+2}|y(t)|^{p+2}\biggr)=-\gamma(t) y'(t)^2. \end{equation} \tag{2.7} $$
From this equality we see that the total energy $E(t)$ is not oscillatory, and, moreover, if we fix a small enough time interval $t\in[0,\varepsilon]$, we get
$$ \begin{equation} E(t)\approx E(0),\quad \text{that is, }\quad |E(t)-E(0)|\leqslant C\varepsilon,\qquad t\in[0,\varepsilon]. \end{equation} \tag{2.8} $$
In contrast, both the kinetic energy $E_k(t):=|y'(t)|^2/2$ and the potential one $E_{p}(t):=E(t)-E_k(t)$ are rapidly oscillatory, and so the right-hand size of (2.7) can be averaged (if the initial energy $E(0)$ is large enough and $\varepsilon$ is fixed). Hence
$$ \begin{equation} E(\varepsilon)-E(0)=-\int_0^\varepsilon\gamma(t)y'(t)^2\,dt\approx -2\int_0^\varepsilon\gamma(t)\,dt\langle E_k\rangle, \end{equation} \tag{2.9} $$
where $\langle E_k\rangle:=(1/\varepsilon)\int_0^\varepsilon E_k(t)\,dt$ is the mean of the kinetic energy on the interval $t\in[0,\varepsilon]$. To find this mean, we multiply (2.6) by $y(t)$ and integrate with respect to time to get
$$ \begin{equation} (p+2)\langle E_p\rangle-2\langle E_k\rangle=-\langle\gamma y'y\rangle+\frac{y'(0)y(0)-y'(\varepsilon)y(\varepsilon)}\varepsilon:=H. \end{equation} \tag{2.10} $$
Using now the fact that the potential energy is super-linear, from (2.8) and the Young inequality, we get
$$ \begin{equation} |H|\leqslant \beta E(0)+C_\beta, \end{equation} \tag{2.11} $$
where $\beta>0$ is arbitrary and $C_\beta$ is independent of $E(0)$. So, if the initial energy $E(0)$ is large enough, then
$$ \begin{equation*} (p+2)\langle E_p\rangle\approx 2\langle E_k\rangle+C,\qquad C=C_{\beta,\varepsilon}, \end{equation*} \notag $$
which together with the nonlinear energy balance $\langle E_k\rangle+\langle E_p\rangle\approx E(0)$ gives the fundamental relation
$$ \begin{equation} \langle E_k\rangle\approx\frac{p+2}{p+4}\langle E\rangle+C \approx \frac{p+2}{p+4}E(0)+C. \end{equation} \tag{2.12} $$
Substituting this relation into the energy identity (2.9), we finally get
$$ \begin{equation*} E(\varepsilon)\approx \biggl(1-\frac{2(p+2)}{p+4}\int_0^\varepsilon \gamma(t)\,dt\biggr)E(0)+C. \end{equation*} \notag $$
Repeating these arguments on the time interval $t\in[n\varepsilon,(n+1)\varepsilon]$, we arrive at
$$ \begin{equation} E\bigl((n+1)\varepsilon\bigr)\approx \biggl(1-\frac{2(p+2)}{p+4}\int_{n\varepsilon}^{(n+1)\varepsilon} \gamma(t)\,dt\biggr)E(n\varepsilon)+C. \end{equation} \tag{2.13} $$
If we now assume that
$$ \begin{equation} \lim_{\varepsilon\to0}\sup_{n\in\mathbb N} \int_{n\varepsilon}^{(n+1)\varepsilon}\gamma(t)\,dt=0 \end{equation} \tag{2.14} $$
and
$$ \begin{equation} \liminf_{T\to\infty}\inf_{t\geqslant0}\frac1T\int_t^{t+T}\gamma(s)\,ds>0, \end{equation} \tag{2.15} $$
we may fix $\varepsilon > 0$ small enough, and hence, since $\ln(1+x)\approx x$, we have
$$ \begin{equation*} E(n\varepsilon)\leqslant CE(0)e^{-\alpha n}+C_* \end{equation*} \notag $$
for some positive constants $\alpha$, $C$ and $C_*$. This gives the desired dissipative estimate. Similarly, if
$$ \begin{equation} \limsup_{T\to\infty}\sup_{t\geqslant0}\frac1T\int_t^{t+T}\gamma(s)\,ds<0, \end{equation} \tag{2.16} $$
the solutions of (2.6) will grow exponentially at least if the initial energy is large enough. Thus, the mean value of the dissipation coefficient indeed controls dissipativity of the corresponding equation.

Remark 2.2. Assumption (2.14) is also crucial for dissipativity. Indeed, it guarantees that the dissipation rate oscillates not too rapidly, and makes possible averaging with respect to internal oscillations (in what follows, we replace it by a bit stronger assumption that $\gamma$ is translation compact in $L^1_b(\mathbb{R})$). It is also not difficult to see that if this condition is violated, we can destabilize equation (2.6) similarly to the linear case (see the proof of Proposition 2.1), but using the sequence of kicks whose positions ($t=t_n$) and parameters $h=h_n$ depend on the energy of the unstable solution (which we are constructing).

We now give a rigorous proof for dissipativity of equation (2.6) under a bit stronger (in comparison with (2.14)) assumption that $\gamma$ has a bounded derivative. This assumption will be relaxed later.

Proposition 2.2. Let $\gamma(t)$ satisfy assumption (2.15), and let, in addition,

$$ \begin{equation} |\gamma'(t)|+|\gamma(t)|\leqslant C,\qquad t\geqslant0. \end{equation} \tag{2.17} $$
Then, for every solution $y(t)$ of (2.6), the following dissipative estimate holds:
$$ \begin{equation} E(t)\leqslant C E(0)e^{-\alpha t}+C_*; \end{equation} \tag{2.18} $$
here the positive constants $\alpha$, $C$ and $C_*$ are independent of $t$ and $E(0)$.

Proof. Although the given heuristic arguments above can be made rigorous, we prefer to verify (2.18) in a more straightforward way using a proper adaptation of the standard energy type estimates. Namely, we multiply equation (2.6) by $y'(t)+(2/(p+4))\gamma(t)y(t)$. Now, after elementary transformations, we get
$$ \begin{equation} \begin{aligned} \, &\frac d{dt}\biggl(E(t)+\frac2{p+4}\gamma(t)y'(t)y(t)\biggr)+\frac{2(p+2)}{p+4}\gamma(t)E(t) \nonumber \\ &\qquad=\frac2{p+2}\bigl(\gamma'(t)-\gamma^2(t)\bigr)y'(t)y(t). \end{aligned} \end{equation} \tag{2.19} $$
Let $\mathcal E(t):=E(t)+(2/(p+4))\gamma(t)y'(t)y(t)$. Using assumption (2.17) and the fact that $p>0$ (analogously to (2.11)), we deduce that
$$ \begin{equation} C_2(E(t)-1)\leqslant\mathcal E(t)\leqslant C_1 (E(t)+1) \end{equation} \tag{2.20} $$
for some positive numbers $C_1$ and $C_2$ and
$$ \begin{equation*} \frac d{dt}\mathcal E(t)+\biggl(\frac{2(p+2)}{p+4}\gamma(t)-\kappa\biggr)\mathcal E(t)\leqslant C_\kappa, \end{equation*} \notag $$
where $\kappa>0$ is arbitrary. Integrating this inequality, we arrive at
$$ \begin{equation} \begin{aligned} \, \mathcal E(t) &\leqslant \mathcal E(0)\exp\biggl\{-\int_0^t\biggl(2\frac{p+2}{p+4}\gamma(\tau)-\kappa\biggr)\,d\tau\biggr\} \nonumber \\ &\qquad+ C_\kappa\int_0^t\exp\biggl\{-\int_s^t\biggl(2\frac{p+2}{p+4}\gamma(\tau)-\kappa\biggr)\,d\tau \biggr\}\,ds. \end{aligned} \end{equation} \tag{2.21} $$
By assumption (2.15), there exist $T>0$ and $\alpha>0$ such that
$$ \begin{equation*} 2\frac{p+2}{p+4}\int_s^{s+nT}\gamma(\tau)\,d\tau\geqslant 2\alpha nT,\qquad s\geqslant0,\quad n\in\mathbb N. \end{equation*} \notag $$
Together with assumption (2.17), this gives
$$ \begin{equation*} 2\frac{p+2}{p+4}\int_s^t\gamma(\tau)\,d\tau\geqslant 2\alpha(t-s)+C,\qquad t\geqslant s\geqslant0, \end{equation*} \notag $$
for some positive $C$, which is independent of $t$ and $s$. Fixing now $\kappa=\alpha$ and inserting this estimate into (2.21), we arrive at the required estimate
$$ \begin{equation*} \mathcal E(t)\leqslant C\mathcal E(0)e^{-\alpha t}+C_*, \end{equation*} \notag $$
which completes the proof of the proposition.

2.3. Non-linear PDE. Key observation

We now turn to the model PDE

$$ \begin{equation} \partial_t^2 u+\gamma(t)\, \partial_t u-\Delta_x u+u|u|^p=0,\qquad x\in\Omega,\quad u\big|_{\partial\Omega}=0, \end{equation} \tag{2.22} $$
in a bounded domain $\Omega$ of $\mathbb{R}^3$. The global well-posedness of this problem will be discussed in the next section and here we concentrate on the conditions for dissipativity. Similarly to the case of an ODE, by formally multiplying the equation by $\partial_t u$ and integrating over $x\in\Omega$, we get the energy identity
$$ \begin{equation} \frac d{dt}E(t):=\frac d{dt}\biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|_{L^{p+2}}^{p+2}\biggr)=-2\gamma(t)E_k(t), \end{equation} \tag{2.23} $$
where
$$ \begin{equation*} E_k(t):=\frac12\|\partial_t u\|^2_{L^2},\qquad E_p(t):=\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}. \end{equation*} \notag $$
Therefore, arguing as in the case of an ODE, we get
$$ \begin{equation} E(\varepsilon)-E(0)\approx-2\biggl( \frac{\langle E_k\rangle}{\langle E\rangle}\int_0^\varepsilon\gamma(\tau)\,d\tau\biggr)E(0). \end{equation} \tag{2.24} $$
However, in contrast to the case of an ODE, the potential energy $E_p$ is no more homogeneous with respect to $u$ (due to the presence of the extra quadratic term $\|\nabla_x u\|^2_{L^2}$/2), and so the ratio between the averaged kinetic and total energy is no more a constant, but may essentially depend on the trajectory considered. Indeed, multiplying equation (2.22) by $u$ and integrating with respect to $x$ and $t$, we get the following analogue of (2.10):
$$ \begin{equation*} \langle\|u\|^{p+2}_{L^{p+2}}\rangle+\langle\|\nabla_x u\|^2_{L^2}\rangle-2\langle E_k\rangle=-\langle\gamma (\partial_t u,u)\rangle+\frac{(\partial_t u(0),u(0))-(\partial_t u(\varepsilon),u(\varepsilon))}\varepsilon; \end{equation*} \notag $$
here $(f,g)$ is the standard inner product in $L^2(\Omega)$. Thus,
$$ \begin{equation} 2\langle E_k\rangle\approx \langle\|\nabla_x u\|^2_{L^2}\rangle +\langle\|u\|^{p+2}_{L^{p+2}}\rangle. \end{equation} \tag{2.25} $$
Since
$$ \begin{equation*} 2\langle E_p\rangle\leqslant \langle\|\nabla_x u\|^2_{L^2}\rangle +\langle\|u\|^{p+2}_{L^{p+2}}\rangle \leqslant (p+2)\langle E_p\rangle, \end{equation*} \notag $$
we finally get
$$ \begin{equation} \frac12 \lessapprox \frac{\langle E_k\rangle}{\langle E\rangle} \lessapprox \frac{p+2}{p+4}. \end{equation} \tag{2.26} $$
Thus, when iterating inequality (2.24) in the worst possible scenario, the ratio $\langle E_k\rangle/\langle E\rangle$ will be close to $1/2$ if $\gamma$ is positive and close to $(p+2)/(p+4)>1/2$ if $\gamma$ is negative. By this reason, the dissipativity condition (2.15) should be naturally replaced by the stronger one
$$ \begin{equation} \liminf_{T\to\infty}\inf_{t\geqslant0}\frac1T\int_t^{t+T} \biggl(\frac12\gamma(s)_+-\frac{p+2}{p+4}\gamma(s)_-\biggr)\, ds>0, \end{equation} \tag{2.27} $$
which coincides with the assumption stated in the introduction.

Remark 2.3. On the one hand, we do not know how to build up an explicit example showing that assumption (2.15) is not enough for equation (2.22) to be dissipative. On the other hand, we also do not know of any mechanism that would prevent the above worst scenario Indeed, the lower bound in (2.26) is “attained” if the term $\langle\|\nabla_x u\|_{L^2}^2\rangle$ in the averaged potential energy is dominating (that is, if the solution oscillates rapidly in space and remains not very big). For the upper bound, we need large (in amplitude) solution that oscillates not too rapidly in space (in this case, the term $\langle\|u\|^{p+2}_{L^{p+2}}\rangle$ will dominate). Solutions of these two different types can be constructed in the Hamiltonian case $\gamma=0$ (for example, in the class of time periodic solutions using the perturbation technique). If we assume in addition that this Hamiltonian system is chaotic on any energy level $E$, then this worst scenario becomes indeed natural. This leads us to the conjecture that assumption (2.15) is not enough for dissipativity, and so this condition should be replaced by (2.27).

§ 3. Statement of the problem and dissipativity

In this section, we recall well-known facts on existence and uniqueness of solutions for the damped wave equation, and give a rigorous proof for dissipative estimates discussed in § 2. Recall that we study the damped wave equation

$$ \begin{equation} \partial_t^2u+\gamma(t)\, \partial_t u-\Delta_x u+f(u)=g, \qquad u\big|_{\partial\Omega}=0,\quad u\big|_{t=\tau}=u_\tau,\quad \partial_t u\big|_{t=\tau}=u'_\tau, \end{equation} \tag{3.1} $$
in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary. We assume that $g\in L^2(\Omega)$ and that the non-linearity $f\in C^1(\mathbb{R},\mathbb{R})$ has the following structure:
$$ \begin{equation} f(u)=u|u|^p+f_0(u),\qquad \lim_{|u|\to\infty}\frac{|f_0'(u)|}{|u|^{p}}=0,\quad f_0(0)=0, \end{equation} \tag{3.2} $$
for some $p>0$. Thus, the leading term of the non-linearity is $u|u|^p$, exactly as in § 2. Concerning the dissipation coefficient $\gamma$, we assume that it belongs to the uniformly local space $L^1_b(\mathbb{R})$,
$$ \begin{equation} \gamma\in L^1_b(\mathbb{R}),\qquad \|\gamma\|_{L^1_b}:=\sup_{t\in\mathbb{R}}\|\gamma\|_{L^1((t,t+1))}<\infty, \end{equation} \tag{3.3} $$
and is translation compact in it, that is,
$$ \begin{equation} \gamma\in L^1_{\textrm{tr-c}}(\mathbb{R}):=[C^\infty_b(\mathbb{R})]_{L^1_b(\mathbb{R})}, \end{equation} \tag{3.4} $$
where $[\,\cdot\,]$ stands for the closure. We recall that a function $\gamma\in L^1_b(\mathbb{R})$ is translation compact if and only if it possesses a uniform $L^1$-modulus of continuity,
$$ \begin{equation} \lim_{h\to0}\sup_{t\in\mathbb{R}}\int_t^{t+1}|\gamma(\tau+h)-\gamma(\tau)|\,d\tau=0 \end{equation} \tag{3.5} $$
(see [16] for more details). In addition, we assume that the uniform analogue of dissipativity condition (2.27) is satisfied:
$$ \begin{equation} \liminf_{T\to\infty}\inf_{t\in\mathbb{R}}\frac1T\int_t^{t+T} \biggl(\frac12\gamma_+(s)-\frac{p+2}{p+4}\gamma_-(s)\biggr)\, ds>0. \end{equation} \tag{3.6} $$
As usual, we set $\xi_u(t):=\{u(t),\partial_t u(t)\}$ and introduce the energy space
$$ \begin{equation*} E:=[H^1_0(\Omega)\cap L^{p+2}(\Omega)]\times L^2(\Omega), \end{equation*} \notag $$
where $H^1_0(\Omega)$ stands for the subspace of the Sobolev space $H^1(\Omega)$ with the extra condition $u\big|_{\partial\Omega}=0$.

We first define the energy solutions of (3.1).

Definition 3.1. A function $u(t)$, $t\geqslant\tau$, is a weak energy solution of (3.1) if

$$ \begin{equation*} \xi_u\in L^\infty(\tau,\infty;E), \end{equation*} \notag $$
and the equation is satisfied in the sense of distributions. The latter means that, for every test function $\varphi\in C_0^\infty((\tau,\infty)\times\Omega)$,
$$ \begin{equation} \begin{aligned} \, &-\int_\mathbb{R}\bigl(\partial_t u(t),\partial_t\varphi(t)\bigr)\,dt +\int_R\gamma(t)\bigl(\partial_t u(t),\varphi(t)\bigr)\,dt \nonumber \\ &\qquad+\int_\mathbb{R}\bigl(\nabla_x u(t),\nabla_x\varphi(t)\bigr)\,dt +\int_\mathbb{R}\bigl(f(u(t)),\varphi(t)\bigr)\,dt=\int_\mathbb{R}\bigl(g,\varphi(t)\bigr)\,dt. \end{aligned} \end{equation} \tag{3.7} $$
Note that, since $\xi_u(t)\in E$, we have $f(u(t))\in L^q(\Omega)$, $q=(p+2)/(p+1)$. Thus, taking into the account that $\gamma\in L^1_b(\mathbb{R})$, we have, for the second distributional derivative,
$$ \begin{equation*} \partial_t^2u\in L^1\bigl(\tau,\tau+T;H^{-1}(\Omega)+L^q(\Omega)\bigr),\qquad T>0, \end{equation*} \notag $$
and, therefore, $\partial_t u\in C(\tau,\infty;H^{-1}(\Omega)+L^q(\Omega))$, which shows that the initial condition for $\partial_t u$ at $t=\tau$ is well-posed. The situation with the initial data for $u$ is simpler, since $\partial_t u\in L^\infty(\tau,\infty;L^2)$. The above arguments also imply, in a standard way, that the trajectory $\xi_u(t)$ is continuous with respect to time in the weak topology of $E$,
$$ \begin{equation*} \xi_u\in C(\tau,\infty;E_w) \end{equation*} \notag $$
(see [16] for more details).

It is well-known that the weak energy solutions are well-posed if $p\leqslant2$. For the case $2<p\leqslant4$, well-posedness still holds in a slightly stronger class of solutions based on the so-called Strichartz estimates.

Definition 3.2. A weak energy solution $u(t)$ is called a Shatah–Struwe solution (an SS-solution) if, in addition,

$$ \begin{equation} u\in L^4\bigl(\tau,\tau+T;L^{12}(\Omega)\bigr),\qquad T\geqslant0. \end{equation} \tag{3.8} $$

The following proposition summarizes the known results on existence and uniqueness of solutions to problem (3.1).

Proposition 3.1. Let functions $f$ and $\gamma$ satisfy the above assumptions and let $\xi_u(\tau):=\{u_\tau,u'_\tau\}\in E$. Then:

1) there exists at least one weak energy solution of problem (3.1) (irrespective of the value of the exponent $p$);

2) if, in addition, $0\leqslant p\leqslant2$, then the weak energy solution is unique, the function

$$ \begin{equation*} t\to\|\xi_u(t)\|_{\mathcal E}^2:=\frac12\|\partial_t u(t)\|^2_{L^2}+\frac12\|\nabla_x u(t)\|^2_{L^2}+(F(u(t)),1) \end{equation*} \notag $$
is absolutely continuous (here and below $F(u):=\int_0^uf(v)\,dv$), and the energy identity
$$ \begin{equation*} \frac d{dt}\|\xi_u(t)\|^2_{\mathcal E}=-\gamma(t)\|\partial_t u(t)\|^2_{L^2} \end{equation*} \notag $$
holds for almost all $t$;

3) if $0\leqslant p\leqslant 4$, then there exists a unique SS-solution of problem (3.1); this solution also satisfies the energy identity in the above sense.

Remark 3.1. Indeed, the proof of the first statement is a standard application of the Galerkin approximation method, and can be found, for example, in [16]. The second result is also classical, (see [15], [16] and the references there).

The third result is more recent and is a bit more delicate. The local existence of SS-solutions can be verified using the perturbation arguments from the Strichartz estimate for the linear equation. Namely, let $V$ solves

$$ \begin{equation*} \partial_t^2V-\Delta_x V=h(t),\qquad V\big|_{\partial\Omega}=0,\quad \xi_v\big|_{t=0}=\xi_0, \end{equation*} \notag $$
with $\xi_0\in E:=H_0^1(\Omega)\times L^2(\Omega)$ and $h\in L^1(0,T;L^2(\Omega))$. Then
$$ \begin{equation} \|\xi_V\|_{C(0,T;E)}+\|V\|_{L^4(0,T;L^{12}(\Omega))}\leqslant C_T(\|\xi_0\|_{E}+\|h\|_{L^1(0,T;L^2(\Omega))}) \end{equation} \tag{3.9} $$
(see [21] for more details).

Here and below, we will use the fact that, by the Sobolev embedding $H^1\subset L^6$ and the assumption $p\leqslant4$, the term $L^{p+2}$ in the definition of the energy space $E$ is not necessary and can be omitted. We will use the following truncated energy norm

$$ \begin{equation*} \|\xi_u(t)\|_{E}^2: =\frac12\|\partial_t u(t)\|^2_{L^2}+\frac12\|\nabla_x u(t)\|^2_{L^2}, \end{equation*} \notag $$
and the notation $\|\xi_u(t)\|^2_{\mathcal E}$ will be used for the full energy (including the $L^{p+2}$-norm).

For the subcritical case $p<4$, the global solvability follows in a straightforward way from the local one and the so-called energy-to-Strichartz estimate for solutions of (3.1),

$$ \begin{equation} \|u\|_{L^4(t,t+1;L^{12})}\leqslant Q(\|\xi_u(t)\|_E)+Q(\|g\|_{L^2}), \end{equation} \tag{3.10} $$
for some monotone function $Q$ independent of $u$ and $t$. This estimate (which also follows from Strichartz estimate (3.9) via the perturbation arguments; see, for example, [33]) is crucial since it allows us to control the Strichartz norm of the solution through its energy norm. In particular, the dissipativity in the Strichartz norm will follow immediately if the dissipativity in the energy norm is established; we will utilize this fact in the next section.

We also mention that, in the subcritical case $p<4$, the function $Q(z)$ is polynomial with respect to $z$,

$$ \begin{equation} Q(z)\leqslant C_p(1+z^{N_p}), \end{equation} \tag{3.11} $$
where the exponent $N_p$ may tend to infinity when $p\to4$.

However, the energy-to-Strichartz estimate (3.10) is problematic in the critical case $p=4$. To the best of our knowledge, it is proved for the periodic boundary conditions only, and its validity for other boundary conditions is an open problem (see [34]). In the critical case, the global existence of SS-solutions is verified using the so-called Pohozhaev–Morawetz inequality and related non-concentration estimates (see [9], [10], [12]). Note that, in contrast to the subcritical case, these arguments give only the global existence of an SS-solution without any quantitative bounds on its Strichartz norm. In particular, this norm may a priori grow uncontrollably as $t\to\infty$, and this hinders construction of the attractor theory. At present, this issue has been circumvented only in the autonomous case (see [33]). By this reason, we will construct the attractor theory for equation (3.1) for the subcritical case $p<4$ (see [19] for uniqueness of energy solutions for $p<4$ in the whole space).

Finally, the validity of the energy identity for SS-solutions (and $p\leqslant4$) is straightforward, see, for example, [33]. Note that the uniqueness of energy solutions for $p>2$ and SS-solutions for $p>4$ is not known yet.

Corollary 3.1. Let $0<p\leqslant 4$, and let functions $\gamma$ and $f$ satisfy the above assumptions. Then equation (3.1) generates a dynamical process $U_\gamma(t,\tau)$, $t\geqslant\tau$, in the energy space $E$ via

$$ \begin{equation} U_\gamma(t,\tau)\xi_\tau:=\xi_u(t),\qquad t\geqslant\tau,\quad \tau\in\mathbb{R},\quad \xi_\tau\in E, \end{equation} \tag{3.12} $$
where $\xi_u(t)=\{u(t),\partial_t u(t)\}$, and $u(t)$ is an SS-solution of (3.1) corresponding to the initial data $\xi_u\big|_{t=\tau}=\xi_\tau$.

We are now ready to state and prove the main result of this section.

Theorem 3.1. Let functions $f$, $\gamma$, $g$ satisfy the assumptions stated at the beginning of this section, and let $p\leqslant4$. Then, for any SS-solutions of problem (3.1),

$$ \begin{equation} \|\xi_u(t)\|_{\mathcal E}\leqslant C\bigl(1+\|\xi_u(\tau)\|_{\mathcal E}\bigr)e^{-\alpha(t-\tau)} +C(1+\|g\|_{L^2}),\qquad t\geqslant\tau, \end{equation} \tag{3.13} $$
for some positive constants $C$ and $\alpha$ independent of $\tau\in\mathbb{R}$, $t\geqslant \tau$ and $\xi_u(\tau)$.

Proof. We first use the fact that $\gamma$ is translation compact in $L^1_b(\mathbb{R})$, and so, for every $\varepsilon>0$, we may find a function $\overline\gamma\in C^1_b(\mathbb{R})$ such that
$$ \begin{equation} \|\gamma-\overline\gamma\|_{L^1_b}\leqslant\varepsilon. \end{equation} \tag{3.14} $$
This function can be constructed using the standard mollifiers
$$ \begin{equation} \overline\gamma(t):=\int_0^\infty k_\nu(s)\gamma(t-s)\,ds,\qquad k_\nu(s):=\nu^{-1}k\biggl(\frac{s}{\nu}\biggr), \end{equation} \tag{3.15} $$
with a kernel $k\in C_0^\infty(\mathbb{R}_+)$ is such that $\int_0^\infty k(s)\,ds=1$, and $\nu=\nu(\varepsilon)$ is small enough.

Note also that the function $\overline\gamma$ also satisfies (3.6) if $\varepsilon>0$ is small enough. Moreover, since $p\leqslant4$, the SS-solution satisfies the energy identity

$$ \begin{equation} \begin{aligned} \, &\frac d{dt}\biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}+(F_0(u(t)),1)\biggr) \nonumber \\ &\qquad+\overline\gamma(t)\|\partial_t u\|^2_{L^2}=-\widetilde\gamma(t)\|\partial_t u\|^2_{L^2}, \end{aligned} \end{equation} \tag{3.16} $$
where $\widetilde\gamma(t):=\gamma(t)-\overline\gamma(t)$ and $F_0(u):=\int_0^uf_0(v)\,dv$. At the next step, we multiply equation (3.1) by $(1/2)\overline\gamma_+(t)u-(2/(p+4))\overline\gamma_-(t)u$, which gives
$$ \begin{equation} \begin{aligned} \, &\frac d{dt}\biggl(\biggl(\frac12\,\overline\gamma_+(t)-\frac2{p+4}\overline\gamma_-(t)\biggr)(\partial_t u,u)\biggr)-\biggl(\frac12\,\overline\gamma_+(t)-\frac2{p+4}\overline\gamma_-(t)\biggr) \|\partial_t u\|^2_{L^2} \nonumber \\ &\qquad\qquad+\biggl(\frac12\,\overline\gamma_+(t)-\frac2{p+4}\overline\gamma_-(t)\biggr) \bigl(\|\nabla_x u\|^2_{L^2}+\|u\|^{p+2}_{L^{p+2}}\bigr) \nonumber \\ &\qquad=\biggl(\frac12\,\overline\gamma_+(t)-\frac2{p+4}\overline\gamma_-(t)\biggr) \bigl((g,u)-(f_0(u),u)-\gamma(t)(\partial_t u,u)\bigr) \nonumber \\ &\qquad\qquad+\biggl(\frac12\,\overline\gamma_+^{\,\prime}(t) -\frac2{p+4}\overline\gamma_-^{\,\prime}(t)\biggr) (\partial_t u,u):=H_u(t). \end{aligned} \end{equation} \tag{3.17} $$
Analogously to (2.25), we have
$$ \begin{equation*} \begin{aligned} \, 2\biggl(\frac12\|\nabla_x u\|^2_{L^2} +\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}\biggr) &\leqslant \|\nabla_x u\|^2_{L^2}+\|u\|^{p+2}_{L^{p+2}} \\ &\leqslant(p+2)\biggl(\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}\biggr) \end{aligned} \end{equation*} \notag $$
and using that $\overline\gamma_\pm(t)\geqslant0$ together with $\overline\gamma_+(t)\overline\gamma_-(t)\equiv0$, we get
$$ \begin{equation*} \biggl(\frac12\,\overline\gamma_+(t)-\frac2{p+4}\overline\gamma_-(t)\biggr) \bigl(\|\nabla_x u\|^2_{L^2}+\|u\|^{p+2}_{L^{p+2}}\bigr) \geqslant 2\biggl(\frac12\,\overline\gamma_+(t) -\frac{p+2}{p+4}\overline\gamma_-(t)\biggr) E_p(t) \end{equation*} \notag $$
with $E_p(t):=(1/2)\|\nabla_x u\|^2_{L^2}+(1/(p+2))\|u\|^{p+2}_{L^{p+2}}$. Summing (3.17) and (3.16) and since $\overline\gamma(t) =\overline\gamma_+(t)-\gamma_-(t)$, we have
$$ \begin{equation} \begin{aligned} \, &\frac d{dt}\mathcal E_u(t)+ 2\biggl(\frac12\,\overline\gamma_+(t)-\frac{p+2}{p+4}\overline\gamma_-(t)\biggr) \nonumber \\ &\qquad\times \biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}\biggr) \leqslant H_u(t)-\widetilde\gamma(t)\|\partial_t u\|^2_{L^2}, \end{aligned} \end{equation} \tag{3.18} $$
where
$$ \begin{equation*} \begin{aligned} \, \mathcal E_u(t) &:=\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}} \\ &\qquad+\bigl(F_0(u(t)),1\bigr)+ \biggl(\frac12\,\overline\gamma_+(t) -\frac2{p+4}\overline\gamma_-(t)\biggr)(\partial_t u,u). \end{aligned} \end{equation*} \notag $$
Since $\overline\gamma\in C^1_b(\mathbb{R})$, $p>0$, and $F_0(u)$ is subordinated to $u|u|^p$, we have
$$ \begin{equation} \begin{aligned} \, &\frac12\biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}\biggr)-C_\varepsilon \nonumber \\ &\qquad\leqslant\mathcal E_u\leqslant 2\biggl(\frac12\|\partial_t u\|^2_{L^2}+\frac12\|\nabla_x u\|^2_{L^2}+\frac1{p+2}\|u\|^{p+2}_{L^{p+2}}\biggr)+C_\varepsilon. \end{aligned} \end{equation} \tag{3.19} $$
Analogously, using again that $p>0$ and since $f_0$ is subordinated to $u|u|^p$, we the term $H_u(t)$ is estimated as
$$ \begin{equation} \begin{aligned} \, H_u(t) &\leqslant C_\varepsilon\bigl(|(|g|,|u|)+|(f_0(u)),u)|+(1+|\gamma(t))|(|\partial_t u|,|u|)\bigr) \nonumber \\ &\leqslant \kappa\bigl(|\gamma(t)|+1\bigr)\mathcal E_u(t) +C_{\kappa,\varepsilon}\bigl(1+|\gamma(t)|+\|g\|^2_{L^2}\bigr), \end{aligned} \end{equation} \tag{3.20} $$
where $\varepsilon>0$ and $\kappa>0$ can be taken arbitrarily small. Inserting the above estimates to (3.18), we finally arrive at
$$ \begin{equation} \frac d{dt}\mathcal E(t)+\beta_\varepsilon(t)\mathcal E(t)\leqslant C_{\kappa,\varepsilon}\bigl(1+\|g\|^2_{L^2}+|\gamma(t)|\bigr), \end{equation} \tag{3.21} $$
where
$$ \begin{equation} \beta_\varepsilon(t):=2\biggl(\frac12\,\overline\gamma_+(t)-\frac{p+2}{p+4}\overline\gamma_-(t) -|\widetilde\gamma(t)|-\kappa|\gamma(t)|-\kappa\biggr). \end{equation} \tag{3.22} $$
Since $\gamma\in L^1_b(\mathbb{R})$ satisfies (3.6) and since $\widetilde\gamma$ is of order $\varepsilon$ (see (3.14)), we may fix positive constants $\varepsilon$ and $\kappa$ so that assumption (3.6) will be satisfied also for the function $\beta_\varepsilon(t)$. Namely,
$$ \begin{equation*} \liminf_{T\to\infty}\frac1T\inf_{t\in\mathbb{R}}\int_{t}^{t+T}\beta_\varepsilon(s)\,ds>0. \end{equation*} \notag $$
Analogously to the proof of Proposition 2.2, this gives
$$ \begin{equation} \int_s^t\beta_\varepsilon(\tau)\,d\tau\geqslant 2\alpha(t-\tau)+C,\qquad t\geqslant s, \end{equation} \tag{3.23} $$
where the positive constants $\alpha$ and $C$ are independent of $t$ and $\tau$. Integrating inequality (3.21) with respect to time, we get
$$ \begin{equation} \mathcal E(t)\leqslant \mathcal E(\tau)\exp\biggl\{-\int_\tau^t\beta_\varepsilon(s)\,ds\biggr\}+ C\int_\tau^t(1+ \|g\|_{L^2} +|\gamma(s)|)\exp\biggl\{-\int_s^t\beta_\varepsilon(l)\,dl\biggr\}\,ds, \end{equation} \tag{3.24} $$
which together with (3.23) gives the required estimate (3.13) and completes the proof of the theorem.

Remark 3.2. The proof shows that the derivation of the dissipative estimate depends neither on the fact that $\Omega$ is three-dimensional, nor on the conditions on the growth rate $p$. In the actual fact, these assumptions are posed only in order to have global well-posedness and validity of the energy identity. Thus, they both can be removed, but in this case, we will be unable to verify the dissipative estimate for all weak energy (or SS) solutions and may only claim that, for all initial data from the energy space, there is a solution satisfying the above dissipative estimate.

Corollary 3.2. Under the conditions of Theorem 3.1, let, in addition, $p<4$. Then, in addition to (3.13), the dissipativity of the Strichartz norm also holds. Namely, for any SS-solution $u(t)$ of equation (3.1)

$$ \begin{equation} \int_{t-1}^t\|u(s)\|^4_{L^{12}}\,ds\leqslant Q(\|\xi_u(\tau)\|_E)e^{-\alpha(t-\tau)}+Q(\|g\|_{L^2}),\qquad t\geqslant\tau+1, \end{equation} \tag{3.25} $$
for some positive $\alpha$ and monotone increasing function $Q$.

Indeed, estimate (3.25) follows from (3.13) and the energy-to-Strichartz estimate (3.10).

Corollary 3.3. Under the conditions of Corollary 3.2, the dynamical process $U(t,\tau)$ is Lipschitz continuous uniformly on bounded sets of the energy space $E$. Namely, for any two SS-solutions $u(t)$ and $v(t)$ of equation (3.1),

$$ \begin{equation} \begin{aligned} \, &\|\partial_t u(t)-\partial_t v(t)\|^2_{L^2}+\|\nabla_x u(t)-\nabla_x v(t)\|^2_{L^2} \nonumber \\ &\ \leqslant Ce^{L(t-\tau)}\bigl(\|\partial_t u(\tau)-\partial_t v(\tau)\|^2_{L^2} +\|\nabla_x u(\tau)-\nabla_x v(\tau)\|^2_{L^2}\bigr),\qquad t\geqslant\tau, \end{aligned} \end{equation} \tag{3.26} $$
where the constant $L$ depends only on $\gamma$, $f$ and $g$, and the constant $C$ depends also on the energy norms of $\xi_u(\tau)$ and $\xi_v(\tau)$.

Proof. Let $w(t):=u(t)-v(t)$. This function solves
$$ \begin{equation} \partial_t^2 w+\gamma(t)\, \partial_t w-\Delta_x w+l(t)w=0, \qquad \xi_w\big|_{t=\tau}=\xi_u(\tau)-\xi_v(\tau), \end{equation} \tag{3.27} $$
where $l(t):=\int_0^1f'(su(t)+(1-s)v(t))\,ds$. Since both $u$ and $v$ satisfy the energy identity, we may multiply the above equation by $\partial_t w$ and integrate over $x\in\Omega$. This gives
$$ \begin{equation*} \frac12\frac d{dt}(\|\partial_t w\|^2_{L^2}+\|\nabla_x w\|^2_{L^2})=-\gamma(t)\|\partial_t w\|^2_{L^2}-\bigl(l(t)w,\partial_t w\bigr), \end{equation*} \notag $$
and we only need to estimate the last term in the right-hand side of this equality. Since $f'(u)$ grows as $|u|^p$ with $p<4$, we have
$$ \begin{equation} \begin{aligned} \, \bigl(l(t)w,\partial_t w\bigr) &\leqslant\|l(t)\|_{L^3}\|w\|_{L^6}\|\partial_t w\|_{L^2} \nonumber \\ &\leqslant C\bigl(1+\|u(t)\|^4_{L^{12}}+\|v\|^4_{L^{12}}\bigr)(\|\partial_t w\|^2+\|\nabla_x w\|^2) \end{aligned} \end{equation} \tag{3.28} $$
and, therefore,
$$ \begin{equation} \begin{aligned} \, &\frac12\frac d{dt}(\|\partial_t w\|^2_{L^2}+\|\nabla_x w\|^2_{L^2}) \nonumber \\ &\qquad\leqslant \bigl(|\gamma(t)|+C(1+\|u(t)\|^4_{L^{12}}+\|v\|^4_{L^{12}})\bigr) (\|\partial_t w\|^2_{L^2}+\|\nabla_x w\|^2_{L^2}). \end{aligned} \end{equation} \tag{3.29} $$
Integrating this inequality and using estimate (3.25) together with the fact that $\gamma\in L^1_b(\mathbb{R})$, we arrive at (3.26), and complete the proof of the corollary.

§ 4. Asymptotic regularity

In this section, we will prove that, similarly to the case of constant dissipation rate, there is an asymptotic smoothing property for SS-solutions of (3.1). Namely, we will split a solution $u(t)$ of this equation into two parts

$$ \begin{equation*} u(t)=v(t)+w(t), \end{equation*} \notag $$
where $w(t)$ is more regular and $v(t)$ is exponentially decaying. However, in contrast to the standard situation, we cannot take $v$ as a solution of the linear equation with $f=g=0$ (as explained in § 2, this solution may be unstable) and should proceed in a more delicate way. The key technical tool for this purpose is the following proposition.

Proposition 4.1. Let a function $\gamma(t)$ satisfy (3.3), (3.4) and (3.6) with $p=0$. Assume also that $h\in L^1_{\mathrm{loc}}(\mathbb{R},L^2(\Omega))$. Then, there exists a positive constant $L=L(\gamma)$ such that the energy solution $v(t)$ of the equation

$$ \begin{equation} \partial_t^2 v+\gamma(t)\, \partial_t v-\Delta_x v+Lv=h,\qquad \xi_v\big|_{t=\tau}=\xi_\tau,\quad v\big|_{\partial\Omega}=0, \end{equation} \tag{4.1} $$
satisfies the dissipative estimate
$$ \begin{equation} \begin{aligned} \, &\|v(t)\|^2_{H^1}+\|\partial_t v(t)\|^2_{L^2} +\biggl(\int_{\max\{\tau,t-1\}}^t\|u(s)\|^4_{L^{12}}\,ds\biggr)^{1/2} \nonumber \\ &\qquad\leqslant C\bigl(\|v(t)\|^2_{H^1}+\|\partial_t v(t)\|^2_{L^2}\bigr)e^{-\alpha(t-\tau)}+ C\biggl(\int_\tau^te^{-\alpha(t-s)}\|h(s)\|_{L^2}\,ds\biggr)^2, \end{aligned} \end{equation} \tag{4.2} $$
where the positive constants $C$ and $\alpha$ are independent of $t$, $\tau$ and $\xi_\tau$.

Proof. We first note that it is sufficient to prove (4.2) for the energy norm only. The estimate for the Strichartz norm will then follow from estimate (3.9). Moreover, it is sufficient to verify estimate (4.2) only for the case $h=0$; the general case follows then from the variation of constants formula.

Analogously to the proof of Theorem 3.1, we split $\gamma(t):=\overline\gamma(t)+\widetilde \gamma(t)$, where $\overline\gamma\in C^1_b(\mathbb{R})$, and $\widetilde\gamma$ satisfies (3.14). Multiplying equation (4.1) by $\partial_t u+(1/2)\overline\gamma u$, we get, after some algebra,

$$ \begin{equation} \frac d{dt}\mathcal E_v(t)+\overline\gamma(t)\mathcal E_v(t)=H_v(t), \end{equation} \tag{4.3} $$
where
$$ \begin{equation*} \mathcal E_v(t):=\frac12\|\partial_t v\|^2_{L^2}+\frac12\|\nabla_x v\|^2_{L^2}+\frac L2\|v\|^2_{L^2}+\frac12\,\overline\gamma(t)(\partial_t v,v) \end{equation*} \notag $$
and
$$ \begin{equation*} H_v(t):=\frac12\,\overline\gamma^{\,\prime}(t)(\partial_t v,v) +\frac12\,\overline\gamma^{\,2}(t)(\partial_t v,v)-\widetilde\gamma(t)\|\partial_t v\|^2_{L^2}. \end{equation*} \notag $$
Since $\overline\gamma$ is globally bounded, we have
$$ \begin{equation} \begin{aligned} \, &\frac12\biggl(\frac12\|\partial_t v\|^2_{L^2}+\frac12\|\nabla_x v\|^2_{L^2}+\frac L2\|v\|^2_{L^2}\biggr) \nonumber \\ &\qquad\leqslant \mathcal E_v(t)\leqslant 2\biggl(\frac12\|\partial_t v\|^2_{L^2}+\frac12\|\nabla_x v\|^2_{L^2}+\frac L2\|v\|^2_{L^2}\biggr) \end{aligned} \end{equation} \tag{4.4} $$
if $L>L_0:=L_0(\varepsilon)$. Analogously, since $\overline\gamma$ and $\overline\gamma^{\,\prime}$ are bounded, for every $\kappa>0$, there exist $L_0=L_0(\kappa,\varepsilon)$ such that
$$ \begin{equation*} H_v(t)\leqslant \kappa\mathcal E_v(t)+2|\widetilde\gamma(t)| \mathcal E_v(t) \end{equation*} \notag $$
and, therefore,
$$ \begin{equation} \frac d{dt}\mathcal E_v(t)+\bigl(\overline\gamma(t)-\kappa-2|\widetilde\gamma(t)|\bigr) \mathcal E_v(t)\leqslant0 \end{equation} \tag{4.5} $$
if $L>L_0(\kappa,\varepsilon)$. We fix now $\kappa>0$ and $\varepsilon>0$ so small enough that
$$ \begin{equation*} \liminf_{T\to\infty}\frac1{T}\inf_{\tau\in\mathbb{R}}\int_\tau^{\tau+T} \bigl(\overline\gamma(t)-\kappa-2|\widetilde\gamma(t)|\bigr)\,dt>0 \end{equation*} \notag $$
and integrate (4.5). This gives us the required estimate for the energy norm, completing the proof.

We are now ready to state and prove the main result of this section on existence of a smooth exponentially attracting set for the dynamical process associated with equation (3.1). Unfortunately, we cannot do this for the critical case of quintic non-linearity, and so we have to impose the subcriticality assumption:

$$ \begin{equation} p<4. \end{equation} \tag{4.6} $$

Theorem 4.1. Under the conditions of Theorem 3.1, let the non-linearity $f$ be subcritical ($p<4$). Then, if $R=R(\gamma,f,g)$ is large enough, the closed ball $\mathcal B_R^1$ of radius $R$ in the space

$$ \begin{equation*} E^1:=[H^2(\Omega)\cap H^1_0(\Omega)]\times H^1_0(\Omega) \end{equation*} \notag $$
is a uniformly attracting set for the dynamical process $U_\gamma(t,\tau)$ associated with equation (3.1). Namely, there are a positive constant $\alpha$ and a monotone increasing function $Q$ such that, for any bounded set $B\subset E$,
$$ \begin{equation} \operatorname{dist}_E\bigl(U_\gamma(t,\tau)B,\mathcal B^1_R\bigr)\leqslant Q(\|B\|_E)e^{-\alpha(t-\tau)},\qquad t\geqslant\tau, \end{equation} \tag{4.7} $$
uniformly with respect to $\tau\in\mathbb{R}$. Here and below,
$$ \begin{equation*} \operatorname{dist}_E(U,V):=\sup_{u\in U}\inf_{v\in V}\|u-v\|_E \end{equation*} \notag $$
is the asymmetric Hausdorff distance between sets $U$ and $V$ in the space $E$.

Proof. We will use the standard bootstrapping arguments together with transitivity of exponential attraction (see, for example, [60]). Let $G=G(x)$ solve the linear elliptic problem
$$ \begin{equation*} -\Delta_x G=g,\qquad G\big|_{\partial\Omega}=0. \end{equation*} \notag $$
By the elliptic regularity,, we have $G\in H^2(\Omega)\cap H^1_0(\Omega)$, and
$$ \begin{equation*} \|G\|_{H^2}\leqslant C\|g\|_{L^2}. \end{equation*} \notag $$
Let $\xi_u(t):=U_\gamma(t,\tau)\xi_\tau$ be a solution of equation (3.1). Then, without loss of generality, we may assume that $\xi_\tau$ belongs to the uniformly absorbing ball $\mathcal B^0_R$ in the energy space $E$. Indeed, such a ball exists by estimate (3.13). Thus, using also (3.25), we may assume without loss of generality that
$$ \begin{equation} \|\xi_u(t)\|^2_E+\int_t^{t+1}\|u(s)\|^4_{L^{12}}\,ds\leqslant Q(\|g\|_{L^2}). \end{equation} \tag{4.8} $$
Let $v(t)$ solve the linear problem
$$ \begin{equation} \partial_t^2 v+\gamma(t)\, \partial_t v-\Delta_x v+Lv=0,\qquad \xi_v\big|_{t=\tau}=\{u(\tau)-G,\partial_t u(\tau)\}, \end{equation} \tag{4.9} $$
where $L>0$ is such that the conditions of Proposition 4.1, are satisfied, and the remainder $w(t)$ satisfies
$$ \begin{equation} \partial_t^2 w+\gamma(t)\,\partial_t w-\Delta_x w+Lw =-f(u(t))+L(u(t)-G):=h_u(t),\qquad \xi_w\big|_{t=\tau}=0. \end{equation} \tag{4.10} $$
Obviously, we have $u(t)=v(t)+G+w(t)$. Moreover, by Proposition 4.1,
$$ \begin{equation} \|\partial_t v(t)\|^2_{L^2}+\|\nabla_x v(t)\|^2_{L^2}\leqslant Ce^{-\alpha (t-\tau)}, \end{equation} \tag{4.11} $$
where the positive constants $C$ and $\alpha$ are independent of $\xi_\tau\in \mathcal B^0_R$ and $t\geqslant\tau$. In order to get the estimate for smoother component $w$, we use the fact that $|f'(u)|\leqslant C(1+|u|^p)$ with $p<4$. Indeed, by the interpolation inequality,
$$ \begin{equation*} \|f(u)\|_{L^2}\leqslant C(1+\|u\|_{L^{10}}^5)\leqslant C(1+\|u\|^4_{L^{12}}\|u\|_{L^6})\leqslant C_1(1+\|u\|^4_{L^{12}}) \end{equation*} \notag $$
and, therefore,
$$ \begin{equation*} \|f(u)\|_{L^1(t,t+1;L^2)}\leqslant C,\qquad t\geqslant\tau. \end{equation*} \notag $$
Analogously, using the Hölder inequality, we get
$$ \begin{equation} \|\nabla_x f(u)\|_{L^\kappa}=\|f'(u)\nabla_x u\|_{L^\kappa}\leqslant C\|(1+|u|^p)|\nabla_x u|\|_{L^\kappa}\leqslant C(1+\|u\|_{L^{12}}^p)\|\nabla_x u\|_{L^2}, \end{equation} \tag{4.12} $$
where $1/\kappa=1/2+p/12$. Thus, we have
$$ \begin{equation*} \|f(u)\|_{L^1(t,t+1;W^{1,\kappa})}\leqslant C,\qquad t\geqslant\tau. \end{equation*} \notag $$
Using the embedding $W^{1,\kappa}\subset H^\beta$, where $1/2=1/\kappa-(1-\beta)/3$, that is,
$$ \begin{equation*} \beta=\beta_1:=1-\frac p4>0, \end{equation*} \notag $$
we arrive at the estimate
$$ \begin{equation} \|f(u)\|_{L^1(t,t+1;H^\beta)}\leqslant C,\qquad t\geqslant\tau. \end{equation} \tag{4.13} $$
Applying now the operator $(-\Delta_x)^{\beta/2}$ to both sides of equation (4.10) and using Proposition 4.1, we get
$$ \begin{equation} \|\partial_t w(t)\|_{H^\beta}^2+\|\nabla_x w(t)\|^2_{H^{\beta}}\leqslant C, \end{equation} \tag{4.14} $$
where we have also implicitly used that
$$ \begin{equation*} \|L(u-G)\|_{L^1(t,t+1;H^1)}\leqslant C,\qquad t\geqslant\tau. \end{equation*} \notag $$
Estimates (4.11) and (4.14) show that the bal $\mathcal B^\beta_R$ in the higher energy space
$$ \begin{equation*} E^\beta:=[H^{\beta+1}\cap H^1_0]\times H^\beta \end{equation*} \notag $$
is a uniformly exponentially attracting set for the dynamical process $U_\gamma(t,\tau)$. This completes the first (the most difficult, in a sense) step of the proof.

To initiate the next step of bootstrapping, we note that if we take $\xi_u(\tau)\in \mathcal B_R^\beta$ from the very beginning, we may apply the operator $(-\Delta_x)^{\beta/2}$ to the equation for $v(t)$ as well, which would give

$$ \begin{equation*} \|\partial_t v(t)\|^2_{H^\beta}+\|\nabla_x v(t)\|^2_{H^\beta}\leqslant Ce^{-\alpha(t-\tau)},\qquad t\geqslant\tau. \end{equation*} \notag $$
This together with (4.14) shows that the dynamical process $U_\gamma(t,\tau)$ is well defined and dissipative in the higher energy space $E^\beta$ as well. In particular, we now have the control of $\nabla_x u(t)$ not only in $L^2$, but also in the more regular space $H^\beta$. In view of the Sobolev embedding this gives the control of $\nabla_x u$ in the space $L^q$ where $1/q=1/2-\beta/3$. In turn, arguing as before, but using this better control for the term $\nabla_x u$, we improve estimate (4.13) as
$$ \begin{equation} \|f(u)\|_{L^1(t,t+1;H^{\beta_1})}\leqslant C, \end{equation} \tag{4.15} $$
where
$$ \begin{equation} \beta_1=1-\frac p4+\beta. \end{equation} \tag{4.16} $$
This gives us the analogue of estimate (4.14) with $\beta$ replaced by $\beta_1>\beta$. Thus, the dynamical process $U_\gamma(t,\tau)$ on $\mathcal B_R^\beta$ has an exponentially attracting ball $\mathcal B^{\beta_1}_R$ in the space $E^{\beta_1}$ if $R$ is large enough. Moreover, since $U_\gamma(t,\tau)$ is globally Lipschitz on $\mathcal B^0_R$ (see Corollary 3.3), the transitivity of exponential attraction gives that $\mathcal B^{\beta_1}_R$ is a uniformly exponentially attracting set for $U_\gamma(t,\tau)$, in the initial energy space as well.

Finally, iterating the above procedure, we get the exponentially attracting ball in the space $E^1$, which corresponds to $\beta=1$. Indeed, from (4.16) we conclude that $\beta_n=n(1-p/4)$, and this guarantees that we reach the value $\beta=1$ in finitely many steps. This proves the theorem.

§ 5. Attractors

The aim of this section is to apply the above results for constructing global and exponential attractors for the dynamical process $U_\gamma(t,\tau)$ associated with equation (3.1). The assumptions on the non-autonomous symbol $\gamma(t)$ posed in § 3 are well adapted for usage of the so-called uniform attractor. In order to build up such an object, following the general scheme (see [16] for the details), we need to consider not only equation (3.1), but also all time shifts of this equation together with their limit in a proper topology. Namely, let us consider the hull $\mathcal H(\gamma)$ of the initial symbol $\gamma$, defined by

$$ \begin{equation} \mathcal H(\gamma):=[T_h\gamma,\, h\in\mathbb{R}]_{L^1_{\mathrm{loc}}(\mathbb{R})},\qquad (T_h\gamma)(t):=\gamma(t+h). \end{equation} \tag{5.1} $$
Then, obviously,
$$ \begin{equation*} T_h\mathcal H(\gamma)=\mathcal H(\gamma). \end{equation*} \notag $$
Moreover, since $\gamma$ is translation compact, the hull $\mathcal H(\gamma)$ is a compact set in $L^1_{\mathrm{loc}}(\mathbb{R})$ (see [16] for details).

From now on, we endow the hull $\mathcal H(\gamma)$ with the $L^1_{\mathrm{loc}}(\mathbb{R})$-topology (no other topologies on this hull will be considered). Given any $\eta\in\mathcal H(\gamma)$, we consider the associated wave equation

$$ \begin{equation} \partial_t^2u+\eta(t)\, \partial_t u-\Delta_x u+f(u)=g,\qquad u\big|_{\partial\Omega}=0,\quad \xi_u\big|_{t=0}=\xi_\tau. \end{equation} \tag{5.2} $$
We denote by $U_\eta(t,\tau)\colon E\to E$, $t\geqslant\tau$, the solution operator of this equation. It is easily checked that estimates (3.6), (3.3) and (3.14) hold uniformly with respect to $\eta \in\mathcal H(\gamma)$ and, therefore, all of the estimates obtained in § 3 and § 4, also hold uniformly with respect to $\eta\in\mathcal H(\gamma)$. We also note that the family of processes $\{U_\eta(t,\tau),\, \eta\in\mathcal H(\gamma)\}$ possesses the so-called translation identity
$$ \begin{equation*} U_\eta(t+h,\tau+h)= U_{T_h\eta}(t,\tau), \end{equation*} \notag $$
which in turn allows us to reduce the considered non-autonomous dynamical system to the autonomous semigroup $\mathbb S(t)$ acting on the extended phase space $\mathbb E:= E\times\mathcal H(\gamma)$ by
$$ \begin{equation} \mathbb S(t)\{\xi,\eta\}:=\{U_\eta(t,0)\xi,\, T_t\eta\} \end{equation} \tag{5.3} $$
(see [16] for more details). Thus, we may define a global attractor $\mathbb A$ for the semigroup $\mathbb S(t)$ in the extended phase space $\mathbb E$.

Definition 5.1. A set $\mathbb A$ is a global attractor for a semigroup $\mathbb S(t)\colon \mathbb E\to\mathbb E$ if

a) $\mathbb A$ is compact in $\mathbb E$;

b) $\mathbb A$ is strictly invariant, that is, $\mathbb S(t)\mathbb A= \mathbb A$ for all $t\geqslant0$;

c) $\mathbb A$ attracts bounded subsets of $\mathbb E$ as $t\to\infty$, that is, for any bounded $\mathbb B\subset \mathbb E$ and any neighbourhood $\mathcal O(\mathbb A)$, there exists $T = T(\mathbb B, \mathcal O)$ such that

$$ \begin{equation} \mathbb S(t)\mathbb B\subset\mathcal O(\mathbb A),\qquad t\geqslant T. \end{equation} \tag{5.4} $$

If an attractor $\mathbb A\subset\mathbb E$ exists, its projection $\mathcal A_{\mathrm{un}} := \Pi_1\mathbb A\subset E$ to the first component of the Cartesian product is called a uniform attractor for the family $U_\eta(t,\tau)$, $\eta\in\mathcal H(\gamma)$, of the dynamical processes associated with equation (5.2).

The existence of a global/uniform attractor is usually tested using the following standard result (see, for example, [16] for more details).

Proposition 5.1. Let a semigroup $\mathbb S(t)\colon\mathbb E\to\mathbb E$

a) be continuous as a map from $\mathbb E$ to $\mathbb E$ for every fixed $t\geqslant0$;

b) possess a compact attracting set $\mathcal B\subset\mathbb E$, that is, for every bounded $\mathbb B\subset\mathbb E$ and every neighbourhood $\mathcal O(\mathbb B)$, there exists $T = T(\mathbb B, \mathcal O)$ such that (5.4) is satisfied (with $\mathbb A$ replaced by $\mathcal B$).

Then the semigroup $\mathbb S(t)$ possesses a global attractor $\mathbb A$. Moreover, if the semigroup $\mathbb S(t)$ is an extended semigroup for the family of processes $U_\eta(t,\tau)\colon E\to E$, $\eta\in\mathcal H(\gamma)$, then the uniform attractor $\mathcal A_{\mathrm{un}}=\Pi_1\mathbb A$ possesses the following description:

$$ \begin{equation} \mathcal A_{\mathrm{un}} =\bigcup_{\eta\in\mathcal H(\gamma)}\mathcal K_\eta\big|_{t=0}, \end{equation} \tag{5.5} $$
where
$$ \begin{equation} \mathcal K_\eta:= \{\xi_u\in L^\infty(\mathbb{R},E)\colon \xi_u(t)= U_\eta(t,\tau)\xi_u(\tau),\ \tau\in\mathbb{R},\, t\geqslant\tau\} \end{equation} \tag{5.6} $$
is the set of all complete (defined for all $t\in\mathbb{R}$) bounded solutions of equation (5.2) with fixed symbol $\eta$ (the so-called kernel of equation (5.2) in the terminology of Chepyzhov and Vishik; see [16]).

Using the estimates obtained in previous two sections, we get the following result.

Theorem 5.1. Under the conditions of Theorem 4.1, the family $U_\eta(t,\tau)$ of dynamical processes associated with the wave equation (5.2) possesses a uniform attractor $\mathcal A_{\mathrm{un}}$ which is a bounded subset of $E^1$. Moreover, this attractor is generated by all complete bounded trajectories of equations (5.2), that is, the representation formula (5.5) holds.

Proof. Indeed, the continuity of the extended semigroup $\mathbb S(t)$ can be verified exactly as in Corollary 3.3 (we left the proof of continuity with respect to the symbol $\eta\in\mathcal H(\gamma)$ to the reader). Moreover, estimate (4.7) shows that the set $\mathbb B:= \mathcal B_R^1\times\mathcal H(\gamma)$ is a compact attracting set for the semigroup $\mathbb S(t)$. Here, we have also used that the embedding $E^1\subset E$ is compact and the hull $\mathcal H(\gamma)$ endowed with the $L^1_{\mathrm{loc}}(\mathbb{R})$ topology is also compact. Thus, all assumptions of Proposition 5.1 are verified and, therefore, the existence of a uniform attractor $\mathcal A_{\mathrm{un}}$ is also verified and the theorem is proved.

Remark 5.1. There is an alternative equivalent (at least in the case where $\mathbb S(t)$ is continuous, see [16]) definition of a uniform attractor, which does not refer explicitly to the reduction to autonomous extended semigroup. Namely, the set $\mathcal A_{\mathrm{un}}\subset E$ is a uniform attractor for the dynamical process $U_\gamma(t,\tau)\colon E\to E$ if

1) $\mathcal A_{\mathrm{un}}$ is compact in $E$;

2) $\mathcal A_{\mathrm{un}}$ attracts bounded sets $B\subset E$ uniformly with respect to $\tau\in\mathbb{R}$, that is, for every bounded $B\subset E$ and every neighbourhood $\mathcal O(\mathcal A_{\mathrm{un}})$, there exists $T=T(B, \mathcal O)$ such that

$$ \begin{equation*} U_\gamma(t,\tau)B\subset \mathcal O(\mathcal A_{\mathrm{un}})\quad\text{if}\quad t-\tau\geqslant T; \end{equation*} \notag $$

3) $\mathcal A_{\mathrm{un}}$ is the minimal set satisfying properties 1) and 2).

However, in order to get the representation formula (5.5), we need in any case to introduce the hull of the non-autonomous symbol $\gamma$. Since this representation formula is crucial for the attractors theory, we prefer to introduce the uniform attractor using the extended semigroup from the very beginning.

Remark 5.2. For every $\eta\in\mathcal H(\gamma)$, we may define the so-called kernel sections

$$ \begin{equation} \mathcal K_\eta(\tau):= \mathcal K_\eta\big|_{t=\tau}. \end{equation} \tag{5.7} $$
Then it is easily checked that the sets $\mathcal K_\eta(\tau)$ are compact and strictly invariant, that is, $\mathcal K_\eta(t)= U_\eta(t,\tau)\mathcal K_\eta(\tau)$, $t\geqslant\tau\in\mathbb{R}$.

Moreover, as proved in [16], [54], for every fixed $\eta$, these kernel sections possess the so-called pullback attraction property, that is, for any bounded set $B\subset E$, any fixed $t\in\mathbb{R}$ and any neighbourhood $\mathcal O(\mathcal K_\eta(t))$, there exists $T = T(t,B,\mathcal O)$ such that

$$ \begin{equation} U_\eta(t,\tau)B\subset O(\mathcal K_\eta(t))\quad \text{if}\quad t-\tau>T. \end{equation} \tag{5.8} $$
By this reason, the time dependent family $t\to\mathcal K_\eta(t)$ is often referred as a pullback attractor associated with equation (5.2) (see [55], [56] and the references there). Note that, in contrast to the case of a uniform attractor, the rate of convergence in (5.8) is not uniform with respect to $t\in\mathbb{R}$ and by this reason the forward in time convergence may fail to exist.

We now turn to exponential attractors. This concept was introduced in [57] for the autonomous case in order to overcome the major drawback of global attractors, namely, the fact that the convergence to a global attractor may be arbitrarily slow and there is no way to control this rate in terms of physical parameters of the considered system. This leads to sensitivity of the attractor to perturbations and makes it in a sense unobservable in experiments. Roughly speaking, the idea of an exponential attractor is to add some extra points to the global attractor in such a way that, on the one hand, the rate of attraction to this object becomes exponential and controllable, and, on the other hand, the size of the new attractor does not grow drastically, for instance, it should remain finite dimensional. The price to pay is that an exponential attractor is only positively invariant and, as a consequence, is not unique (see [59], [60] and the references there).

The situation is more delicate when non-autonomous equations are considered, since new essential drawbacks of global attractors come into play. Indeed, a uniform attractor is usually huge (infinite-dimensional) even when the “real” attractor consists of the only exponentially stable trajectory and kernel sections (pullback attractors) do not attract in a natural sense (forward in time). All these drawbacks can be overcome using the concept of a non-autonomous exponential attractor, which is, on the one hand, remains finite-dimensional, and, on the other hand, not only pullback, but also forward in time. We will require the construction proposed in [59].

Definition 5.2. Let $U(t,\tau)\colon E\to E$ be a dynamical process in a Banach space $E$. Then the family of sets $\mathcal M(t)\subset E$, $t\in\mathbb{R}$, is a non-autonomous exponential attractor for $U(t,\tau)$ if

1) the sets $\mathcal M(t)$ are compact in $E$;

2) they are positively semi-invariant: $U(t,\tau)\mathcal M(\tau)\subset\mathcal M(t)$;

3) the fractal dimension of $\mathcal M(t)$ is finite and uniformly bounded,

$$ \begin{equation*} \dim_f (\mathcal M(t),E)\leqslant C<\infty \end{equation*} \notag $$
for all $t\in\mathbb{R}$;

4) they are uniformly exponentially attracting, that is, there exists a positive constant $\alpha$ and a monotone increasing function $Q$ such that, for any bounded $B\subset E$,

$$ \begin{equation*} \operatorname{dist}_H\bigl(U(t,\tau)B, \mathcal M(t)\bigr)\leqslant Q(\|B\|_E)e^{-\alpha (t-\tau)} \end{equation*} \notag $$
uniformly with respect to $t\geqslant\tau$ and $\tau\in\mathbb{R}$.

We are now ready to state and prove the main result of this section.

Theorem 5.2. Under the conditions of Theorem 4.1, let, in addition, $f\in C^2(\mathbb{R})$ and

$$ \begin{equation} \gamma\in L^{1+\varepsilon}_b(\mathbb{R}) \end{equation} \tag{5.9} $$
for some positive $\varepsilon$. Then, the dynamical process $U_\gamma(t,\tau)\colon E\to E$ associated with the wave equation (3.1), possesses a non-autonomous exponential attractor $t\to\mathcal M(t)=\mathcal M_\gamma(t)$ which is uniformly bounded in $E^1$ and Hölder-continuous with respect to time,
$$ \begin{equation} \operatorname{dist}^{\mathrm{sym}}_E\bigl(\mathcal M(t), \mathcal M(\tau)\bigr)\leqslant C\biggl(\int_0^\infty e^{-\alpha s}|\gamma(t-s)-\gamma (\tau-s)| \,ds\biggr)^\kappa, \end{equation} \tag{5.10} $$
where the positive constant $C$ and $0<\kappa<1$ are independent of $t$ and $\tau$, and $\operatorname{dist}^{\mathrm{sym}}$ is the symmetric Hausdorff distance between sets in $E$.

Proof. We follow the approach developed in [59]. As a first step, we note that it is sufficient to construct an exponential attractor not on the whole space $E$, but only on the exponentially attracting ball $\mathcal B^1_R$ of the smoother space $E^1$ constructed in Theorem 4.1. Indeed, the already mentioned transitivity of exponential attraction will give then the result for the whole space $E$.

As the next step, we introduce the family of discrete dynamical processes

$$ \begin{equation} \overline U_\tau(m,n):=U_\gamma(nT+\tau,\, mT+\tau),\qquad n,m\in\mathbb Z,\quad n\geqslant m, \end{equation} \tag{5.11} $$
depending on a parameter $\tau\in\mathbb{R}$. Here, $T$ is a sufficiently large positive number, which will be fixed later. If we construct exponential attractors $\mathcal M_{d,\tau}(n)$ for these processes, the required “continuous” exponential attractor will be obtained by the standard expression
$$ \begin{equation} \mathcal M(\tau):=\bigcup_{s\in[0,T]}U_\gamma(\tau,\, \tau-s)\mathcal M_{d,\tau-s}(0) \end{equation} \tag{5.12} $$
(see [59], [60] for details). To verify the existence of the attractors $\mathcal M_{d,\tau} (n)$, we need to check a number of assumptions of the abstract exponential attractor existence theorem stated in [59]. Namely, we first need to verify that
$$ \begin{equation} \overline U_\tau(n +1,n)\bigl( \mathcal O^{E^1}_\varepsilon(\mathcal B_R^1)\bigr)\subset\mathcal B_R^1,\qquad \tau\in\mathbb{R}, \end{equation} \tag{5.13} $$
for sufficiently small $\varepsilon>0$. Here, we denote by $\mathcal O^{E^1}_\varepsilon (V )$ the $\varepsilon$-neighbourhood of a set $V\subset E^1$ in the topology of $E^1$. Indeed, as follows from the proof of Theorem 4.1, the dynamical process $U_\gamma(t,\tau)$ is well posed in $E^1$ and possesses a uniformly absorbing ball in it. By this reason, taking $\varepsilon>0$ small enough and increasing the radius $R$ if necessary, we get (5.13) for all $\tau\in\mathbb{R}$ if $T> 0$ is large enough.

Second, we need to verify the squeezing property (asymptotic smoothing property for differences of solutions) for the operators $\overline U_\tau(n,m)$. Let $u_1(t)$ and $u_2(t)$ be two solutions of problem (3.1) starting at $t=\tau$ and from $\mathcal O^{E^1}_\varepsilon(\mathcal B^1_R)$. According to the dissipative estimate in $E^1$,

$$ \begin{equation} \|\partial_t u_i(t)\|_{H^1} + \|u_i(t)\|_{H^2}\leqslant C,\qquad t\geqslant\tau,\quad i=1,2, \end{equation} \tag{5.14} $$
where the constant $C$ is independent of $t$ and $u_i$. Let $v(t):= u_1(t)-u_2(t)$. Then this function solves equation (3.27). In particular, estimate (3.26) holds. We now split $v(t)= v_1(t)+ v_2(t)$, where the function $v_1(t)$ solves
$$ \begin{equation} \partial_t^2 v_1+\gamma(t)\, \partial_t v_1-\Delta_x v_1+Lv_1=0,\qquad \xi_{v_1}\big|_{t=\tau}=\xi_v\big|_{t=\tau}, \end{equation} \tag{5.15} $$
where $L$ is large enough so as to satisfy the hypotheses of Proposition 4.1. Using estimate (4.2), we have
$$ \begin{equation} \|\partial_t v_1(t)\|_{L^2}^2+ \|\nabla_x v_1(t)\|^2_{L^2}\leqslant Ce^{-\alpha(t-\tau)}\bigl( \|\partial_t v(\tau)\|^2_{L^2}+ \|\nabla_x v(\tau)\|^2_{L^2}\bigr). \end{equation} \tag{5.16} $$
The remainder $v_2(t)$ solves the equation
$$ \begin{equation} \partial_t^2v_2+\gamma(t)\, \partial_t v_2-\Delta_x v_2+Lv_2 = Lv-l(t)v:=h_v(t),\qquad \xi_{v_2}\big|_{t=\tau}=0. \end{equation} \tag{5.17} $$
Since $u_i(t)$ are bounded in $H^2$, and since $f\in C^2$, it follows from (3.26) that
$$ \begin{equation} \|h_v(t)\|_{H^1_0}^2\leqslant C\|\nabla_x v(t)\|^2_{L^2}\leqslant Ce^{K(t-\tau)}\bigl(\|\partial_t v(\tau)\|^2_{L^2}+\|\nabla_x v(\tau)\|^2_{L^2}\bigr). \end{equation} \tag{5.18} $$
Therefore, applying the operator $(-\Delta_x)^{1/2}$ to both sides of equation (5.17) and using estimate (4.2), we have
$$ \begin{equation} \|\partial_t v(t)\|_{H^1}^2+\|\nabla_x v(t)\|^2_{H^1}\leqslant Ce^{K(t-\tau)}\bigl(\|\partial_t v(\tau)\|^2_{L^2}+\|\nabla_x v(\tau)\|^2_{L^2}\bigr), \end{equation} \tag{5.19} $$
where the constants $C$ and $K$ are independent of $u_1$, $u_2$, $t$ and $\tau$.

Let now $\xi_v:= \overline U_\tau(n +1,n)\xi_1-\overline U_\tau(n +1,n)\xi_2$. Then, according to the above estimates, $\xi_v$ can be written as $\xi_v = \xi_{v_1}+\xi_{v_2}$, where

$$ \begin{equation*} \|\xi_{v_1}\|_E\leqslant Ce^{-\alpha T}\|\xi_1-\xi_2\|_E, \qquad \|\xi_{v_2}\|_{E^1}\leqslant Ce^{KT}\|\xi_1-\xi_2\|_E. \end{equation*} \notag $$
Now fixing $T>0$ so that $Ce^{-\alpha T}\leqslant 1/2$, we get the required squeezing property.

Third, we need to estimate the difference between the processes corresponding to different symbols from the hull $\mathcal H(\gamma)$ (say, $\gamma$ and $T_{-s}\gamma$). Let $\xi_{u_1}(t):=U_\gamma(t,\tau)\xi_\tau$ and $\xi_{u_2}(t):=U_{T_{-s}\gamma}(t,\tau)\xi_\tau$. Then these functions also satisfy the uniform estimate (5.14). Let $v(t):=u_1(t)-u_2(t)$. This function solves the equation

$$ \begin{equation*} \partial_t^2 v+\gamma(t)\partial_t v-\Delta_x v+l(t)v=(\gamma(t)-\gamma(t-s))\,\partial_t u_2(t),\qquad \xi_v\big|_{t=\tau}=0. \end{equation*} \notag $$
Now, arguing as in the proof of estimate (3.26), we end up with the required estimate
$$ \begin{equation} \|\xi_v(t)\|_E^2\leqslant C\int_\tau^t e^{K(t-l)}|\gamma(l)-\gamma(l-s)|\,dl. \end{equation} \tag{5.20} $$
Thus, all sthe conditions of the abstract theorem on the existence of an exponential attractor are verified, and, therefore, the discrete exponential attractors $\mathcal M_{d,\tau}(n)$ are constructed (see [59] for more details). Finally, in order to pass from discrete to continuous exponential attractors via (5.12), we need to vereify that the trajectories of the dynamical process $U_\gamma(t,\tau)$ (started from $\mathcal O^{E^1}_\varepsilon(\mathcal B^1_R))$ are uniformly Hölder continuous with respect to time (with values in $E$). The continuity (and even Lipschitz continuity) of $u(t)$ in $H^1$ is obvious since we have the control of $\partial_t u$ in $H^1$, but the Hölder continuity of $\partial_t u$ in $L^2$ is a bit more delicate. Indeed, if we know that $\gamma\in L^1_b(\mathbb{R})$, from equation (3.1) and (5.14) we will get only that $\partial_t^2 u\in L^1_b(L^2)$, and this is not sufficient to get the Hölder continuity with respect to time for $\partial_t u$. To circumvent this issue, we put an extra condition (5.9), which guarantees that $\partial_t^2 u\in L^{1+\varepsilon}_b(L^2)$. This will give us the required Hölder continuity of $\partial_t u$, and the proof will complete. Thus, the above mentioned Hölder continuity is verified and the theorem is proved.

Remark 5.3. The Hölder continuity (5.10) is crucial for consistency with the autonomous case. Indeed, it guarantees that in the autonomous case $\gamma\equiv \mathrm{const}$, the non-autonomous exponential attractor coincides with the autonomous one. Moreover, if $\gamma$ is periodic, quasi-periodic, etc., the same will be true for the non-autonomous attractor as well. Actually, exactly in order to guarantee this Hölder continuity, we need to assume the extra regularity (5.9). This Hölder continuity sometimes may be looked upon as an essential restriction, but it cannot be relaxed without losing the consistency with the autonomous case. In particular, without this assumption, we may have a “pathological” dependence on time of the exponential attractor corresponding to the autonomous case $\gamma\equiv \mathrm{const}$.

Remark 5.4. Since the sections of the kernel $\mathcal K_\gamma(\tau)$ are always subsets of the corresponding exponential attractors,

$$ \begin{equation*} \mathcal K_\gamma(\tau)\subset\mathcal M(\tau), \end{equation*} \notag $$
the kernel sections $\mathcal K_\gamma(\tau)$ are have finite-dimensional for all $\tau\in\mathbb{R}$.

§ 6. Random dissipation rate

In this section, we will relax the assumptions on the dissipation coefficient $\gamma(t)$ in order to be able to deal with random dissipation. Indeed, although assumption (3.6) is well adapted to the case of periodic or almost periodic dissipation, the uniformity of averaging with respect to $\tau\in\mathbb{R}$, as postulated there, is too restrictive if we want to consider chaotic or random dissipation rates. So, the assumption should be relaxed. In this section, we assume that there exists a Borel probability measure $\mu$ on the hull $\mathcal H(\gamma)$ such that the group of shifts $\{T_s$, $s\in\mathbb{R}\}$ is measure preserving and ergodic,

$$ \begin{equation} T_s\mu=\mu,\qquad s\in\mathbb{R}. \end{equation} \tag{6.1} $$
So, we replace (3.6) by the weaker assumption
$$ \begin{equation} \int_{\eta\in\mathcal H(\gamma)}\biggl(\int_0^1\biggl(\frac12\gamma_+(s)-\frac{p+2}{p+4}\gamma_-(s)\biggr)\,ds\biggr) \, \mu(d\eta)=\overline\beta>0. \end{equation} \tag{6.2} $$
Now, by Birkhoff ergodic theorem, for any $\tau\in\mathbb{R}$
$$ \begin{equation} \lim_{T\to\infty}\frac1{2T}\int_{\tau-T}^{\tau+T}\beta(s)\,ds= \lim_{T\to\infty}\frac1{T}\int_{\tau-T}^{\tau}\beta(s)\,ds=\overline\beta \end{equation} \tag{6.3} $$
for almost all $\eta\in\mathcal H(\gamma)$, where $\beta(t)=\beta_\eta(t):=(1/2)\eta_+(s)-((p+2)/(p+4))\eta_-(s)$. However, in contrast to the previous sections, this limit is not uniform with respect to $\tau\in\mathbb{R}$, and this leads to essential changes in the theory. In particular, as we will see below, assumption (6.3) does not guarantee the dissipativity of equation (5.2) forward in time; moreover, most of the trajectories may be unbounded as $t\to\infty$. To overcome this difficulty, we will use the pullback attraction property and the theory of pullback/random attractors theory, as developed in [55], [61], [63] (see also the references cited there). We start with necessary definitions and straightforward results.

Definition 6.1. A function $t\to\varphi(t)\in\mathbb{R}$, $t\in\mathbb{R}$, is called tempered if

$$ \begin{equation} \lim_{t\to-\infty}e^{\theta t}|\varphi(t)|=0\quad \forall\, \theta>0. \end{equation} \tag{6.4} $$
Analogously, a family of bounded sets $B(t)\subset E$, $t\in\mathbb{R}$, is called tempered if the function $\varphi_B(t):=\|B(t)\|_{E}$ is tempered.

We now state “the tempered” analogues of absorbing and attracting sets as well as the tempered pullback attractors.

Definition 6.2. A family $\mathbb B(t)$, $t\in\mathbb{R}$, of bounded sets in $E$ is called tempered (pullback) absorbing set for a process $U_\eta(t,\tau)\colon E\to E$ if it is tempered and for every other tempered family $B(t)$, $t\in\mathbb{R}$, of bounded sets and every fixed $t\in\mathbb{R}$,

$$ \begin{equation*} U_\eta(t,t-s)B(t-s)\subset \mathbb B(t) \end{equation*} \notag $$
if $s\geqslant S(B,t)$ is large enough.

A family $\mathcal B(t)$, $t\in\mathbb{R}$, of bounded sets in $E$ is called tempered attracting set if it is tempered and, for every other tempered family $B(t)$ of bounded sets and every fixed $t\in\mathbb{R}$,

$$ \begin{equation*} \lim_{s\to\infty}\operatorname{dist}_H\bigl(U_\eta(t,t-s)B(t-s),\mathcal B(t)\bigr)=0. \end{equation*} \notag $$

We are now ready to define the “tempered” analogues of the kernel sections (pullback attractors).

Definition 6.3. A tempered family of bounded sets $\mathcal K_\eta(t)$ is called a tempered pullback attractor if

1) $\mathcal K_\eta(t)$ is compact in $E$ for every fixed $t\in\mathbb{R}$;

2) the strict invariance property holds: $U_\eta(t,\tau)\mathcal K_\eta(\tau)=\mathcal K_\eta(t)$;

3) the family $\mathcal K_\eta(t)$ is a tempered (pullback) attracting set.

The next proposition is an analogue of Proposition 5.1 for the tempered case (see [55] for details).

Proposition 6.1. Let $U_\eta(t,\tau)$ be a dynamical process in $E$ consisting of continuous operators in $E$ and possesses a compact (for every $t\in\mathbb{R}$) tempered attracting set. Then $U_\eta(t,\tau)$ possesses a tempered (pullback) attractor $\mathcal K_\eta(t) $ with the following description:

$$ \begin{equation} \mathcal K_\eta(\tau)=\mathcal K_\eta\big|_{t=\tau}, \end{equation} \tag{6.5} $$
where $\mathcal K_\eta$ is the set of all complete tempered trajectories of $U_\eta(t,\tau)$ (the tempered kernel of $U_\eta(t,\tau)$).

Remark 6.1. The assumption that the attracting set is tempered can be relaxed if we assume the existence of a tempered absorbing set. Then, as usual, only asymptotic compactness is necessary to get the existence of an attractor. However, the assumption that the absorbing set is tempered is important in order to have the representation formula (6.5). This property may be lost if the absorbing ball is not tempered; this, in turn, leads to many pathological effects (non-uniqueness of the attractor, etc.).

The next theorem is the main result of this section.

Theorem 6.1. Let a non-linearity $f\in C^1(\mathbb{R})$ satisfy (3.2) with $p<4$, and let $\gamma\in L^1_{\textrm{tr-c}}(\mathbb{R})$ satisfy (6.2). Then, for almost all $\eta\in\mathcal H(\gamma)$, the dynamical process $U_\eta(t,\tau)$ associated with equation (5.2) possesses a pullback attractor $\mathcal K_\eta(t)$ which is tempered in $E^\beta$ for some $\beta>0$.

Proof. To verify the conditions of Proposition 6.1 we need to adapt the proofs of Theorems 3.1 and 4.1 to the random case. This adaptation is almost straightforward, and so we briefly indicate below the main differences related with verification that the obtained absorbing/attracting sets are tempered. We start with Theorem 3.1, which gives us the existence of an absorbing set.

Indeed, arguing exactly as in the proof of Theorem 3.1, we derive estimate (3.24) with $\beta_\varepsilon(t)=\beta_{\varepsilon,\eta}$ defined by (3.22) (where $\gamma$ is replaced by $\eta\in\mathcal H(\gamma)$). Moreover, defining $\overline\gamma$ (respectively, $\overline\eta$) and using (3.15), we see that the function $\eta\to\int^1_0\beta_{\varepsilon,\eta}(s)\,ds$ is continuous on the compact set $\mathcal H(\gamma)$. Hence there exists its mean value

$$ \begin{equation*} \int_{\eta\in\mathcal H(\gamma)} \biggl(\int_0^1\beta_{\varepsilon,\eta}(s)\,ds\biggr)\, \mu(d\eta) =\overline\beta_\varepsilon<\infty \end{equation*} \notag $$
and now, fixing $\varepsilon>0$ and $\kappa>0$, we may assume that $\overline\beta_\varepsilon>0$ (in view of assumption (6.2)). The Birkhoff ergodic theorem now gives that, for every $\tau\in\mathbb{R}$,
$$ \begin{equation} \lim_{T\to\infty}\frac1T\int_{\tau-T}^\tau\beta_{\varepsilon,\eta}(s)\,ds= \lim_{T\to\infty}\frac1T\int_{\tau}^{\tau+T}\beta_{\varepsilon,\eta}(s)\,ds =\overline\beta_\varepsilon>0 \end{equation} \tag{6.6} $$
for almost all $\eta\in\mathcal H(\gamma)$. Of course, $T_s\eta$ also satisfies (6.6) if $\eta$ does, and so there is a full measure set $\mathcal H_{\mathrm{erg}}\subset \mathcal H(\gamma)$ invariant with respect to $T_s$ such that (6.6) is satisfied for all $\eta\in\mathcal H_{\mathrm{erg}}$.

The following technical lemma is crucial for further analysis.

Lemma 6.1. Let $\phi\in L^1_{\mathrm{loc}}(\mathbb{R})$ be such that $t\to\int_t^{t+1}|\phi(s)|\,ds$ is tempered and let $\beta_\varepsilon\in L^1_b(\mathbb{R})$ satisfy (6.6). Then the function

$$ \begin{equation} R(t):=\int_{-\infty}^t\phi(s)\exp\biggl\{-\int_s^t\beta_\varepsilon(l)\,dl\biggr\}\,ds \end{equation} \tag{6.7} $$
is well-defined and tempered. Moreover, if $A(t)$ is tempered, then there exists $\alpha>0$ such that
$$ \begin{equation} \lim_{s\to\infty}e^{-\alpha (t-s)}A(t-s)\exp\biggl\{-\int_s^t\beta_\varepsilon(l)\,dl\biggr\}=0 \end{equation} \tag{6.8} $$
for all $t\in\mathbb{R}$.

Proof. Without loss of generality we may assume that $t\leqslant0$. Then we split
$$ \begin{equation*} \int_s^t\beta_\varepsilon(l)\,dl=\int_s^0\beta_\varepsilon(l)\,dl -\int_t^0\beta_\varepsilon(l)\,dl \end{equation*} \notag $$
and write
$$ \begin{equation*} |R(t)|\leqslant \exp\biggl\{\int_t^0\beta_\varepsilon(l)\,dl\biggr\} \int_{-\infty}^t|\phi(s)|\exp\biggl\{-\int^0_s\beta_\varepsilon(l)\,dl\biggr\}\,ds. \end{equation*} \notag $$
Moreover, using (6.6), for every $\nu>0$ we have
$$ \begin{equation} -(\overline\beta_\varepsilon-\nu)s-C_\nu\leqslant\int_s^0\beta_\varepsilon(l)\,dl\leqslant -(\overline\beta_\varepsilon+\nu)s+C_\nu \end{equation} \tag{6.9} $$
(for some positive $C_\nu$), and now the previous estimate reads
$$ \begin{equation*} |R(t)|\leqslant C'_\nu e^{-(\overline\beta_\varepsilon+\nu)t} \int_{-\infty}^t|\phi(s)|e^{(\overline\beta_\varepsilon-\nu)s}\,ds. \end{equation*} \notag $$
Finally, since the integral $\int_t^{t+1}|\phi(s)|\,ds$ is tempered, we have
$$ \begin{equation*} \int_{t}^{t+1}|\phi(s)|\,ds\leqslant C_\nu e^{-t\nu/2} \end{equation*} \notag $$
and
$$ \begin{equation*} |R(t)|\leqslant C_\nu'e^{-(\overline\beta_\varepsilon+\nu)t}\int_{-\infty}^t e^{(\overline\beta_\varepsilon-3\nu/2)s}\,ds\leqslant C_\nu''e^{-5\nu t/2}. \end{equation*} \notag $$
Since $\nu>0$ is arbitrary, the function $R(t)$ is indeed tempered.

To verify the second assertion, we note that it is enough to check (6.8) for $t=0$ (the uniformity with respect to $t\in\mathbb{R}$ is not assumed in (6.8)). Now estimate (6.9) immediately gives us (6.8) for any $\alpha<\overline\beta_\varepsilon$. This proves the lemma.

We are now ready to complete the proof of the theorem. Let $\eta\in\mathcal H_{\mathrm{erg}}$ and

$$ \begin{equation} R_\eta(t):=2C\int_{-\infty}^t(1+\|g\|^2_{L^2}+|\eta(s)|) \exp\biggl\{-\int_s^t\beta_{\varepsilon,\eta}(l)\,dl\biggr\}\,ds, \end{equation} \tag{6.10} $$
where all the constants and functions are the same as in (3.24). Now, by estimate (3.24) and Lemma 6.1, the set
$$ \begin{equation*} \mathcal B^0_E(R_\eta(t)):=\{\xi\in E,\, \mathcal E(\xi)\leqslant R_\eta(t)\} \end{equation*} \notag $$
is a tempered absorbing set for the process $U_\eta(t,\tau)$ associated with the wave equation (5.2), Moreover, the fact that $R_{\eta}(t)$ solves the integral equation
$$ \begin{equation*} \begin{aligned} \, R_\eta(t) &=R_\eta(\tau) \exp\biggl\{-\int_\tau^t\beta_{\varepsilon,\eta}(s)\,ds\biggr\} \\ &\qquad+2C\int_{\tau}^t \bigl(1+\|g\|^2_{L^2}+|\eta(s)|\bigr) \exp\biggl\{-\int_s^t\beta_{\varepsilon,\eta}(l)\,dl\biggr\}\,ds \end{aligned} \end{equation*} \notag $$
for all $\tau\leqslant t$ (compare with (3.24)), gives that this absorbing set is invariant,
$$ \begin{equation*} U_{\eta}(t,\tau)\mathcal B^0_E(R_{\eta}(\tau))\subset\mathcal B^0_E(R_{\eta}(t)). \end{equation*} \notag $$
To obtain the tempered compact attracting set, we need to get random analogues of estimates from in § 4. We first observe that the analogue of estimate (4.2) for the random case reads
$$ \begin{equation*} \mathcal E_v(t)\leqslant C\mathcal E_v(\tau) \exp\biggl\{-\int_\tau^t\beta_{\varepsilon,\eta}(l)\,dl\biggr\}+ C\biggl(\int_s^te^{-\beta_{\varepsilon,\eta}(l)\,dl}\|h(s)\|_{L^2}\,ds\biggr)^2, \end{equation*} \notag $$
and large constant $L$ can be chosen uniformly with respect to $\eta\in\mathcal H(\gamma)$ (see (4.3) and (4.5)). Thus, the solution $\xi_v(t)$ of (4.9) starting from a tempered absorbing set $\xi_u(\tau)\in\mathcal B^0_E(R_\eta(\tau))$, will be exponentially decaying as $\tau\to-\infty$ according to Lemma 6.1.

Hence, to verify the existence of a tempered attracting set, it is sufficient to check that the function $h_u(t)$ in the right-hand side of (4.10) is tempered as $\tau\to -\infty$. To be more precise, we need to show that the function $\tau\to \|h_u\|_{L^1(\tau,\tau+1;H^\beta)}$ is tempered as $\tau\to-\infty$. But this is an immediate corollary of the fact that $\xi_u(t)$ belongs to a tempered absorbing ball and estimate (3.11) (see the proof of Theorem 4.1).

Thus, the tempered compact attracting set for $U_\eta(t,\tau)$ is constructed (as a tempered ball in smoother space $E^\beta$, $\beta>0$) and the theorem is proved.

Remark 6.2. Arguing as in the proof of Theorem 4.1, we may show that the attractor $\mathcal K_\eta(t)$ is a tempered set in $E^1$, but to this end we need the tempered version for transitivity of exponential attraction. Since we need not this regularity for what follows, we prefer not to discuss this topic here.

We also note that the radius $R_\eta(t)$ of the absorbing ball is measurable (for every fixed $t$ as a function of $\eta\in\mathcal H(\gamma)$). Indeed, it can be presented as an almost everywhere limit of continuous functions:

$$ \begin{equation*} R_\eta(t):=\lim_{n\to \infty}\int_{-n}^t\bigl(1+\|g\|^2_{L^2}+|\eta(s)|\bigr) \exp\biggl\{-\int_s^t\beta_{\varepsilon,\eta}(l)\,dl\biggr\}\,ds. \end{equation*} \notag $$
This shows that the set-valued function $\eta\to \mathcal B_E^0(R_\eta(t))$ is measurable. Now by standard arguments (see, for example, [61], [65]) the attractor $\mathcal K_\eta(t)$ is also measurable as a set-valued function $\eta\to\mathcal K_\eta(t)$ for every fixed $t$.

We are now ready to complete the construction of a random attractor for equation (5.2). We first recall its definition adapted to our case (see [55] for more details). First, by definition, a random tempered set is a measured set-valued function $\eta\to B(\eta)\subset E$ such that the function $t\to \|B(T_t\eta)\|_E$ is tempered for almost all $\eta$.

Definition 6.4. A random tempered set $\eta\to\mathcal A(\eta)\subset E$ is a random attractor for the family of processes $U_\eta(t,\tau)$, $\eta\in\mathcal H(\gamma)$ if

1) $\mathcal A(\eta)$ is compact for almost all $\eta\in\mathcal H(\gamma)$;

2) the strict invariance property holds: $U_{\eta}(t,0)\mathcal A(\eta)=A(T_t\eta)$ for all $t\geqslant0$ and almost all $\eta$;

3) for any tempered random set $\eta\to B(\eta)$ and almost all $\eta\in\mathcal H(\gamma)$,

$$ \begin{equation} \lim_{\tau\to-\infty}\operatorname{dist}_E(U_\eta(0,\tau)B(T_\tau\eta),\mathcal A(\eta))=0. \end{equation} \tag{6.11} $$

Corollary 6.1. Let the conditions of Theorem 6.1 be met. Then the family of processes $U_\eta(t,\tau)\colon E\to E$, $\eta\in\mathcal H(\gamma)$, possesses a random attractor $\mathcal A(\eta)$ which is a tempered random set in $E^\beta$ for some $\beta>0$.

Proof. Indeed, we may define
$$ \begin{equation} \mathcal A(\eta):=\begin{cases} \mathcal K_\eta(0), &\eta\in\mathcal H_{\mathrm{erg}}(\gamma), \\ \varnothing, &\eta\notin\mathcal H_{\mathrm{erg}}(\gamma), \end{cases} \end{equation} \tag{6.12} $$
where $\mathcal K_\eta(t)$ is the tempered pullback attractor constructed in Theorem 6.1. All the conditions of Definition 6.4 were in fact verified above. So, the required attractor is constructed.

Remark 6.3. As already mentioned, a pullback attractor in general fails to attract bounded sets forward in time. The situation is much better for the case of random attractors for which the forward convergence in measure usually holds (see [61]). Indeed, the Lebesgue dominated convergence theorem allows us to derive from almost everywhere convergence (6.11) that

$$ \begin{equation*} \int_{\mathcal H(\gamma)}\frac{\operatorname{dist}_E(U_\eta(0,\tau)B(T_\tau\eta),\mathcal A(\eta))}{1+\operatorname{dist}_E(U_\eta(0,\tau)B(T_\tau\eta),\mathcal A(\eta))}\, \mu(d\eta)\to0 \end{equation*} \notag $$
as $\tau\to-\infty$. Using now the translation identity and the fact that $T_\tau$ is measure preserving, after the change of variable $\eta\to T_\tau\eta$, we have
$$ \begin{equation*} \int_{\mathcal H(\gamma)} \frac{\operatorname{dist}_E(U_\eta(\tau,0)B(\eta),\mathcal A(T_\tau\eta))} {1+\operatorname{dist}_E(U_\eta(\tau,0)B(\eta),\mathcal A(T_\tau\eta))}\, \mu(d\eta)\to0 \end{equation*} \notag $$
as $\tau\to+\infty$. It remains to note that the last convergence is equivalent to the required forward convergence in measure
$$ \begin{equation} \mu{-}\!\!\lim_{\tau\to\infty} \operatorname{dist}_E\bigl(U_\eta(0,\tau)B(\eta),\mathcal A(T_\tau\eta)\bigr)=0. \end{equation} \tag{6.13} $$

We complete this section by considering a natural model example when the dynamics $T_s\colon\mathcal H(\gamma)\to\mathcal H(\gamma)$ is determined by the Bernoulli shift dynamics. Namely, let $\Gamma:=\{a,-b\}^{\mathbb Z}$ be a two-sided Bernoulli scheme with two symbols $\{a,-b\}$ and let

$$ \begin{equation*} (T_l\gamma)(n)=\gamma(n+l),\qquad l\in\mathbb Z,\quad \gamma=(\dots,\gamma_{-n},\dots,\gamma_n,\dots)\in\Gamma, \end{equation*} \notag $$
be the associated Bernoulli process. We endow the set $\Gamma$ with the Tichonoff topology and by the Borel probability product measure $\mu$ generated by the probability measure on the cross section
$$ \begin{equation*} \mu(\{a\})=q,\quad \mu(\{-b\})=1-q,\qquad q\in(0,1). \end{equation*} \notag $$
It is known (see, for example, [51]) that the dynamical system $(T_l,\Gamma,\mu)$ is transitive and ergodic. We extend any element $\gamma\in\Gamma$ to a function $\mathcal R(\gamma)\in L^\infty(\mathbb{R})$ by
$$ \begin{equation} \mathcal R(\gamma)(t):=\gamma_{[t]}, \end{equation} \tag{6.14} $$
where $[t]$ is an integer part of $t$. It is clear that
$$ \begin{equation*} T_l\mathcal R(\gamma)=\mathcal R(T_l\gamma) \end{equation*} \notag $$
for all $l\in\mathbb Z$. It is easily checked that the Tychonoff topology on $\Gamma$ induces the $L^1_{\mathrm{loc}}$-topology on $\mathcal R(\gamma)$, and so $\mathcal R(\gamma)$ is translation compact for any $\gamma\in\Gamma$. Moreover, if we take any transitive trajectory $\gamma\in\Gamma$, the hull $\mathcal H(\mathcal R(\gamma))$ will contain the image of the whole Bernoulli scheme $\Gamma$.

Thus, the elements of the hull $\mathcal H(\gamma):=\mathcal H(\mathcal R(\gamma))$ are parameterized by the elements $\eta\in\Gamma$, and the dynamics on the hull is equivalent to the classical Bernoulli shift dynamics. Being pedantic, we first need to extend the discrete Bernoulli shifts to the continuous ones acting on the extended space $\Gamma\times[0,1]$ (to parameterize the non-integer shifts), and only after this, establish the equivalence, but to avoid the technicalities we omit this step and will identify (with a slight abuse of rigor) the element $\gamma\in\Gamma$ with $\mathcal R(\gamma)\in L^\infty(\mathbb{R})$, as well as discrete shifts on $\Gamma$ with continuous shifts on $\mathcal H(\gamma)$.

The key condition (6.2) now reads

$$ \begin{equation} \int_{\eta\in\mathcal H(\gamma)} \int_0^1\biggl(\frac12\gamma_+(s)-\frac{p+2}{p+4}\gamma_-(s)\biggr)\, ds\,\mu(d\eta)=\frac12 aq-\frac{p+2}{p+4}b(1-q)>0. \end{equation} \tag{6.15} $$

Thus, we have established the following result.

Theorem 6.2. Let exponents $a$ and $b$ and the probability $q$ satisfy condition (6.15). Then the wave equation (5.2) with damping rate $\eta$ generated by the above Bernoulli process possesses a random attractor $\mathcal A(\eta)$ in the energy space $E$.

§ 7. Infinite-dimensionality of random attractors: a toy example

The results on the existence of random attractors from § 6, look more or less standard (see, for example, [55] and the references there). Nevertheless, there is an essential difference between this case and the random attractors considered in the above mentioned works. This difference becomes transparent if we try to compute the mean of the size of tempered absorbing set constructed in Theorem 6.1. Indeed, taking the mean of the expression (6.10), we get

$$ \begin{equation} \begin{aligned} \, &\int_{\eta\in\mathcal H(\gamma)}R_\eta(0)\, \mu(d\eta) \nonumber \\ &\qquad\sim C\int_{\eta}\biggl(\sum_{k=-\infty}^0 \exp\biggl\{-\varepsilon k-2\sum_{l=-k}^0\frac12\eta_+(l)-\frac{p+2}{p+4}\eta_-(l)\biggr\}\biggr)\, \mu(d\eta) \nonumber \\ &\qquad=\sum_{k=-\infty}^0e^{\varepsilon(-k+1)} \int_\eta \prod_{l=-k}^0 \exp\biggl\{-\eta_+(l)+\frac{2(p+2)}{p+4}\eta_-(l)\biggr\}\, \mu(d\eta) \nonumber \\ &\qquad=\sum_{k=-\infty}^0e^{-\varepsilon k} \biggl(\int_\eta \exp\biggl\{-\eta_+(0)+\frac{2(p+2)}{p+4}\eta_-(0)\biggr\} \, \mu(d\eta)\biggr)^{-k+1} \nonumber \\ &\qquad=\sum_{k=0}^\infty \exp\bigl\{(k+1) \bigl(\varepsilon+\ln\bigl(e^{-a}q+e^{2(p+2)/(p+4)b}(1-q)\bigr)\bigr)\bigr\}, \end{aligned} \end{equation} \tag{7.1} $$
where we have used that $\eta(n)$ and $\eta(m)$ are independent as random variables if $m\ne n$. Thus, the expression in the right-hand side will be finite if and only if
$$ \begin{equation*} \ln\bigl(e^{-a}q+e^{2(p+2)/(p+4)b}(1-q)\bigr)<-\varepsilon<0. \end{equation*} \notag $$
So, if the probability $q$ and the exponents $a>0$ and $b>0$ are such that
$$ \begin{equation} \ln\bigl(e^{-a}q+e^{2(p+2)/(p+4)b}(1-q)\bigr)>0>-aq+\frac{2(p+2)}{p+4}b(1-q), \end{equation} \tag{7.2} $$
the obtained energy bound $R_\eta(t)$ will have infinite mean (in contrast to [61], [63], where this mean is always finite).

The absence of the first momentum for the energy may make a drastic impact on the underlying limit dynamics, in particular, making it infinite-dimensional. This observation is based on the following standard counterpart of the Birkhoff ergodic theorem (see, for example, [66] for more delicate results in this direction).

Proposition 7.1. Let $(X,\mu)$ be a compact metric space with Borel probability measure $\mu$ on it and let $T\colon X\to X$ be an ergodic map. Assume that $f\colon X\to\mathbb{R}$ is a non-negative measurable function such that $\int_X f(x)\,\mu(dx)=\infty$. Then

$$ \begin{equation} \lim_{N\to\infty}\frac1N\sum_{n=1}^Nf(T^nx)=\infty \end{equation} \tag{7.3} $$
for almost all $x\in X$.

Proof. Indeed, by the definition of Lebesgue integration, there exist a sequence of simple functions $f_l\in L^1(X,\mu)$ such that
$$ \begin{equation*} 0\leqslant f_l(x)\leqslant f(x),\qquad f_l(x)\to f(x) \text{ almost everywhere} \end{equation*} \notag $$
and $\int_X f_l(x)\,\mu(dx)\to\infty$. Let $x\in X$ be such that Birkhoff ergodic theorem holds for all $l$. Then
$$ \begin{equation*} \lim_{N\to\infty}\frac1N\sum_{n=1}^Nf(T^nx)\geqslant\lim_{N\to\infty}\frac1N\sum_{n=1}^Nf_l(T^nx)= \int_Xf_l(x)\, \mu(dx)\to\infty \end{equation*} \notag $$
as $l\to\infty$, proving the proposition.

This result allows us to expect that, in the case where the momentums of the energy do not exist, the global Lyapunov exponents and Lyapunov dimension of the attractors may also be infinite (since the ergodic sums involving energy are usually present at least in the estimates for these exponents; see [63], [64] for the details). This, in turn, allows us to expect that the Hausdorff and fractal dimension of the attractor may be also infinite. To be more precise, we state the following conjecture.

Conjecture 7.1. Let exponents $p$, $a$, $b$ and probability $q$ satisfy (7.2). Then there are a non-linearity $f$ and a right-hand side $g$ satisfying the conditions of Theorem 6.2 and such that the associated random attractor $\mathcal A(\eta)$ has an infinite Hausdorff and fractal dimensions in $E$.

Up to the moment, we are unable to prove or disprove this conjecture. However, to support this conjecture, we conclude the section by a simplified model example for which this conjecture holds.

Example 7.1. Let $H=l_2$ (the space of square summable sequences). Consider the random dynamical system in $H$ generated by the equations

$$ \begin{equation} \frac d{dt} u_1+\eta(t)u_1(t)=1,\quad \frac d{dt}u_k+k^4u_k=u_1(t)u_k-u_k^3, \qquad k=2,3,\dots, \end{equation} \tag{7.4} $$
where $u=(u_1,u_2,\dots)\in H$, and $\eta\in\Gamma$ is exactly the Bernoulli process used in Theorem 6.2. The first equation of this system models the energy evolution of (5.2) and the remaining equations give some coupling of the first equation with a parabolic PDE.

System of ODEs (7.4) is simple enough to be solved explicitly. In particular, if $aq-(1-q)b>0$, then

$$ \begin{equation} u_1(t)=u_{1,\eta}(t)=\int_{-\infty}^t \exp\biggl\{-\int_s^t\eta(l)\,dl\biggr\}\,ds \end{equation} \tag{7.5} $$
is a unique tempered complete solution of the first equation (for almost all $\eta\in\Gamma$). Moreover, arguing as in (7.1), we get
$$ \begin{equation} \int_{\eta\in\Gamma}u_{1,\eta}(0)\, \mu(d\eta)=\infty \end{equation} \tag{7.6} $$
if
$$ \begin{equation} \ln\bigl(qe^{-a}+(1-q)e^{b}\bigr)>0>-aq+(1-q)b. \end{equation} \tag{7.7} $$
We claim that under this assumption the random attractor associated with (7.4) is infinite-dimensional.

Theorem 7.1. Let exponents $a,b>0$ and probability $q\in(0,1)$ satisfy (7.7). Then the random attractor $\mathcal A(\eta)$ for system (7.7) in $H$ has infinite Hausdorff and fractal dimensions in $H$,

$$ \begin{equation} \dim_H(\mathcal A(\eta),H)=\dim_f(\mathcal A(\eta),H)=\infty \end{equation} \tag{7.8} $$
for almost all $\eta\in\Gamma$.

Proof. We first briefly discuss why this random attractor exists. The existence of a random tempered absorbing set for the first component $u_1$ of system (7.4) follows from the explicit formula for the solution and Lemma 6.1. Assuming that $u_1(\tau)$ already lies in this absorbing set, let us estimate $u_k$. To this end, multiplying the $k$th equation by $\operatorname{sgn}(u_k)$ and summing, we have, after standard estimates,
$$ \begin{equation*} \frac d{dt}\biggl(\sum_{k=2}^\infty|u_k|\biggr)+\sum_{k=2}^\infty k^4|u_k|\leqslant C\biggl(\sum_{k=2}^\infty\frac1{k^2}\biggr)|u_1|\leqslant C|u_1(t)|. \end{equation*} \notag $$
Integrating this inequality and since $u_1(t)$ is tempered, we get a tempered absorbing ball for $u$ in $l_1\subset H$. To obtain a compact absorbing set, it is enough to use the parabolic smoothing property in a standard way. This verifies the existence of a random attractor $\mathcal A(\eta)$.

Recall also that a random attractor consists of all complete tempered trajectories of the system considered, and so it is enough to find all such trajectories. The first equation is linear and is independent of the other equations, and so such a trajectory is unique and is given by (7.5). Thus, to find $\mathcal K_\eta$ we need to fix $u_1(t)=u_{1,\eta}(t)$ in the other equations of (7.5) and find the tempered attractor $\mathcal A^k(\eta)$, $k=2,3,\dots$, for every component of (7.5) separately. Then, the desired attractor $\mathcal A(\eta)$ for the whole system has the product form

$$ \begin{equation} \mathcal A(\eta)=\{u_{1,\eta}(0)\}\times\bigotimes_{k=2}^\infty\mathcal A^k(\eta). \end{equation} \tag{7.9} $$
Moreover, since the attractor is always connected, every $\mathcal A^k(\eta)$ is a closed interval, and so to prove the desired infinite-dimensionality, it is enough to prove that $\mathcal A^k(\eta)\ne\{0\}$ for all $k$. In other words, we need to find a non-zero tempered trajectory $u_k=u_k(t)$ for the equation
$$ \begin{equation} \frac d{dt} u_k+k^4u_k=u_{1,\eta}(t)u_k-u_k^3, \end{equation} \tag{7.10} $$
using that $u_{1,\eta}(t)$ is tempered and have an infinite mean.

To this end, we first note that every solution of equation (7.10) is either tempered as $t\to-\infty$, or blows up backward in time (this can be easily shown by comparison using the fact that $u_{1,\eta}(t)$ is tempere), and so any complete solution $u_k(t)$, $t\in\mathbb{R}$, is automatically tempered.

We construct this solution by solving equation (7.10) explicitly. Namely,

$$ \begin{equation*} u_{k,\eta}(t):=\biggl(2\int_{-\infty}^t\exp\biggl\{2\int_s^t(k^4-u_{1,\eta}(l))\,dl\biggr\} \,ds\biggr)^{-1/2}. \end{equation*} \notag $$
Indeed, the finiteness of the integral is guaranteed by (7.6) and Proposition 7.1 (analogously to the proof of Lemma 6.1). Thus, $\mathcal A^k(\eta)\ne \{0\}$ for almost all $k$, proving the theorem.

Remark 7.1. In the actual fact, we have explicitly found the random attractor in the previous example:

$$ \begin{equation*} \mathcal A(\eta)=\{u_{1,\eta}(0)\}\times\bigotimes_{k=2}^\infty[-u_{k,\eta}(0),u_{k,\eta}(0)]. \end{equation*} \notag $$

It would be interesting to compute (for example, via this expression) the typical Kolmogorov entropy of this infinite-dimensional attractor.

The authors would like to thank D. Turaev and V. Kalantarov for stimulating discussions.


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Citation: Q. Chang, D. Li, Ch. Sun, S. V. Zelik, “Deterministic and random attractors for a wave equation with sign changing damping”, Izv. Math., 87:1 (2023), 154–199
Citation in format AMSBIB
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\by Q.~Chang, D.~Li, Ch.~Sun, S.~V.~Zelik
\paper Deterministic and random attractors for a~wave equation with sign changing damping
\jour Izv. Math.
\yr 2023
\vol 87
\issue 1
\pages 154--199
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