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This article is cited in 4 scientific papers (total in 4 papers) On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds D. V. Tunitsky V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow
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     IntroductionSemilinear second-order parabolic equations of the form where
$$
\begin{equation}
Lq=-\sum_{l,m=1}^n\frac{\partial}{\partial x^l} \biggl(a^{l,m}(x)\, \frac{\partial q}{\partial x^m}\biggr)+ \sum_{l=1}^n b^l(x)\, \frac{\partial q}{\partial x^l}
\end{equation}
\tag{0.2}
$$
is a linear elliptic differential operator, proved widely useful in mathematical modeling of various reaction–diffusion processes. Among numerous studies on these equations, we mention, first of all, the famous papers by Kolmogorov, Petrovskii, Piskunov [1] and Fisher [2], which dealt with a homogeneous operator $L$ and a homogeneous right-hand side $f$,
$$
\begin{equation*}
L=-\sum_{l=1}^n\frac{\partial^2}{(\partial x^l)^2},\qquad f(x,q)=f(q).
\end{equation*}
\notag
$$
We also mention the paper [3], which provides a historical account and a bibliographical survey on the studies of equations (0.1), (0.2). Some applications of these equations can be found in the book [4]. Equations (0.1), (0.2) in which the operator $L$ has periodic coefficients were studied in [3]. This case, which is reduced to an equation on a manifold diffeomorphic to the $n$-dimensional torus $\mathbb{T}^n$, has great applied value. Of special importance here are analogues of equations (0.1), (0.2) on other closed manifolds, and, in particular, on those diffeomorphic the $n$-dimensional sphere $\mathbb{S}^n$. In this regard, it is worth pointing out that, according to the generalized Poincaré conjecture, the sphere $\mathbb{S}^n$ is homeomorphic to each closed manifold homotopically equivalent to $\mathbb{S}^n$. The greatest challenge in this conjecture is in the dimension $n = 3$. In this case, the proof proposed by G. Ya. Perelman is based on the study of Ricci flows on closed three-dimensional manifolds (see [5]–[7]). This very approach proved capable of justifying the Poincaré conjecture, thereby solving one of the “millennium problems”. With the help of the DeTurck trick (see [8]), the problem of finding Ricci flows can be essentially reduced to solution of the corresponding parabolic equations. This clearly demonstrates that the study of solutions of nonlinear parabolic equations on closed manifolds has great applied and mathematical value. In the present paper, we consider existence, uniqueness, and stabilization of solutions of analogues of equation (0.1) on arbitrary closed finite-dimensional manifolds. It is worth pointing out that, in many applied problems (for example, in many control problems), the right-hand sides of equations (0.1), (0.2) may involve terms which are not smooth or even not continuous. So, it is desirable to be able to select a class of admissible solutions based on which a satisfactory theory of solvability of such equations can be constructed under minimal requirements on the regularity of their coefficients. In the present paper, the weak solutions are considered as such a class. In this class, it proves possible to investigate solvability of analogues of the inhomogeneous equation (0.1) on closed manifolds under fairly light requirements on the regularity of their coefficients. In particular, some coefficients of these equations can be generalized function. § 1. Statement of the problem1.1. Function spaces of tensor fieldsBy $X$ we denote an $n$-dimensional smooth closed Riemannian manifold, that is, a connected Hausdorff compact manifold without boundary equipped with metric $g\colon TX\times TX \to \mathbb{R}$. The metrics induced by $g$ on tensor bundles $(TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l}$, $m,l=0,1,2,\dots$, of this manifold will be denoted by the same letter $g$; further, by $(TX)^{\otimes^0}\otimes (T^* X)^{\otimes^0}$ we will denote the trivial bundle $X \times \mathbb{R}$ with $g(r,t)=rt$ for $r,t \in \mathbb{R}$. The metric $g$ induces on the manifold $X$ the measure $V=V_g$ in the local coordinates by $x^1,\dots,x^n$ where $g=\det\bigl(g(\partial/\partial x^m,\partial/\partial x^l)\bigr)$, and the Levi-Civita connection with the covariant differentiation operator $\nabla=\nabla_g$ uniquely defined by this connection.Given real-value functions $u$ and $v$ on $X$, we set
$$
\begin{equation*}
\langle u,v\rangle=\int_X u(x)v(x)\,dV, \qquad \operatorname*{ess\,sup}_{x \in X} u(x)= \inf_{\substack{S\subseteq X\\ V(S)=0}}\, \sup_{x \in X\setminus S} u(x).
\end{equation*}
\notag
$$
From the metric $g$ and the measure $V$, one defines, in the usual way, the function space $L^p(X)$ and the space of tensor fields $L^p\bigl((TX)^{\otimes^m} \otimes (T^* X)^{\otimes^l}\bigr)$, where $p \geqslant 1$ and $m,l=0,1,2,\dots$ . Similarly, with the help of the covariant differentiation operator $\nabla$, one defines the Sobolev spaces $W^{k,p}(X)$ and $W^{k,p}\bigl((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l}\bigr)$, $k=0,1,2,\dots$, and the Hölder spaces $C^{k,\alpha}(X)$ and $C^{k,\alpha}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)$ for $0<\alpha \leqslant 1$ (see [9], § 10.2.4, and [10], § 1). On the tangent bundle of the Cartesian product $[0,T)\times X$, where $T \in (0,+\infty]$, consider the metric
$$
\begin{equation*}
g_{[0,T)\times X}\colon T([0,T)\times X)\ni (\tau,\xi)\mapsto \tau^2+g(\xi,\xi)\in \mathbb{R}
\end{equation*}
\notag
$$
and the corresponding measure $V_{g_{[0,T)\times X}}$ and the covariant differentiation operator $\nabla_{g_{[0,T)\times X}}$, using which, we similarly construct the function spaces
$$
\begin{equation*}
\begin{gathered} \, L^p([0,T)\times X),\qquad L^p\bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr), \\ W^{k,p}([0,T)\times X),\qquad W^{k,p}\bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr), \\ C^{k,\alpha}([0,T)\times X),\qquad C^{k,\alpha} \bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr). \end{gathered}
\end{equation*}
\notag
$$
If $B$ is a Banach space with norm ${\|\,{\cdot}\,\|_B}$, then, for $T \in (0,+\infty]$, one defines in the standard way, the Banach spaces $L^p(T;B)$ with the norms
$$
\begin{equation*}
\|q\|_{L^p(T;B)}=\biggl(\int_0^T\|q(t)\|_B^p\,dt\biggr)^{1/p}, \quad p\geqslant 1, \qquad \|q\|_{L^\infty(T;B)}= \operatorname*{ess\,sup}_{t\in [0,T)}\|q(t)\|_B,
\end{equation*}
\notag
$$
see Chap. III, § 1 in [11] and Chap. II, § 2 in [12]. The spaces $L^2(T;W^{1,2}(X))$ and $W^{1,2}(T;L^2(X))$ are Hilbert spaces equipped with the inner products
$$
\begin{equation*}
\begin{aligned} \, \langle q,p\rangle_{L^2(T;W^{1,2}(X))}&= \int_0^T\langle q(t),p(t)\rangle_{W^{1,2}(X)}\,dt, \\ \langle q,p\rangle_{W^{1,2}(T;L^2(X))}&= \int_0^T\bigl(\langle q(t),p(t)\rangle+ \langle q'(t),p'(t)\rangle\bigr)\,dt; \end{aligned}
\end{equation*}
\notag
$$
their intersection $L^2(T;W^{1,2}(X))\cap W^{1,2}(T;L^2(X))$ is isomorphic to the Hilbert space $W^{1,2}([0,T)\times X)$ with the inner product
$$
\begin{equation*}
\langle q,p\rangle_{W^{1,2}([0,T)\times X)}= \int_0^T\bigl(\langle q(t),p(t)\rangle+\langle dq(t),dp(t)\rangle_{L^2(T^* X)} +\langle q'(t),p'(t)\rangle\bigr)\,dt.
\end{equation*}
\notag
$$
Here and in what follows, $q(t,\,\cdot\,)$ is identified with $q(t)$. Setting we get a Banach space with the norm
$$
\begin{equation*}
\|q\|_{W(T;X)}^2=\operatorname*{ess\,sup}_{t\in [0,T)} \langle q(t),q(t)\rangle+\int_0^T\langle dq(t),dq(t)\rangle_{L^2(T^* X)}\,dt.
\end{equation*}
\notag
$$
As usual, for a linear semilocal subspace $E \subseteq \mathcal{D}'([0,T)\times X)$, we let $E_{\mathrm{loc}}$ denote the smallest local subspace $\mathcal{D}'([0,T)\times X)$ containing $E$ (see § 10.1 in [13]). In particular, for $L^2([0,T)\times X)$ we have $L_{\mathrm{loc}}^2([0,T)\times X)$, for $L^\infty([0,T)\times X)$ we have $L_{\mathrm{loc}}^\infty([0,T)\times X)$, and for $W(T;X)$ we have $W_{\mathrm{loc}}(T;X)$. 1.2. Elliptic equationsIn what follows, a particular property is said to hold almost everywhere if the set of points for which the property fails to hold is a set of measure zero with respect to the measure $V$ (see (1.1)). Let the Riemannian manifold $X$ be equipped, in addition to $g$, with another metric $a$. Assume that this metric is measurable, and there exist positive numbers $a_0$ and $a_1$ such that, almost everywhere,
$$
\begin{equation}
a_0 g(\eta,\eta) \leqslant a(\eta,\eta) \leqslant a_1 g(\eta,\eta)
\end{equation}
\tag{1.2}
$$
for all $\eta \in T^* X$. Consider the operators $d_a^*$ and $d_g^*$ formally adjoint with the exterior differentiation operator $d$ with respect to the metrics $a$ and $g$, respectively (see [14], Chap. VIII, § 1). In particular, $\langle a(du,v),1\rangle=\langle a(u,d_a^* v),1\rangle$ for all differential $k$-forms $u$ and $(k+1)$-forms $v$, $k=0,1,\dots,n-1$, and if the manifold $X$ is orientable, then, $d_a^*=(-1)^{n(k+1)+1}*d\,*$ on $k$-forms, where $* = *_a$ is the Hodge operator induced by the metric $a$. Given a function $u\in C^\infty(X)$, we define the linear second-order differential operator where $\Delta=\Delta_a=d_a^*\circ d$ is the geometric Laplacian (the Laplace–de Rahm operator; see Chap. IV, § 5 in [15]), and $b$, $c$ are measurable bounded (with respect to the metric $g$) vector field and linear differential form on $X$, respectively. In the local coordinates $x^1,\dots,x^n$,
$$
\begin{equation*}
\begin{aligned} \, Lu&=-\frac{1}{\sqrt{a}} \sum_{l,m=1}^n \frac{\partial}{\partial x^l} \biggl(\sqrt{a}\,a(dx^l,dx^m)\, \frac{\partial u}{\partial x^m}\biggr)+ \sum_{l=1}^n b(dx^l)\,\frac{\partial u}{\partial x^l} \\ &\qquad-\frac{1}{\sqrt{g}} \sum_{l,m=1}^n \frac{\partial}{\partial x^l} \biggl(\sqrt{g}\,g(dx^l,dx^m) uc\biggl(\frac{\partial}{\partial x^m}\biggr)\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $a=\det\bigl(a(\partial/\partial x^m,\partial/\partial x^l)\bigr)$ and $g=\det\bigl(g(\partial/\partial x^m,\partial/\partial x^l)\bigr)$; cf. (0.2). In this context, condition (1.2) means that operator (1.3) is uniformly elliptic on the manifold $X$. Consider the function
$$
\begin{equation}
f\colon X \times \mathbb{R} \ni (x,r)\mapsto f(x,r)\in \mathbb{R}
\end{equation}
\tag{1.4}
$$
which is locally Lipschitz continuous almost everywhere in $r$, that is, for any $r\in \mathbb{R}$ there exists a positive constant $\mu_0=\mu_0(r)$ such that, for $r_1,r_2 \in [-r,r]$,
$$
\begin{equation}
|f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_2)| \leqslant \mu_0(r)|r_1-r_2|
\end{equation}
\tag{1.5}
$$
almost everywhere. A weak (or generalized) solution of the equation where $h$ is a measurable bounded linear differential form on the manifold $X$, is a function $u \in W^{1,2}(X)$ such that $f(\,\cdot\, ,u) \in L^2(X$), and
$$
\begin{equation}
\mathcal{L}(u,v)=\langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}
\end{equation}
\tag{1.7}
$$
for each function $v \in C^\infty(X)$, where $\mathcal{L}$ is a continuous bilinear form
$$
\begin{equation}
\mathcal{L}\colon W^{1,2}(X) \times C^\infty(X) \ni (u,v)\mapsto \langle a(du,dv),1\rangle+\langle bu,v\rangle+ \langle uc,dv\rangle_{L^2(T^* X)} \in \mathbb{R}.
\end{equation}
\tag{1.8}
$$
Remark 1. According to (1.8), $\mathcal{L}(r,v) = r\langle c,dv\rangle_{L^2(T^* X)}$ for each $r \in \mathbb{R}$, and hence by definition (1.7) a constant function $u=r$ is a weak solution of equation (1.6) if and only if $\theta(r,v)=0$ for any function $v \in C^\infty(X)$, where Remark 2. For a fixed $r$, the functional $\theta(r,\,\cdot\,)$ is a linear form on $C^\infty(X)$. The space $C^\infty(X)$ is dense in $W^{1,2}(X)$, and hence both the linear form $\theta(r,\,\cdot\,)$ (see (1.9)) and the bilinear form $\mathcal{L}$ (see (1.8)) can be uniquely extended by continuity to functions $v \in W^{1,2}(X)$. Moreover, if either of $\theta(r,v)=0$ or (1.7) holds for any function $v \in C^\infty(X)$, then it also holds for all functions $v \in W^{1,2}(X)$. It is clear that the fulfillment of any of these equalities for all $v \in W^{1,2}(X)$ is equivalent to that for only non-negative $v \in W^{1,2}(X)$. 1.3. Parabolic equationsThe operator $L$ (1.3) is elliptic, and hence the evolutionary equation is parabolic. By a weak (or generalized) solution of equation (1.10) on the half-open interval $[0,T)$, $T \in (0,+\infty]$, with initial value we mean a function $q \in W_{\mathrm{loc}}(T;X)$ such that $f(\,\cdot\, ,q) \in L_{\mathrm{loc}}^2([0,T) \times X)$, and
$$
\begin{equation}
\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)=\langle q_0,p(0)\rangle+ \int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2 (T^* X)}\bigr)\,d\tau
\end{equation}
\tag{1.12}
$$
for each $p \in C^\infty([0,T) \times X)$ and $t \in [0,T)$, where
$$
\begin{equation}
\mathcal{L}^t \colon W(T;X) \times C^\infty([0,T)\times X) \ni (q,p) \mapsto \int_0^t\bigl(\mathcal{L}(q(\tau),p(\tau))- \langle q(\tau),p'(\tau)\rangle\bigr)\,d\tau \in \mathbb{R}.
\end{equation}
\tag{1.13}
$$
Remark 3. By definitions (1.7) and (1.12), if the initial value $q_0$ (see (1.11)) is a solution of equation (1.6), then $q=q_0$ is a solution of the Cauchy problem (1.10), (1.11) on the infinite interval $[0,+\infty)$. Hence, in the case of a constant initial value $q_0=r \in \mathbb{R}$, according to Remark 1, the constant function $q=r$ is a weak solution of the Cauchy problem (1.10), (1.11) if and only if $\theta(r,v)=0$ for any function $v \in C^\infty(X)$. Remark 4. The space $C^\infty([0,T) \times X)$ is dense in $W^{1,2}([0,T) \times X)$, and hence, as in Remark 2, the bilinear form $\mathcal{L}^t$ (1.13) can be uniquely extended by continuity to functions $p \in W^{1,2}([0,T) \times X)$ with traces $p(0),p(t) \in L^2(X)$. In addition, the fulfillment of (1.12) for $p \in W^{1,2}([0,T) \times X)$ is equivalent to that for only non-negative $p \in W^{1,2}([0,T) \times X)$. § 2. Main results2.1. Existence, uniqueness, and regularityLet us study conditions under which a weak solution of the Cauchy problem (1.10), (1.11) exists and is unique. A function $u \in W^{1,2}(X)$ is a weak (or generalized) subsolution (supersolution) of equation (1.6) if $f(\,\cdot\, ,u) \in L^2(X)$ and if
$$
\begin{equation*}
\mathcal{L}(u,v) \leqslant \langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\qquad \bigl(\mathcal{L}(u,v)\geqslant \langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\bigr)
\end{equation*}
\notag
$$
for any non-negative function $v \in C^\infty(X)$. A function $q \in W_{\mathrm{loc}}(T;X)$ is a weak (or generalized) subsolution (supersolution) of the Cauchy problem (1.10), (1.11) on the half-open interval $[0,T)$, $T \in (0,+\infty]$, if $f(\,\cdot\, ,q)\in L_{\mathrm{loc}}^2([0,T) \times X)$ and, if, for any non-negative function $p \in C^\infty([0,T) \times X)$,
$$
\begin{equation}
\begin{aligned} \, &\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p) \\ &\quad\leqslant \langle q_0,p(0)\rangle \,{+}\int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle \,{+}\,\langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \\ &\biggl(\langle q(t),p(t)\rangle+ \mathcal{L}^t(q,p) \\ &\quad\geqslant\langle q_0,p(0)\rangle \,{+}\int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle \,{+}\,\langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau\biggr),\qquad t \in [0,T). \end{aligned}
\end{equation}
\tag{2.1}
$$
A weak subsolution (supersolution) of problem (1.10), (1.11) which is not a weak solution will be called strong. In what follows, unless otherwise explicitly stipulated, we will assume that all solutions, subsolutions, and supersolutions are weak, and the adjective “weak” will be dropped for brevity. Remark 5. It is clear that each solution is simultaneously a subsolution and a supersolution. Conversely, if a function is simultaneously a subsolution and a supersolution, then by the concluding lines of Remarks 2 and 4, this function is a solution. It is easily checked that if $w \in W^{1,2}(X)$ is a subsolution (supersolution) of equation (1.6) and $w \leqslant q_0$ ($w \geqslant q_0$) almost everywhere, then $q=w$ is a subsolution (supersolution) of problem (1.10), (1.11) on the infinite interval $[0,+\infty)$. Next, in analogy with Remark 1, the constant function $u=r$, $r \in \mathbb{R}$, is a subsolution (supersolution) of equation (1.6) if and only if $\theta(r,v) \leqslant 0$ ($\theta(r,v) \geqslant 0$) for each non-negative function $v \in C^\infty(X)$. Correspondingly, in analogy with Remark 3, if $r \leqslant q_0$ ($r \geqslant q_0$) almost everywhere and $\theta(r,v) \leqslant 0$ ($\theta(r,v) \geqslant 0$) for each non-negative function $v \in C^\infty(X)$, then $q=r$ is a subsolution (supersolution) of problem (1.10), (1.11). The following result holds. Theorem 1 (existence and uniqueness of a solution). Assume that a metric $a\in L^\infty\bigl((T^* X)^{\otimes^2}\bigr)$ satisfies estimate (1.2), a vector field $b$ lies in $L^\infty(TX)$, differential forms $c,h $ lie in $L^\infty(T^* X)$, there exists a number $\mu \in \mathbb{R}$ such that For a proof, see § 3. It is known that solutions of the Cauchy problem (1.10), (1.11) are locally Hölder-continuous. More precisely, the following result holds. Theorem 2 (regularity of the solution). Let the coefficients and right-hand side of (1.10) satisfy all the conditions of Theorem 1 except (2.2). Let $q$ be a solution of the Cauchy problem (1.10), (1.11) from $L^\infty([0,T) \times X)$, $T \in (0,+\infty)$. Then $q \in C(T;L^2(X))$ and, for each $(t,x) \in (0,T] \times X$, there exist its neighbourhood $U$ and a number $0<\alpha<1$ such that $q_{|U\cap [0,T) \times X}\in C^{0,\alpha}(U \cap [0,T)\times X)$. Under the above constraints on the coefficients of the differential operator $L$ (see (1.3)) and the function $f$ (see (1.4)), the result of Theorem 2 follows from the known properties of solutions of linear parabolic equations (see Chap. VI, § 7 in [16] and § 1.5 in [17]). It is natural that a further increase of regularity of the coefficients of equation (1.10) increases correspondingly the regularity of its solutions (see Chap. VI, § 2 in [16]). The following result is clear from Theorems 1 and 2. Theorem 3 (global existence and uniqueness). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1. If $q_1$ and $q_2$ are, respectively, a subsolution and a supersolution of problem (1.10), (1.11) on $[0,T)$, $T \in (0,+\infty]$, and $q_1,q_2 \in L_{\mathrm{loc}}^\infty ([0,T) \times X)$, then on $[0,T)$ problem (1.10), (1.11) has a unique solution $q$, which is locally Hölder-continuous on $(0,T) \times X$, $q \in C(T;L^2(X))$, and $q_1(t) \leqslant q(t) \leqslant q_2(t)$ almost everywhere for $t \in [0,T)$. 2.2. Stabilization of solutionsLet us find conditions under which the solution $q$ of the Cauchy problem (1.10), (1.11) tends as $t\to +\infty$ to the solution $u$ of the stationary equation (1.6). A function $f$ (see (1.4)) is called strictly right (left) concave at a point $r_0 \in \mathbb{R}$ if
$$
\begin{equation}
(1-\alpha)f(\,\cdot\, ,r_0)+\alpha f(\,\cdot\, ,r)< f\bigl(\,\cdot\, ,(1-\alpha) r_0+\alpha r\bigr)
\end{equation}
\tag{2.3}
$$
almost everywhere for $r>r_0$ ($r<r_0$) and $0<\alpha<1$. Similarly, a function $f$ is called strictly right (left) convex at a point $r_0 \in \mathbb{R}$ if
$$
\begin{equation}
f\bigl(\,\cdot\, ,(1-\alpha)r_0+\alpha r\bigr)<(1-\alpha)f(\,\cdot\, ,r_0)+ \alpha f(\,\cdot\, ,r)
\end{equation}
\tag{2.4}
$$
almost everywhere for $r>r_0$ ($r<r_0$) and $0<\alpha<1$. The solutions of problem (1.10), (1.11) have the following asymptotic properties. Theorem 4 (stabilization to a stationary solution). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let $r_0,N \in \mathbb{R}$, $r_0 \ne N$, and let functions $q_0 \in L^2(X)$ and $w \in W^{1,2}(X)\cap L^\infty(X)$ be such that $w \geqslant 0$ almost everywhere, $V(x \in X\mid w \ne 0)>0$ and $V(x \in X \mid q_0 \ne r_0)>0$. Then the following hold. (a) Let $f$ be strictly right concave at $r_0$, let $r_0$ and $N$ be, respectively, a subsolution and supersolution of equation (1.6), let the initial value (1.11) satisfy $r_0 \leqslant q_0 \leqslant N$ almost everywhere, and let there exist $\delta$ such that $r_0+\alpha w$ is a subsolution of equation (1.6) for $0<\alpha<\delta$. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions $u$ and $q$, which are continuous, $r_0< u, q(t) \leqslant N$ for $0<t<+\infty$, and satisfy the limit relation (b) Let $f$ be strictly left convex at $r_0$, let $N$ and $r_0$ be, respectively, a subsolution and supersolution of equation (1.6), let the initial value (1.11) satisfy $N \leqslant q_0 \leqslant r_0$ almost everywhere, and let there exist $\delta$ such that $r_0-\alpha w$ is a supersolution of equation (1.6) for $0<\alpha<\delta$. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions $u$ and $q$, which are continuous, $N \leqslant u,q(t)<r_0$ for $0<t<+\infty$, and satisfy (2.5). For a proof, see § 4. Remark 6. The idea of the proof is to construct a non-increasing (in $t$) function $\widetilde{q} \in C([0,+\infty)\times X)$ such that Theorem 5 (stabilization to a constant solution). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let $r_0,A \in \mathbb{R}$, $A>0$, let $q_0 \in L^2(X)$, $w \in W^{1,2}(X) \cap L^\infty(X)$, and let $w \geqslant 0$ almost everywhere Then the following hold. (a) Let the initial value (1.11) satisfy $r_0 \leqslant q_0 \leqslant r_0+Aw$ almost everywhere, let $r_0$ be a solution and $r_0+\alpha w$ be a strong supersolution of equation (1.6) for $0<\alpha \leqslant A$. Then the Cauchy problem (1.10), (1.11) has a unique solution $q$, which is continuous, $r_0 \leqslant q \leqslant r_0+Aw$ almost everywhere for $0<t<+\infty$, and a unique solution $u$ of equation (1.6) satisfying $r_0 \leqslant u \leqslant r_0+Aw$ almost everywhere is $u=r_0$, and the limit relation holds (b) Let the initial value (1.11) satisfy $r_0-Aw \leqslant q_0 \leqslant r_0$ almost everywhere, let $r_0$ be a solution and $r_0-\alpha w$ be a strong subsolution of equation (1.6) for $0<\alpha \leqslant A$. Then the Cauchy problem (1.10), (1.11) has a unique solution $q$, which is continuous, $r_0-Aw \leqslant q \leqslant r_0$ almost everywhere for $0<t<+\infty$, and a unique solution $u$ of equation (1.6) satisfying $r_0-Aw \leqslant u \leqslant r_0$ almost everywhere is $u=r_0$, and the limit relation (2.7) holds. For a proof, see § 4; moreover, Remark 6 with $u=r_0$ applies as in Theorem 4. Remark 7. Under the conditions of Theorems 4 and 5, using standard a priori estimates of solutions of linear second-order parabolic equations (see Chap. VI, § 1 in [16] and § 1.5 in [17]) and applying Theorems 4 and 5, it can be shown that the solutions of problem (1.10), (1.11) are asymptotically stable. 2.3. The eigenvalue problemAssume that the function $f$ (see (1.4)) has the right (left) derivative
$$
\begin{equation*}
\begin{gathered} \, \frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)=\lim_{r \to r_0+0} \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0} \\ \biggl(\frac{\partial f}{\partial r}(\,\cdot\, ,r_0-0)= \lim_{r\to r_0-0}\frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}\biggr), \end{gathered}
\end{equation*}
\notag
$$
almost everywhere at a point $r_0 \in \mathbb{R}$. For the differential operator $L$ (see (1.3)), consider the eigenvalue problem with the potential
$$
\begin{equation}
F(x)=\frac{\partial f}{\partial r}(x,r_0+0)\qquad \biggl(F(x)=\frac{\partial f}{\partial r}(x,r_0-0)\biggr).
\end{equation}
\tag{2.9}
$$
We will say that $w \in W^{1,2}(X)$ is an eigenfunction belonging to an eigenvalue $\lambda \in \mathbb{R}$ if it satisfies equality (2.8) in the weak sense, that is, for all functions $v \in C^\infty(X)$. The following result is known. Theorem 6. Let the coefficients of the operator $L$ (see (1.3)) satisfy the conditions of Theorem 1. If $F \in L^\infty(X)$, then problem (2.8)–(2.10) has a unique simple eigenvalue $\lambda_1$ to which there belongs a real eigenfunction $w \in C^{0,\alpha}(X)$, $0<\alpha<1$, which does not vanish on $X$. The required result follows from the Krein–Rutman theorem (see § 6.5.2 in [18], § 6.4 in [19], Chap. 11, § C in [20], and Chap. I, § 7 in [21]) and from the regularity of solutions of elliptic equations (see § 8.9 in [22]). The eigenvalue $\lambda_1$ is known as the principal eigenvalue. From definition (2.3) it easily follows that the function $f$ (see (1.4)) is strictly right (left) concave at a point $r_0 \in \mathbb{R}$ if and only if
$$
\begin{equation}
\frac{f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_0)}{r_1-r_0}< \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}
\end{equation}
\tag{2.11}
$$
almost everywhere for $r_0<r<r_1$ ($r_1<r<r_0$). Similarly, from definition (2.4) it easily follows that a function $f$ is strictly right (left) convex at a point $r_0 \in \mathbb{R}$ if and only if
$$
\begin{equation}
\frac{f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_0)}{r_1-r_0}> \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}
\end{equation}
\tag{2.12}
$$
almost everywhere for $r_0<r<r_1$ ($r_1<r<r_0$). The monotonicity properties of the difference relations (2.11) and (2.12) imply that the derivatives in the right-hand sides of (2.9) exist and lie in $L^\infty(X)$. As a corollary, problem (2.8)–(2.10) is well posed. The following results hold. Theorem 7. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, and let $\lambda_1<0$ be the principal eigenvalue. Then the following hold. (a) Let $f$ be strictly right concave at $r_0 \in \mathbb{R}$, let $r_0$ and $N$, $r_0<N$, be, respectively, a subsolution and supersolution of equation (1.6), and let the initial value (1.11) satisfy $r_0\leqslant q_0\leqslant N$ almost everywhere and $V(x\in X\mid q_0 \ne r_0)>0$. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions $u$ and $q$, which are continuous, $r_0<u,q(t)\leqslant N$ for $0<t<+\infty$, and satisfy the limit relation (2.5). (b) Let $f$ be strictly left convex at $r_0 \in \mathbb{R}$, let $r_0$ and $N$, $N<r_0$, be, respectively, a supersolution and subsolution of equation (1.6), and let the initial value (1.11) satisfy $N\leqslant q_0\leqslant r_0$ almost everywhere and $V(x\in X\mid q_0 \ne r_0)>0$. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions $u$ and $q$, which are continuous, $N \leqslant u,q(t)<r_0$ for $0<t\leqslant +\infty$, and satisfy the limit relation (2.5). For a proof, see § 5; moreover, Remark 6 applies as in Theorem 4. Theorem 8. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let a number $r_0 \in \mathbb{R}$ be a solution of equation (1.6), and let $q_0 \in L^\infty(X)$ and $\lambda_1 \geqslant0$. Then the following hold. (a) Let $f$ be strictly right concave at $r_0$ and the initial value (1.11) satisfy the inequality $q_0 \geqslant r_0$ almost everywhere. Then the Cauchy problem (1.10), (1.11) has a unique solution $q$, which is continuous, $r_0 \leqslant q(t)$ for $0<t\leqslant +\infty$, and equation (1.6) has a unique solution $u$ such that $r_0 \leqslant u$ almost everywhere, $u=r_0$, and which satisfies (2.7). (b) Let $f$ be strictly left concave at $r_0$ and let the initial value (1.11) satisfy $q_0 \leqslant r_0$ almost everywhere. Then the Cauchy problem (1.10), (1.11) has a unique solution $q$, which is continuous, satisfies $q(t)\leqslant r_0$ for $0<t\leqslant +\infty$, and equation (1.6) has a unique solution $u$ such that $u\leqslant r_0$ almost everywhere, $u=r_0$, and which satisfies (2.7). For a proof, see § 5. In addition, Remark 6 with $u=r_0$ applies as in Theorem 4. Moreover, similarly to Remark 7, under the conditions of Theorems 7 and 8, the solutions of problem (1.10), (1.11) are asymptotically stable. § 3. Existence and uniqueness of solutions3.1. Auxiliary resultsFor a proof of Theorem 1, we will require two technical results. The first one is the following comparison test. Lemma 1. Let the coefficients of the differential operator $L$ (see (1.3)) and the function $f$ (see (1.4)) satisfy the conditions of Theorem 1. If $q_1,q_2\in W(T;X)\cap L^\infty([0,T) \times X)$, $T\in (0,+\infty]$, and non-negative function $p\in C^\infty([0,T) \times X)$ satisfy Proof. By the Lipschitz condition (1.5), if
$$
\begin{equation*}
\mu_0=\mu_0(\max\{\|q_1\|_{L^\infty([0,T) \times X)}, \|q_2\|_{L^\infty([0,T) \times X)}\})
\end{equation*}
\notag
$$
then
$$
\begin{equation*}
f(\,\cdot\, ,q_1(\tau))-f(\,\cdot\, ,q_2(\tau))\leqslant \mu_0\bigl(q_1(\tau)-q_2(\tau)\bigr) \operatorname{sgn}\bigl(q_1(\tau)-q_2(\tau)\bigr)
\end{equation*}
\notag
$$
almost everywhere. Hence, by the assumption, for $q=q_1-q_2$ and for a non-negative $p\in C^\infty([0,T) \times X)$, we have
$$
\begin{equation*}
\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)-\langle q(0),p(0)\rangle- \int_0^t\langle \mu_0 q(\tau)\operatorname{sgn}q(\tau), p(\tau)\rangle\,d\tau \leqslant 0.
\end{equation*}
\notag
$$
By definition of $\mathcal{L}$ (see (1.8)), $\mathcal{L}(e^{\mu t}q,p)=\mathcal{L}(q,e^{\mu t}p)$. Hence, by definition of $\mathcal{L}^t$ (see (1.13)) and substituting $q=e^{\mu t}\widetilde{q}$ and $\widetilde{p}=e^{\mu t}p$, the last inequality assumes the form
$$
\begin{equation*}
\langle \widetilde{q}(t),\widetilde{p}(t)\rangle+ \mathcal{L}^t(\widetilde{q},\widetilde{p})- \langle q(0),\widetilde{p}(0)\rangle+\int_0^t \langle(\mu-\mu_0\operatorname{sgn}q(\tau)) \widetilde{q}(\tau),\widetilde{p}(\tau)\rangle\,d\tau \leqslant0
\end{equation*}
\notag
$$
with non-negative $\widetilde{p}\in C^\infty([0,T) \times X)$. For sufficiently large $\mu>0$, the conclusion of the lemma follows from inequality (2.2) and the strong parabolic maximum principle (see Chap. VI, § 7 in [16] and § 1.5 in [17]). Lemma is proved. Note that that conclusion of Lemma 1 is true, in particular, if $q_1(0)=q_2(0)$ almost everywhere, and also if $f \equiv 0$. The second technical result we require is the following a priori estimate. Lemma 2. Let the coefficients of the differential operator $L$ (see (1.3)) satisfy the conditions of Theorem 1. If $q_0 \in L^2(X)$, $q,v \in W(T;X)$, $T\in (0,+\infty]$, and $p\in C^\infty([0,T) \times X)$ satisfy then there exists a constant $C>0$, independent of $q_0$ and $v$, such that A proof of Lemma 2 follows from the known a priori estimates for solutions of linear second-order parabolic equations (see Chap. VI, § 1 in [16] and § 1.5 in [17]). Note that, in particular, Lemma 2 holds with $q_0 \equiv 0$. 3.2. Proof of Theorem 1By the assumption of the theorem and definition (2.1),
$$
\begin{equation*}
\begin{aligned} \, &\langle q_1(t),p(t)\rangle+\mathcal{L}^t(q_1,p)-\langle q_0,p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_1(\tau)),p(\tau)\rangle\,d\tau \\ &\qquad\leqslant \langle q_2(t),p(t)\rangle+ \mathcal{L}^t(q_2,p)-\langle q_0,p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_2(\tau)),p(\tau)\rangle\,d\tau \end{aligned}
\end{equation*}
\notag
$$
for non-negative $p \in C^\infty([0,T) \times X)$ and $t \in [0,T)$. By Lemma 1, we have $q_1(t) \leqslant q_2(t)$ almost everywhere for $t\in[0,T)$. The function $f$ (see (1.4)) satisfies condition (1.5) almost everywhere. Hence, if
$$
\begin{equation*}
\mu \geqslant \mu_0(\max\{\|q_1\|_{L^\infty([0,T) \times X)}, \|q_2\|_{L^\infty([0,T) \times X)}\})
\end{equation*}
\notag
$$
and if functions $v_1$ and $v_2$ are such that $q_1\leqslant v_1 \leqslant v_2 \leqslant q_2$ almost everywhere, then
$$
\begin{equation*}
f(\,\cdot\, ,v_1)+\mu v_1 \leqslant f(\,\cdot\, ,v_2)+\mu v_2
\end{equation*}
\notag
$$
almost everywhere (cf. the appendix to Chap. IV, § 2 in [23] and [24]), and, therefore,
$$
\begin{equation}
\langle f(\,\cdot\, ,v_1)+\mu v_1,v\rangle \leqslant \langle f(\,\cdot\, ,v_2)+\mu v_2,v\rangle
\end{equation}
\tag{3.1}
$$
for any non-negative function $v \in C^\infty(X)$. It is clear that (1.10) is equivalent to the equation
$$
\begin{equation*}
\frac{\partial q}{\partial t}+Lq+\mu q=f(\,\cdot\, ,q)+d_g^* h+\mu q.
\end{equation*}
\notag
$$
Let us construct successive approximations $\{q_{1,l}\}$ and $\{q_{2,l}\}$ by induction. We set $q_{1,0}=q_1$, $q_{2,0}=q_2$, and as $q_{j,l}$ for $j=1,2$ and $l=1,2,\dots$ we take the solution of the Cauchy problem
$$
\begin{equation*}
\frac{\partial q_{j,l}}{\partial t}+Lq_{j,l}+\mu q_{j,l}= f(\,\cdot\, ,q_{j,l-1})+d_g^* h+\mu q_{j,l-1},\qquad q_{j,l}(0)=q_0,
\end{equation*}
\notag
$$
$l=1,2,\dots$, that is, by definition (1.12), for any function $p\in C^\infty([0,T) \times X)$ and $t\in [0,T)$, we have
$$
\begin{equation}
\begin{aligned} \, &\langle q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l},p)+ \mu\int_0^t\langle q_{j,l}(\tau),p(\tau)\rangle\,d\tau \nonumber \\ &\ =\langle q_0,p(0)\rangle+\int_0^t \bigl(\langle f(\,\cdot\, ,q_{j,l-1}(\tau))+\mu q_{j,l-1}(\tau),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau. \end{aligned}
\end{equation}
\tag{3.2}
$$
The linear equation (3.2) has a unique solution $q_{j,l}\in W(T;X)$, see Chap. VI, § 1 in [16] and § 1.5 in [17]. Hence, by monotonicity (3.1) and by construction (3.2),
$$
\begin{equation*}
\begin{aligned} \, &\langle q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l},p)+ \mu\int_0^t\langle q_{j,l}(\tau),p(\tau)\rangle\,d\tau \\ &\qquad\leqslant \langle q_{j,l-1}(t),p(t)\rangle+ \mathcal{L}^t(q_{j,l-1},p)+ \mu\int_0^t\langle q_{j,l-1}(\tau),p(\tau)\rangle\,d\tau. \end{aligned}
\end{equation*}
\notag
$$
Hence by Lemma 1, for $t\in[0,T)$,
$$
\begin{equation}
q_1(t)\leqslant q_{1,1}(t) \leqslant q_{1,2}(t)\leqslant\dots\leqslant q_{2,2}(t) \leqslant q_{2,1}(t) \leqslant q_2(t)
\end{equation}
\tag{3.3}
$$
almost everywhere. By monotonicity and since $\{q_{1,l}\}$ and $\{q_{2,l}\}$ are bounded, the limits
$$
\begin{equation}
q_{1,\infty}=\lim_{l \to +\infty}q_{1,l},\qquad q_{2,\infty}=\lim_{l\to +\infty}q_{2,l}
\end{equation}
\tag{3.4}
$$
exist almost everywhere and satisfy
$$
\begin{equation}
q_1(t) \leqslant q_{1,1}(t) \leqslant q_{1,2}(t)\leqslant\dots\leqslant q_{1,\infty}(t) \leqslant q_{2,\infty}(t)\leqslant\dots\leqslant q_{2,2}(t) \leqslant q_{2,1}(t) \leqslant q_2(t)
\end{equation}
\tag{3.5}
$$
almost everywhere. Hence by the Lebesgue dominated convergence theorem, the limit relations (3.4) also hold in the norm $\|\,{\cdot}\,\|_{L^2([0,t)\times X)}$ for $t\in [0,T)$. Next, by construction (3.2)
$$
\begin{equation*}
\begin{aligned} \, &\langle q_{j,l+m}(t)-q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l+m}- q_{j,l},p)+\mu\int_0^t\langle q_{j,l+m}(\tau)-q_{j,l}(\tau),p(\tau)\rangle\,d\tau \\ &\quad =\int_0^t\langle f(\,\cdot\, ,q_{j,l-1+m}(\tau))- f(\,\cdot\, ,q_{j,l-1}(\tau))+\mu(q_{j,l-1+m}(\tau) -q_{j,l-1}(\tau)),p(\tau)\rangle\,d\tau, \end{aligned}
\end{equation*}
\notag
$$
and hence by Lemma 2 there exists $C>0$ such that, for $j=1,2$ and $l,m=1,2,\dots$,
$$
\begin{equation*}
\begin{aligned} \, \|q_{j,l+m}-q_{j,l}\|_{W(t;X)} &\leqslant Ce^{Ct}\|f(\,\cdot\, ,q_{j,l-1+m})- f(\,\cdot\, ,q_{j,l-1})\|_{L^2([0,t)\times X)} \\ &\qquad+\mu Ce^{Ct}(\|q_{j,l-1+m}-q_{j,l-1}-q_{j,l+m}+ q_{j,l}\|_{L^2([0,t) \times X)}). \end{aligned}
\end{equation*}
\notag
$$
By the already established $\|\,{\cdot}\,\|_{L^2([0,t)\times X)}$-convergence of the approximations $\{q_{1,l}\}$ and $\{q_{2,l}\}$ and also from (3.3) and (1.5), it follows that $\{q_{1,l}\}$ and $\{q_{2,l}\}$ are Cauchy sequences also in the norm $\|\,{\cdot}\,\|_{W(t;X)}$ (see (1.2)). Since $W(t;X)$ is complete, these sequences converge to the functions $q_{1,\infty}$ and $q_{2,\infty}$ (see (3.4)). Therefore, by passing to a limit in (3.2), we get
$$
\begin{equation*}
\begin{aligned} \, &\langle q_{j,\infty}(t),p(t)\rangle+L^t(q_{j,\infty},p)= \langle q_0,p(0)\rangle \\ &\qquad\qquad+\int_0^t\bigl(\langle f(\,\cdot\, ,q_{j,\infty}(\tau)),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \end{aligned}
\end{equation*}
\notag
$$
for $j=0,1$, $p\in C^\infty([0,T) \times X)$ and $t\in[0,T)$, that is, $q_{j,\infty}$ are solutions of problem (1.10), (1.11), and, by (3.5), $q_1(t) \leqslant q_{1,\infty}(t) \leqslant q_{2,\infty}(t) \leqslant q_2(t)$ almost everywhere. By Lemma 1, any solution $q$ of problem (1.10), (1.11) satisfies $q_{1,\infty}=q=q_{2,\infty}$ almost everywhere on the half-open interval $[0,T)$. Theorem 1 is proved. § 4. Stabilization of solutions4.1. Auxiliary resultIn the proof of Theorems 4 and 5 we will require the following auxiliary result. Lemma 3. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, and let $w_1$ and $w_2$ be, respectively, a subsolution and supersolution of equation (1.6) such that $w_1,w_2 \in L^\infty(X)$ and $w_1 \leqslant w_2$ almost everywhere. Then there exist unique solutions $q_1,q_2 $ of the problems Proof. By the conditions of the lemma and in view of Remark 5, the functions $w_1$ and $w_2$ are, respectively, a subsolution and a supersolution of problem (1.10), (1.11). Therefore, by Theorem 3, on $[0,+\infty)$, there exist unique solutions $q_1$ and $q_2$ of the Cauchy problems (4.1) and (4.2). Moreover, $q_1,q_2 \in C(+\infty;L^2(X))$, $q_1,q_2$ are locally Hölder-continuous on $(0,+\infty)\times X$, and $w_1 \leqslant q_1(t) \leqslant q_2(t) \leqslant w_2$ almost everywhere for $t\in[0,+\infty)$.
By Lemma 1, $w_1 \leqslant q_1(t') \leqslant q_1(t'')$, and $q_2(t'') \leqslant q_2(t') \leqslant w_2$ almost everywhere for $0 \leqslant t' \leqslant t''$. Since $q_1$, $q_2$ are pointwise monotone, the limit functions in (4.3) and (4.4) exist on the manifold $X$. By construction, $q_1$, $q_{1,\infty}$, $q_2$ and $q_{2,\infty}$ obey (4.5). In view of (4.5) and by the Lebesgue dominated convergence theorem, equalities (4.3) and (4.4) also hold in the norm $\|\,{\cdot}\,\|_{L^2(X)}$. The coefficients and right-hand side of equation (1.10) do not depend explicitly on $t$, and hence, by definitions (1.12) and (1.13), we have
$$
\begin{equation}
\begin{aligned} \, &\langle q_j,p\rangle(t_0+t)-\langle q_j,p\rangle(t_0)+ \int_0^t\bigl(L(q_j,p)(t_0+\tau)- \langle q_j,p'\rangle(t_0+\tau)\bigr)\,d\tau \nonumber \\ &\qquad=\int_0^t\bigl(\langle f(\,\cdot\, ,q_j),p\rangle(t_0+\tau)+ \langle h,dp(t_0+\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \end{aligned}
\end{equation}
\tag{4.6}
$$
for $t_0 \geqslant 0$ and $p\in C^\infty([0,+\infty)\times X)$, and hence, for $0 \leqslant t' \leqslant t''$,
$$
\begin{equation*}
\begin{aligned} \, &\langle q_j(t''+t)-q_j(t'+t),p(t)\rangle \\ &\quad\qquad+ \int_0^t\bigl(\mathcal{L}(q_j(t''+\tau)-q_j(t'+\tau),p(\tau)) -\langle q_j(t''+\tau)-q_j(t'+\tau),p'(\tau)\rangle\bigr)\,d\tau \\ &\quad=\langle q_j(t'')-q_j(t'),p(0)\rangle+ \int_0^t\langle f(\,\cdot\, ,q_j)(t''+\tau) -f(\,\cdot\, ,q_j)(t'+\tau),p(\tau)\rangle\,d\tau. \end{aligned}
\end{equation*}
\notag
$$
Therefore, by Lemma 2, there exists a constant $C>0$ such that, for $t \in [0,+\infty)$,
$$
\begin{equation*}
\begin{aligned} \, &\|q_j(t''+\,\cdot\,)-q_j(t'+\,\cdot\,)\|_{W(t;X)} \\ &\ \ \leqslant Ce^{Ct}\bigl(\|f(\,\cdot\, ,q_j)(t''+\,\cdot\,) -f(\,\cdot\, ,q_j)(t'+\,\cdot\,)\|_{L^2([0,t)\times X)}+ \|q_j(t'')-q_j(t')\|_{L^2(X)}\bigr). \end{aligned}
\end{equation*}
\notag
$$
Hence, by the already established $\|\,{\cdot}\,\|_{L^2(X)}$-convergence (see (4.3) and (4.4)) and also by (4.6) and (1.5), we conclude that, as $t_0 \to +\infty$, the functions $q_1(t_0+\,\cdot\,)$ and $q_2(t_0+\,\cdot\,)$ satisfy the Cauchy condition in the norm $\|\,{\cdot}\,\|_{W(t;X)}$. By completeness of $W(t;X)$, they converge in this space to the functions $q_{1,\infty}$ (see (4.3)) and $q_{2,\infty}$ (see (4.4)), respectively. Choosing $p \in W^{1,2}(X)$ in (4.5) and making $t_0\to +\infty$, this gives
$$
\begin{equation*}
\int_0^t \mathcal{L}(q_{j,\infty},p)\,d\tau= \int_0^t\bigl(\langle f(\,\cdot\, ,q_{j,\infty}),p\rangle+ \langle h,dp\rangle_{L^2(T^* X)}\bigr)\,d\tau
\end{equation*}
\notag
$$
for $j=0,1$ and $t\in [0,T)$. The integrands in this equality are independent of $\tau$, and hence $q_{1,\infty}$ and $q_{2,\infty}$ are solutions of equation (1.6). By well-known regularity properties (see § 8.9 in [22] and § 2.2 in [10]), these solutions are continuous, and so the convergence in (4.4) also holds in the norm $\|\,{\cdot}\,\|_{C(X)}$ by Dini’s test. This proves Lemma 3. 4.2. Proof of Theorem 4(a) By conditions of the theorem and in view of Remark 5, the numbers $r_0$ and $N$ are, respectively, a subsolution and a supersolution of problem (1.10), (1.11). Therefore, by Theorem 3, on the infinite interval $[0,+\infty)$ there exists a unique solution $q$ of problem (1.10), (1.11); moreover, $q \in C(+\infty;L^2(X))$, is locally Hölder-continuous on $(0,+\infty)\times X$, and $r_0 \leqslant q(t) \leqslant N$ for $t\in[0,+\infty)$. By definitions (1.12) and (2.1),
$$
\begin{equation*}
\begin{aligned} \, &\langle r_0,p(t)\rangle+\mathcal{L}^t(r_0,p)-\langle r_0,p(0)\rangle- \int_0^t \langle f(\,\cdot\, ,r_0),p(\tau)\rangle\,d\tau \\ &\qquad \leqslant\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)- \langle q(0),p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle\,d\tau \end{aligned}
\end{equation*}
\notag
$$
for all non-negative $p \in C^\infty([0,+\infty)\times X)$, which implies $q(t)>r_0$ for $t \in (0,+\infty)$ by Lemma 1. In particular, $q(1)>r_0$, and hence, by the conditions of the theorem, there exists $\delta_0>0$ such that $r_0+\alpha w$ is a subsolution of equation (1.6) satisfying $r_0+\alpha w \leqslant q(1)$ almost everywhere for $0<\alpha<\delta_0$. By Lemma 3, for $w_1=r_0+\alpha w$ and $w_2=N$, there exist unique solutions $q_1$ and $q_2$ of the Cauchy problems (4.1) and (4.2) on the infinite interval $[0,+\infty)$; moreover, $q_1,q_2\in C(+\infty;L^2(X))$ and $q_1,q_2$ are locally Hölder-continuous on $(0,+\infty)\times X$. The $\|\,{\cdot}\,\|_{C(X)}$-limits $q_{1,\infty}$ (see (4.3)) and $q_{2,\infty}$ (see (4.4)) exist, satisfy equation (1.6), and obey (4.5) almost everywhere. From (4.5) it follows that $q_{1,\infty},q_{2,\infty}>r_0$ by the strong maximum principle for elliptic equations (see § 8.7 in [22]), and, therefore, by the uniqueness theorem, $q_{1,\infty}=q_{2,\infty}$ (see § 2.3 in [10]). Moreover, for $t \in [0,+\infty)$ and $0<\alpha<\delta_0$, by Lemma 1, we have $q_1(t) \leqslant q(t+1)$ and $q(t) \leqslant q_2(t)$, and hence, for $u=q_{1,\infty}=q_{1,\infty}$ and $\widetilde{q}(t)=\max\{u-q_1(t),q_2(t)-u\}$, we have (2.6), and, as a corollary, the limit relation (2.5). Assertion (b) is verified similarly. Theorem 4 is proved. 4.3. Proof of Theorem 5(a) By conditions of the theorem and in view of Remark 5, $r_0$ is a subsolution, and $r_0+Aw$ is a supersolution of problem (1.10), (1.11). Hence an appeal to Theorem 3 and Lemma 3 with $w_1=r_0$ and $w_2=r_0+Aw$ shows that there exist unique solutions $q$ and $q_2$ of problems (1.10), (1.11) and (4.2) on the infinite interval $[0,+\infty)$, where $q,q_2\in C(+\infty;L^2(X))$ and $q,q_2$ are locally Hölder-continuous on $(0,+\infty)\times X$. Next, by Lemma 3, the $\|\,{\cdot}\,\|_{C(X)}$-limit $q_{2,\infty}$ exists (see (4.4)) and is a solution of equation (1.6). Furthermore, by (4.5) $r_0 \leqslant q_{2,\infty} \leqslant r_0+Aw$ almost everywhere. Hence $q_{2,\infty}=r_0$ by the well-known uniqueness theorem (see § 2.3 in [10]). Next, by Lemma 1, $r_0 \leqslant q(t) \leqslant q_2(t)$ almost everywhere for $t\in [0,+\infty)$, and hence $\widetilde{q}(t)=q_2(t)-r_0$ obeys (2.6), and, therefore, and the limit relation (2.7). Assertion (b) is proved similarly. Theorem 5 is proved. § 5. The eigenvalue problem5.1. Proof of Theorem 7(a) By Theorem 6, an eigenfunction $w \in C(X)$ belonging to the eigenvalue $\lambda_1$ can be chosen so as to have $\inf_{x \in X}w(x)\,{>}\,0$. By the assumption $\lambda_1<0$, and hence, by definition of the right derivative there exists $\delta>0$ such that, for all $0<\alpha<\delta$,
$$
\begin{equation*}
\frac{\lambda_1}{2}\alpha w<f(\,\cdot\, ,r_0+\alpha w)- f(\,\cdot\, ,r_0)-\alpha w\,\frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)
\end{equation*}
\notag
$$
almost everywhere. Now from (2.10), (2.9) we have
$$
\begin{equation*}
\begin{aligned} \, 0&=\alpha\mathcal{L}(w,v)-\alpha\biggl\langle\frac{\partial f}{\partial r} (\,\cdot\, ,r_0+0)w,v\biggr\rangle-\lambda_1\alpha\langle w,v\rangle \\ &\geqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\, ,r_0),v\rangle+ \frac{\lambda_1}{2}\alpha\langle w,v\rangle-\lambda_1\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
$$
for non-negative functions $v \in C^\infty(X)$ (cf. § 3.2 in [3]). Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\alpha \mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \\ &\qquad\leqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \frac{\lambda_1}{2}\alpha\langle w,v\rangle. \end{aligned}
\end{equation*}
\notag
$$
Hence by the definitions of $\mathcal{L}$ (see (1.8)) and $\theta$ (see (1.9)), for all $0<\alpha<\delta$, we have
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \leqslant \theta(r_0,v)+\frac{\lambda_1}{2}\alpha\langle w,v\rangle.
\end{equation*}
\notag
$$
By Remark 5, a number $r_0$ is a subsolution of (1.6) if and only if $\theta(r_0,v) \leqslant 0$ for each non-negative function $v \in C^\infty(X)$. Next, since $\inf_{x \in X}w(x)>0$ and $\lambda_1<0$, and hence, for $0<\alpha<\delta$, the function $r_0+\alpha w$ is a subsolution of equation (1.6). So, the conditions of assertion (a) of Theorem 4 are met. Applying this assertion we get the required result. Assertion (b) is verified similarly. Theorem 7 is proved. 5.2. Proof of Theorem 8(a) By Theorem 6, an eigenfunction $w \in C(X)$ belonging to the eigenvalue $\lambda_1$ can be chosen so as to have $\inf_{x\in X}w(x) > 0$. Hence, by monotonicity of the difference relations (2.11), for any $\alpha>0$, we have
$$
\begin{equation}
f(\,\cdot\, ,r_0+\alpha w)-f(\,\cdot\, ,r_0)< \alpha w\,\frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)
\end{equation}
\tag{5.1}
$$
almost everywhere. Now an appeal to (2.10), (2.9) shows that
$$
\begin{equation*}
\begin{aligned} \, 0&=\alpha\mathcal{L}(w,v)- \alpha\biggl\langle\frac{\partial f}{\partial r} (\,\cdot\, ,r_0+0)w,v\biggr\rangle-\lambda_1\alpha\langle w,v\rangle \\ &\leqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\, ,r_0),v\rangle-\lambda_1\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
$$
for non-negative functions $v \in C^\infty(X)$ (cf. § 3.2 in [3]). Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\alpha\mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0+\alpha w),v\rangle-\langle h,dv\rangle_{L^2(T^* X)} \\ &\qquad\geqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \lambda_1 \alpha\langle w,v\rangle, \end{aligned}
\end{equation*}
\notag
$$
and now by definitions of $\mathcal{L}$ (see (1.8)) and $\theta$ (see (1.9)), for any $\alpha>0$, we have
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \geqslant \theta(r_0,v)+\lambda_1 \alpha\langle w,v\rangle.
\end{equation*}
\notag
$$
In addition, since inequality (5.1) is strict, for $v=1$ we have
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,1)-\langle f(\,\cdot\, ,r_0+\alpha w),1\rangle> \theta(r_0,1)+\lambda_1\alpha\langle w,1\rangle.
\end{equation*}
\notag
$$
By Remark 5, a number $r_0$ is a solution of equation (1.10) if and only if $\theta(r_0,v)=0$ for each non-negative function $v \in C^\infty(X)$. Since $\inf_{x\in X}w(x)>0$ and $\lambda_1\geqslant 0$, it follows that, for $\alpha>0$, the function $r_0+\alpha w$ is a strict supersolution of equation (1.6). Let $A>0$ be such that $r_0+Aw \geqslant q_0$ almost everywhere. This $A$ satisfies the conditions of assertion (a) of Theorem 5, which, in turn, secures the required result. Assertion (b) is verified similarly. Theorem 8 is proved. The author is grateful to A. A. Davydov for posing the problem and useful discussions.
Citation in format AMSBIB
\Bibitem{Tun23} |
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