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Sbornik: Mathematics, 2024, Volume 215, Issue 6, Pages 841–860
DOI: https://doi.org/10.4213/sm10029e (Mi sm10029) 		 
On the solvability of the boundary value problem for one class of nonlinear systems of high-order partial differential equations

S. S. Kharibegashviliab, B. G. Midodashvilic

a Andrea Razmadze Mathematical Institute of Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia
b Georgian Technical University, Tbilisi, Georgia
c Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia
English version PDF (577 kB) HTML full-text Russian version article
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DOI: https://doi.org/10.4213/sm10029e
Abstract: For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution.
Bibliography: 18 titles.
Keywords: high-order nonlinear systems; fixed point principles; existence, uniqueness and nonexistence of a solution.
Received: 14.11.2023 and 12.02.2024
Bibliographic databases:
Document Type: Article
MSC: 35G30
Language: English
Original paper language: Russian

§ 1. Statement of the problem

In the Euclidean space $\mathbb{R}^{n+1}$ of the variables $x=(x_1,\dots,x_n)$ and $t$ consider a nonlinear systems of partial differential equations of the form

$$ \begin{equation} L_f u := \frac{\partial^{2(2k+1)}u}{\partial t^{2(2k+1)}}-\Delta^2 u + f(u, \nabla u) =F(x,t), \end{equation} \tag{1.1} $$
where $f=(f_1,\dots,f_N)$ and $F=(F_1,\dots,F_N)$ are given vector functions and $u=(u_1,\dots,u_N)$ is an unknown vector function, $N \geqslant 2$,
$$ \begin{equation*} \nabla := \biggl(\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial t}\biggr), \quad \Delta:= \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}, \qquad n \geqslant 2, \end{equation*} \notag $$
and $k \geqslant 0$ is an integer.

For system (1.1) consider the following setup of a boundary value problem: in a cylindrical domain $D_T:=\Omega \times (0,T)$, where $\Omega$ is an open domain in $\mathbb{R}^n$ with Lipschitz boundary, find a solution $u=u(x,t)$ of (1.1) for the boundary data

$$ \begin{equation} \frac{\partial^i u}{\partial t^i} \bigg|_{\Omega_0 \cup \Omega_T}=0, \qquad i=0,\dots,2k, \end{equation} \tag{1.2} $$
$$ \begin{equation} u|_{\Gamma}=0\quad\text{and} \quad \biggl(\Delta u +A \frac{\partial u}{\partial \nu}\biggr) \bigg|_{\Gamma}=0, \end{equation} \tag{1.3} $$
where $\Gamma :=\partial \Omega \times (0,T)$ is the lateral part of the boundary of $D_T$; $\Omega_0=\{(x,t) \mid {x\in\Omega},\, t=0 \}$ and $\Omega_T=\{(x,t) \mid x\in\Omega,\, t=T \}$ are, respectively, the lowed and upper bases of this cylinder; $A\colon \overline{\Gamma} \to \mathbb{R}^{N \times N}$ is a fixed continuous square matrix function of order $N$; ${\partial}/{\partial \nu}$ is the outward normal derivative on the boundary $\partial D_T$ of $D_T$, here $\nu =(\nu_1,\dots,\nu_n,\nu_{n+1})$ is the unit outward normal vector to $\partial D_T$ and, clearly, $\nu_{n+1}|_{\Gamma}=0$.

Apart from second-order equations of one of the standard types, such as elliptic, hyperbolic, parabolic, mixed and others, for which one states some or other problems and examines their well-posedness, the theory of partial differential equations considers also partial differential equations of high order, which cannot in general be classified with some type. For example, in the scalar case the linear part of the operator on the left-hand side of (1.1) is hypoelliptic in Hörmander’s nomenclature (see [1], Ch. 11, § 1, Definition 11.1.2). Investigations of partial differential equations and systems with nonstandard structure from the standpoint of the existence or absence of solutions, stating well-posed local, nonlocal and other problems is of certain scientific interest in the theory of partial differential equations.

In the scalar case, for equations of the form (1.1) boundary value problems were treated in [2] and [3], where the existence, uniqueness and absence of solutions to them were considered. The analysis of initial-value, boundary-value and mixed problems for nonlinear partial differential equations of high order with structure different from (1.1) was the topic of many papers (for instance, see [4]–[13] and the references there). For instance, in [4] the authors proposed a general approach to a priori estimates for solutions of partial differential equations and systems of high order, which, in particular, enables one to treat questions relating to the nonexistence of solutions. In [5] a universal approach was put forward to the question of the blowup of solutions to four types of nonlinear evolution partial differential equations of high order, namely, parabolic, hyperbolic, dispersion and Schrödinger’s ones. Routine questions of the existence, nonexistence, uniqueness, nonuniqueness and global asymptotic behaviour of solutions were also considered there. Also note [6]–[8], where for nonlinear partial differential equations of high order the authors investigated initial-boundary value problems. In those papers they considered the questions of the existence, nonexistence, uniqueness, and asymptotic behaviour of solutions to these problems. In [9] and [10], for nonlinear equations with iterated multidimensional wave operator in the principal part these authors investigated boundary value problems in which the data in this problems have support on the conical characteristic variety, while in [12], for a class of nonlinear problems with iterated multidimensional wave operator in the principal part, the boundary value problem in a cylindrical domain was considered in the case when the Dirichlet and Neumann conditions are prescribed on the whole boundary. Also note [11] and [13], where boundary value problems in a cylindrical domain were considered for some classes of nonlinear partial differential equations and systems. In those papers, depending on the conditions imposed on the nonlinear terms of the equations under consideration, the uniqueness, existence or nonexistence of solutions to them were established. Note that (1.1) does not belong to the classes of equations considered in [4]–[13].

Let $C^{4,4k+2}(\overline{D}_T)$ denote the space of continuous vector-valued functions $u=(u_1,\dots,u_N)$ in $\overline{D}_T$ that have there partial derivatives $\partial_x^\beta u$ and ${\partial^l u}/{\partial t^l}$, where ${\partial_x^\beta =\partial^{|\beta|}/(\partial x_1^{\beta_1}\dots \partial x_n^{\beta_n})}$, $\beta=(\beta_1,\cdots, \beta_n)$ and $|\beta|=\sum_{i=1}^{n}\beta_i \leqslant 4$, ${l\!=\!1,\dots,4k\!+\!2}$.

Here and throughout, we say that a vector-valued function $v=(v_1,\dots,v_N)$ belongs to a space $X$ if each component $v_i$ of this vector, $1 \leqslant i \leqslant N$, belongs to $X$.

Let

$$ \begin{equation} C_0^{4,4k+2}(\overline D_T):=\biggl\{u \in C^{4,4k+2}(\overline{D}_T)\colon u|_{\Gamma}=0, \, \frac{\partial^i u}{\partial t^i} \bigg|_{\Omega_0\cup\Omega_T}=0,\, i=0,\dots,2k \biggr\}. \end{equation} \tag{1.4} $$

We consider the Hilbert space $W_0^{2,2k+1}(D_T)$ obtained as the completion, with respect to the norm

$$ \begin{equation} \| u \|_{W_0^{2,2k+1}(D_T)}^2 =\int_{D_T} \biggl[u^2+\sum_{i=1}^{n} \biggl(\frac{\partial u}{\partial x_i}\biggr)^2 +\sum_{i,j=1}^n \biggl(\frac{\partial^2 u}{\partial x_i\, \partial x_j} \biggr)^2 +\sum_{i=1}^{2k+1}\biggl(\frac{\partial^i u}{\partial t^i} \biggr)^2 \biggr] \,dx\,dt, \end{equation} \tag{1.5} $$
of the classical space $C_0^{4,4k+2}(\overline D_T)$, $u^2=\sum_{i=1}^N u_i^2$.

Remark 1.1. It follows from (1.5) that if $u \in W_0^{2,2k+1}({D_T})$, then $u \in \mathring{W_2^1} ({D_T})$ and ${\partial^2u}/(\partial x_i\, \partial x_j), \partial^l u/{\partial t^l} \in {L_2}({D_T})$; $i,j=1,\dots,n$; $l=1,\dots,2k+1$. Here $W_2^m({D_T})$ is the well-known Sobolev space of functions in $L_2({D_T})$ that have generalized partial derivatives in $L_2({D_T})$ up to order $m$ inclusive and $\mathring{W_2^1}({D_T})=\{{u \in W_2^1({D_T})}\colon u|_{\partial D_T}=0\}$, where the equality $u|_{\partial D_T}=0$ must be treated in the sense of trace theory (see [14], Ch. I, § 6, Theorem 6.3). Moreover, when $\Omega$ is a convex domain — so that $D_T$ is convex too — we have the following estimate (see [14], Ch. II, § 6, inequality (6.5)):

$$ \begin{equation} \begin{aligned} \, \notag &\int_{D_T}\biggl[\,\sum_{i,j=1}^{n} \biggl(\frac{\partial^2 u}{\partial x_i\, \partial x_j} \biggr)^2+\sum_{i=1}^{n} \biggl(\frac{\partial^2 u}{\partial x_i\, \partial t} \biggr)^2 +\biggl(\frac{\partial^2 u}{\partial t^2} \biggr)^2 \biggr]\,dx\,dt \\ &\qquad \leqslant c \int_{D_T}\biggl[\, \sum_{i=1}^{n} \frac{\partial^2 u}{\partial x_i^2}+\frac{\partial^2 u}{\partial t^2} \biggr]^2 \,dx\,dt \quad \forall\, u \in \mathring{C^2} ({\overline{D}_T}) \end{aligned} \end{equation} \tag{1.6} $$
for some positive constant $c$ independent of $u$ and $D_T$, where $\mathring{C^2}({\overline{D}_T}):=\{u \in C^2(\overline{D}_T)\colon u|_{\partial D_T}=0\}$. It follows from (1.5) and (1.6) that there is a continuous embedding
$$ \begin{equation} W_0^{2,2k+1}(D_T) \subset W_2^2(D_T). \end{equation} \tag{1.7} $$
We assume below that $\Omega$ is a convex domain.

Before we introduce the concept of a weak generalized solution of problem (1.1)–(1.3) in $W_0^{2,2k+1}(D_T)$, let $u \in C_0^{4,4k+2}({\overline{D}_T})$ be a classical solution of this problem. Taking the scalar product of both sides of (1.1) with an arbitrary vector function $\varphi \in C_0^{4,4k+2}({\overline D_T})$ and integrating the resulting equality over ${D_T}$, from (1.2) we obtain

$$ \begin{equation} \begin{aligned} \, \notag &-\int_{D_T} \frac{\partial^{2k+1}u}{\partial t^{2k+1}} \, \frac{\partial^{2k+1}\varphi}{\partial t^{2k+1}}\,dx\,dt +\int_{\partial D_T} \frac{\partial \varphi}{\partial \nu} \, \Delta u\,ds -\int_{\partial D_T} \varphi \, \frac{\partial}{\partial \nu} \Delta u\,ds \\ &\qquad\qquad -\int_{D_T} \Delta u \Delta \varphi \,dx\,dt +\int_{D_T} f(u,\nabla u) \varphi \,dx\,dt =\int_{D_T} F\varphi \,dx\,dt, \end{aligned} \end{equation} \tag{1.8} $$
where $\eta \cdot \xi$ denotes the scalar product of two $N$-dimensional vectors, that is, $\sum_{i=1}^N \eta_i \xi_i$. Taking the second boundary condition in (1.3), as well as the equalities $\varphi|_{\partial D}=0$ and $(\partial \varphi/\partial \nu)|_{\Omega_0 \cup \Omega_T}=0$ (since $\varphi \in C_0^{4,4k+2}({\overline D_T})$), into account, from (1.8) we obtain
$$ \begin{equation} \begin{aligned} \, \notag &\int_{D_T} \biggl[\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \, \frac{\partial^{2k+1} \varphi}{\partial t^{2k+1}}+\Delta u \Delta \varphi \biggr]\,dx\,dt +\int_{\Gamma} A \frac{\partial u}{\partial \nu} \, \frac{\partial \varphi}{\partial \nu}\,ds \\ &\qquad\qquad -\int_{D_T} f(u,\nabla u)\varphi \,dx\,dt =-\int_{D_T} F \varphi \,dx\,dt \quad \forall\, \varphi \in C_0^{4,4k+2}(\overline{D}_T). \end{aligned} \end{equation} \tag{1.9} $$
A converse result also holds in a certain sense, namely, if $ u=(u_1,\dots,u_N) \in C^{4,4k+2}(\overline{D}_T)$ satisfies (1.2), the first boundary condition in (1.3) and integral equality (1.9) for each $\varphi=(\varphi_1,\dots,\varphi_N)\in C_0^{4,4k+2}(\overline{D}_T)$, then it follows by standard arguments that $u$ is a solution of system (1.1) in $D_T$ and satisfies the second boundary condition in (1.3) in the classical sense, that is, $u$ is a classical solution of problem (1.1)–(1.3). We are going to base the definition of a weak generalized solution of problem (1.1)–(1.3) in the space $W_0^{2,2k+1}(D_T)$ on equality (1.9), but to do this we must first of all specify the conditions imposed on the growth exponents of the nonlinearity, the vector function $f(u, \nabla u)$, with respect to the independent variables involved in $f$, to ensure that the integral
$$ \begin{equation*} \int_{D_T} f(u, \nabla u) \varphi \,dx\,dt, \end{equation*} \notag $$
on the left-hand side of (1.9) exist for each $\varphi \in W_0^{2,2k+1}(D_T)$. In what follows we assume about the vector-valued function $f=f(s_0,s_1,\dots,s_{n+1})$, $s_i \in \mathbb{R}^N$, $i=0,1,\dots,n+ 1$, that
$$ \begin{equation} f \in C(\mathbb {R}^{(n+2)N}) \end{equation} \tag{1.10} $$
and
$$ \begin{equation} |f(s_0,s_1,\dots,s_{n+1})| \leqslant M+\sum_{i=0}^{n+1}M_i|s_i|^{\alpha_i} \quad \forall\, s_i \in \mathbb{R}^N, \quad i=0,1,\dots,n+1, \end{equation} \tag{1.11} $$
where $M, M_i, \alpha_i =\mathrm{const} >0$, $i=0, 1, \dots,n+1$, and we have
$$ \begin{equation} 1<\alpha_0<\frac{n+1}{n-3} \quad \text{for}\ n>3, \qquad \alpha_0 >1 \quad \text{for} \ n=2, 3 \end{equation} \tag{1.12} $$
and
$$ \begin{equation} 1<\alpha_i<\frac{n+1}{n-1}, \qquad i=1,\dots,n+1, \quad n \geqslant 2. \end{equation} \tag{1.13} $$

Here and below $|w|$ denotes the standard scalar norm or Euclidean vector norm, depending on whether $w$ is a scalar or a vector.

Remark 1.2. It is known that the space $W_2^2(D_T)$ is continuously and compactly embedded in $L_p(D_T)$ for $p<2(n+1)/(n-3)$ if $n>3$ and for each $p \geqslant 1$ if $n=2,3$; in a similar way $W_2^1(D_T)$ is continuously and compactly embedded in $L_q(D_T)$ for $q<2(n+1)/(n-1)$ (see [14], Ch. I, § 7, Theorem 7.2). Thus, taking account of the continuous embedding of spaces in (1.7) and inequality (1.11) with nonlinearity exponents $\alpha_i$ satisfy conditions (1.12) and (1.13), from the properties of the Nemytskii operators $N_i$, $i=0,1,\dots,n+1$, acting by the formulae $N_i v =|v|^{\alpha_i}$ we see that the nonlinear operator

$$ \begin{equation} N\colon W_0^{2,2k+1}(D_T) \to L_2(D_T), \end{equation} \tag{1.14} $$
acting by
$$ \begin{equation} N u=f(u,\nabla u), \end{equation} \tag{1.15} $$
is continuous and compact (see [15], Ch. III, § 12, Theorem 12.10). Hence it follows, in particular, that if $u \in W_0^{2,2k+1}(D_T)$, then $f(u, \nabla u) \in L_2(D_T)$ and the integral $\displaystyle\int_{D_T} f(u, \nabla u) \varphi \,dx\,dt$ on the left-hand side of (1.9) exists, while if $u_m \to u$ in the space $W_0^{2,2k+1}(D_T)$, then $f(u_m, \nabla u_m) \to f(u,\nabla u)$ in the space $L_2(D_T)$.

Definition 1.1. Let $\Omega$ be a convex domain with Lipschitz boundary, let the vector-valued function $f$ satisfy conditions (1.10)–(1.13), and let $A \!\in\! C(\overline{\Gamma})$ and ${F \!\in\! L_2(D_T)}$. Then a vector function $u \in W_0^{2,2k+1} (D_T)$ is called a weak generalized solution of problem (1.1)–(1.3) if the integral equality (1.9) holds for each vector function $\varphi \in W_0^{2,2k+1} (D_T)$, that is,

$$ \begin{equation} \begin{aligned} \, \notag &\int_{D_T} \biggl[\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \, \frac{\partial^{2k+1} \varphi}{\partial t^{2k+1}}+\Delta u \Delta \varphi \biggr]\,dx\,dt +\int_{\Gamma} A \frac{\partial u}{\partial \nu} \,\frac{\partial \varphi}{\partial \nu}\,ds \\ &\qquad\qquad -\int_{D_T} f(u,\nabla u)\, \varphi \,dx\,dt =-\int_{D_T} F \varphi \,dx\,dt \quad \forall\, \varphi \in W_0^{2,2k+1} (D_T). \end{aligned} \end{equation} \tag{1.16} $$

§ 2. Equivalent norms in the space $W_0^{2,2k+1}(D_T)$

The Hilbert space $W_0^{2,2k+1}(D_T)$ introduced above is brought about by the norm $\|u\|_0= \|u\|_{W_0^{2,2k+1}(D_T)}$ defined by the right-hand side of (1.5), which corresponds to the scalar product

$$ \begin{equation} (u,v)_0 =\int_{D_T}\biggl[u v+\sum_{i=1}^n \frac{\partial u}{\partial x_i} \, \frac{\partial v}{\partial x_i}+ \sum_{i,j=1}^n \frac{\partial^2 u}{\partial x_i\, \partial x_j} \, \frac{\partial^2 v}{\partial x_i\, \partial x_j} + \sum_{i=1}^{2k+1} \frac{\partial^i u}{\partial t^i} \, \frac{\partial^i v}{\partial t^i} \biggr] \,dx\,dt. \end{equation} \tag{2.1} $$

Now, assuming that the matrix $A$ is symmetric, continuous and nonnegative definite, that is,

$$ \begin{equation} A^* =A, \quad A \in C(\overline{\Gamma})\quad\text{and}\quad \quad A(x,t) \xi \cdot \xi \geqslant 0 \quad \forall\,(x,t) \in \overline{\Gamma}, \quad \forall\, \xi \in \mathbb{R}^N, \end{equation} \tag{2.2} $$
we show that, apart from (2.1), the bilinear form
$$ \begin{equation} (u,v)_1=\int_{D_T}\biggl[\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \, \frac {\partial^{2k+1} v}{\partial t^{2k+1}}+\Delta u \Delta v \biggr] \,dx\,dt +\int_{\Gamma} A \, \frac{\partial u}{\partial \nu} \, \frac{\partial v}{\partial \nu}\,ds \end{equation} \tag{2.3} $$
also defines an inner product in $W_0^{2,2k+1}(D_T)$, with the norm
$$ \begin{equation} \|u\|_1^2=\int_{D_T}\biggl[\biggl(\frac{\partial^{2k+1} u}{\partial t^{2k+1}}\biggr)^2+(\Delta u)^2 \biggr] \,dx\,dt +\int_{\Gamma} A \, \frac{\partial u}{\partial \nu} \, \frac{\partial u}{\partial \nu}\,ds. \end{equation} \tag{2.4} $$

Lemma 2.1. Under assumption (2.2), let $\Omega$ be a convex domain with boundary $\partial \Omega$ of class $C^2$. Then the following inequalities hold:

$$ \begin{equation} c_1 \|u\|_0 \leqslant \|u\|_1 \leqslant c_2 \|u\|_0 \quad\forall\, u \in C_0^{4,4k+2}(\overline {D}_T), \end{equation} \tag{2.5} $$
where the positive coefficients $c_1$ and $c_2$ are independent of $u$.

Proof. First we find estimate for the quantities $\|{\partial^i u}/{\partial t^i} \|^2_{L_2(D_T)}$, $i=0,1,\dots,2k$, in terms of $\|{\partial^{2k+1} u}/{\partial t^{2k+1}} \|^2_{L_2(D_T)}$. Because $u \in C_0^{4,4k+2}(\overline {D}_T)$ satisfies (1.2), we have
$$ \begin{equation} \frac{\partial^i u(\cdot, t)}{\partial t^i}=\frac{1}{(2k-i)!} \int_0^t(t-\tau)^{2k-i} \, \frac{\partial^{2k+1} u(\cdot,\tau)}{\partial t^{2k+1}}\,d\tau, \qquad i=0,1,\dots,2k. \end{equation} \tag{2.6} $$
By Cauchy’s inequality, it follows from (2.6) that
$$ \begin{equation*} \begin{aligned} \, \biggl(\frac{\partial^i u(\cdot, t)}{\partial t^i} \biggr)^2 &\leqslant \frac {1}{((2k-i)!)^2}\int_0^t(t-\tau)^{2(2k-i)}\,d\tau \int_0^t\biggl(\frac{\partial^{2k+1} u(\cdot, \tau)}{\partial t^{2k+1}}\biggr)^2\,d\tau \\ &\leqslant T^{4k-2i+1} \int_0^T\biggl(\frac{\partial^{2k+1} u(\cdot, \tau)}{\partial t^{2k+1}}\biggr)^2\,d\tau, \end{aligned} \end{equation*} \notag $$
so that
$$ \begin{equation} \int_0^T\biggl(\frac{\partial^i u(\cdot, t)}{\partial t^i} \biggr)^2\,dt \leqslant T^{4k-2i+2} \int_0^T\biggl(\frac{\partial^{2k+1} u(\cdot, \tau)}{\partial t^{2k+1}}\biggr)^2\,d\tau, \qquad i=0,1,\dots,2k. \end{equation} \tag{2.7} $$

Integrating both sides of (2.7) over $\Omega$ we obtain

$$ \begin{equation} \int_{D_T}\biggl(\frac{\partial^i u}{\partial t^i} \biggr)^2 \,dx\,dt \leqslant T^{4k-2i+2} \int_{D_T}\biggl(\frac{\partial^{2k+1} u}{\partial t^{2k+1}}\biggr)^2 \,dx\,dt, \qquad i=0,1,\dots,2k. \end{equation} \tag{2.8} $$

If $u \in C_0^{4,4k+2}(\overline{D}_T)$, then $u(\cdot, t) \in C^2(\overline{\Omega})$ and $u(\cdot, t)|_{\partial \Omega}=0$ for fixed $t \in [0,T]$, and therefore, by a well-known inequality (see [14], Ch. II, § 3, inequality 3.29)

$$ \begin{equation} \int_{\Omega} \biggl[u^2(\cdot, t)+\sum_{i=1}^{n}\biggl(\frac{\partial u(\cdot, t)}{\partial x_i}\biggr)^2 \biggr]\,dx \leqslant c_0 \int_{\Omega}\bigl(\Delta u(\cdot, t)\bigr)^2\,dx, \end{equation} \tag{2.9} $$
where the positive constant $c_0=c_0(\Omega)$ is independent of $t \in [0,T]$ and $u$. Integrating (2.9) with respect to $t$ we obtain
$$ \begin{equation} \int_{D_T} \biggl[u^2+\sum_{i=1}^n \biggl(\frac{\partial u}{\partial x_i} \biggr)^2 \biggr] \,dx\,dt \leqslant c_0 \int_{D_T}(\Delta u)^2 \,dx\,dt \quad \forall\, u \in C_0^{4,4k+2}(\overline{D}_T). \end{equation} \tag{2.10} $$

If $u \in C_0^{4,4k+2}(\overline{D}_T)$ and therefore $u(\cdot, t) \in C^2(\overline{\Omega})$, $t \in [0,T]$, then, as the boundary $\partial \Omega$ of the domain $\Omega$ is of class $C^2$ by assumption, $\partial u(\cdot, t)/{\partial \nu} |_{\partial \Omega}$ satisfies the estimate ([16], Ch. III, Theorem 3.37)

$$ \begin{equation} \begin{aligned} \, \notag \int_{\partial \Omega} \biggl(\frac{\partial u(\cdot, t)}{\partial \nu} \biggr)^2\,ds &= \biggl\|\frac{\partial u(\cdot, t)}{\partial \nu} \biggr\|^2_{L_2(\partial \Omega)} \leqslant \widetilde{c}_0 \|u(\cdot, t)\|^2_{W_2^2(\Omega)} \\ &=\widetilde{c}_0 \int_{\Omega}\biggl[u^2(\cdot, t)+\sum_{i=1}^{n}\biggl(\frac{\partial u(\cdot, t)}{\partial x_i} \biggr)^2+\sum_{i,j=1}^{n}\biggl(\frac{\partial^2 u(\cdot, t)}{\partial x_i\, \partial x_j} \biggr)^2 \biggr]\,dx, \end{aligned} \end{equation} \tag{2.11} $$
where the positive constant $\widetilde{c}_0=\widetilde{c}_0 (\Omega)$ is independent if $t$ and $u$.

Taking (2.2) into account, set

$$ \begin{equation*} a_0=\max_{(x,t) \in \overline{\Gamma},\, |\xi|_{\mathbb{R}^N}=1} (A(x,t)\xi,\xi)_{\mathbb{R}^N}. \end{equation*} \notag $$
It follows from (2.11) that
$$ \begin{equation} \begin{aligned} \, \notag \int_{\Gamma} A \frac{\partial u}{\partial \nu} \frac{\partial u}{\partial \nu}\,ds &\leqslant a_0 \int_{\Gamma} \biggl(\frac{\partial u}{\partial \nu} \biggr)^2\,ds \\ \notag &=a_0 \int_0^T \biggl[\int_{\partial \Omega} \biggl(\frac{\partial u(\cdot, t)}{\partial \nu} \biggr)^2\,ds \biggr]\,dt \leqslant a_0 \widetilde {c}_0 \int_0^T \|u(\cdot, t)\|^2_{W_2^2(\Omega)}\,dt \\ &= a_0 \widetilde {c}_0 \int_{D_T} \biggl[u^2+\sum_{i=1}^{n} \biggl(\frac{\partial u}{\partial x_i} \biggr)^2+\sum_{i,j=1}^n \biggl(\frac{\partial^2 u}{\partial x_i\, \partial x_j} \biggr)^2 \biggr] \,dx\,dt. \end{aligned} \end{equation} \tag{2.12} $$

Now, since $\Omega$ is convex by assumption and therefore (1.6) holds, we easily obtain (2.5) from (1.5), (2.1)–(2.4), (2.8)–(2.10) and (2.12).

Lemma 2.1 is proved.

Remark 2.1. By Lemma 2.1 the completion of $C_0^{4,4k+2}(\overline{D}_T)$ in the norm (2.4) produces the same Hilbert space $W_0^{2,2k+1}(D_T)$ with the equivalent inner products (2.1) and (2.3).

§ 3. Equivalent reduction of problem (1.1)–(1.3) to the nonlinear functional equation $u=Ku$ in the space $W_0^{2,2k+1}(D_T)$ and an estimate for $\|Ku\|_1$

We start with the linear case of problem (1.1)–(1.3), that is, with $f=0$. Then for $F \in L_2(D_T)$ we can introduce in a similar way the concept of a weak generalized solution $u \in W_0^{2,2k+1}(D_T)$ of this problem, which, in view of (1.16) and (2.3), satisfies the integral equality

$$ \begin{equation} \begin{aligned} \, \notag (u,\varphi)_1 &=\int_{D_T}\biggl[\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \, \frac {\partial^{2k+1} \varphi}{\partial t^{2k+1}}+\Delta u \Delta \varphi \biggr] \,dx\,dt+ \int_{\Gamma} A \frac{\partial u}{\partial \nu} \, \frac{\partial \varphi}{\partial \nu}\,ds \\ &=-\int_{D_T}F \varphi \,dx\,dt \quad \forall\, \varphi \in W_0^{2,2k+1}(D_T). \end{aligned} \end{equation} \tag{3.1} $$

From (2.5) it is easy to see that

$$ \begin{equation} \begin{aligned} \, \notag \biggl|\int_{D_T}F \varphi \,dx\,dt\biggr| &\leqslant \|F\|_{L_2(D_T)}\|\varphi\|_{L_2(D_T)} \\ &\leqslant \|F\|_{L_2(D_T)}\|\varphi\|_0 \leqslant c^{-1}_1\|F\|_{L_2(D_T)}\|\varphi\|_1. \end{aligned} \end{equation} \tag{3.2} $$

Taking Remark 2.1, as well as (3.1) and (3.2), into account, it follows from Riesz’s theorem that there exists a unique vector-valued function $u \in W_0^{2,2k+1}(D_T)$ such that (3.1) holds for each $\varphi \in W_0^{2,2k+1}(D_T)$ and its norm has the estimate

$$ \begin{equation} \|u\|_1 \leqslant c_1^{-1}\|F\|_{L_2(D_T)}. \end{equation} \tag{3.3} $$

Thus, setting $u=L_0^{-1} F$ we see that the linear problem (1.1)–(1.3), that is, the case $f=0$, corresponds to a linear bounded operator

$$ \begin{equation*} L_0^{-1} \colon L_2(D_T)\to W_0^{2,2k+1} (D_T), \end{equation*} \notag $$
whose norm, owing to (3.2), has the estimate
$$ \begin{equation} \|L_0^{-1}\|_{L_2(D_T) \to W_0^{2,2k+1}(D_T)} \leqslant c_1^{-1}. \end{equation} \tag{3.4} $$

Remark 3.1. By Definition 1.1 of a weak generalized solution of problem (1.1)–(1.3) and by the above definition of the operator $L_0^{-1}$ the integral identity (1.16), which is equivalent to this problem, can be written as the nonlinear functional equation

$$ \begin{equation} u=L_0^{-1}[f(u,\nabla u)-F] \end{equation} \tag{3.5} $$
in the Hilbert space $W_0^{2,2k+1}(D_T)$. Taking (1.15) into account we write (3.5) in the form
$$ \begin{equation} u=K u:= L_0^{-1}(Nu-F), \end{equation} \tag{3.6} $$
where, by (3.4) and Remark 1.2, if the nonlinear function $f$ satisfies (1.10)–(1.13), then $K\colon W_0^{2,2k+1}(D_T) \to W_0^{2,2k+1}(D_T)$ in (3.6) is a continuous compact operator.

If $v \in \mathring{W_2^1}(D_T, \Omega_0 \cup \Omega_T) :=\{v \in W_2^1(D_T)\colon v|_{\Omega_0 \cup \Omega_T}=0 \}$, then taking the structure of the domain $D_T=\Omega \times (0,T)$ into account, we have the multiplicative inequality (see [14], Ch. I, § 7, Theorem 7.1)

$$ \begin{equation} \begin{gathered} \, \|v\|_{Lp(D_T)} \leqslant \beta \|\nabla v\|_{L_m(D_T)}^{\widetilde{\alpha}} \|v\|_{L_r(D_T)}^{1-\widetilde{\alpha}} \quad \forall\, v \in \mathring{W_2^1}(D_T, \Omega_0 \cup \Omega_T), \\ \notag \widetilde{\alpha} =\biggl(\frac{1}{r} -\frac{1}{p} \biggr) \biggl(\frac{1}{r}-\frac {1}{\widetilde{m}} \biggr)^{-1}, \qquad \widetilde{m}=\frac{(n+1)m}{n+1-m}, \qquad r \leqslant p, \end{gathered} \end{equation} \tag{3.7} $$
where the positive constant $\beta=\beta (\Omega)$ is independent of $v$ and $T$, where $p \in [1,(n+ 1)/(n-1)]$ for $r=1$ and $m=2$.

Taking the well-known inequality (see [15], Ch. III, § 12, inequality (5))

$$ \begin{equation*} \int_{D_T}|v|\,dx\,dt \leqslant (\operatorname{mes}D_T)^{1-1/p}\|v\|_{L_p(D_T)}, \qquad p \geqslant 1, \end{equation*} \notag $$
into account, it follows from (3.7) that
$$ \begin{equation} \|v\|_{L_p(D_T)} \leqslant \beta_0 (\operatorname{mes}D_T)^{1/p+1/(n+1)-1/2}\|v\|_{W_2^1(D_T)} \quad \forall\, v \in \mathring{W_2^1}(D_T, \Omega_0 \cup \Omega_T), \end{equation} \tag{3.8} $$
where the positive constant $\beta_0=\beta_0(\Omega)$ is independent of $v$ and $T$.

Since $\operatorname{mes}D_T=T \operatorname{mes}\Omega$, from (3.8) we obtain

$$ \begin{equation} \|v\|_{L_p(D_T)} \leqslant \beta_1 T^{1/p+1/(n+1)-1/2}\|v\|_{W_2^1(D_T)} \quad\forall\, v \in \mathring{W_2^1}(D_T, \Omega_0 \cup \Omega_T), \end{equation} \tag{3.9} $$
where $\beta_1=\beta_0 (\operatorname{mes}\Omega)^{1/p+1/(n+1)-1/2}$; also, it is easy to verify that the condition $1/p+1/(n+1)-1/2>0$ is equivalent to $p<2(n+1)/(n-1)$.

By (1.4) and the definition of the space $W_0^{2,2k+1}(D_T)$ as the completion of the classical space $C_0^{4,4k+2}(\overline{D}_T)$ in the norm (1.5), in view of the existence of traces of elements of $W_2^1(D_T)$ on $(\Omega_0 \cup \Omega_1) \subset \partial D_T$ and taking the embedding (1.7) into account we have

$$ \begin{equation} u,u_t,u_{x_i} \in \mathring{W_2^1}(D_T, \Omega_0 \cup \Omega_T), \qquad i=1,\dots,n. \end{equation} \tag{3.10} $$
Therefore, if the $\alpha_i$, $i=1,\dots,n+1$, satisfy inequalities (1.13), by (1.5), (1.7) and (3.9), (3.10) we have
$$ \begin{equation} \begin{aligned} \, \notag &\biggl[\int_{D_T}|u_{x_i}|^{2\alpha_i}\,dx\,dt \biggr]^{1/2} =\|u_{x_i}\|_{L_{2\alpha_i}(D_T)}^{\alpha_i} \leqslant \beta_1^{\alpha_i}T^{\alpha_i(1/(2\alpha_i)+1/(n+1)-1/2)} \|u_{x_i}\|_{W_2^1(D_T)}^{\alpha_i} \\ \notag &\qquad\leqslant \beta_1^{\alpha_i}T^{\alpha_i (1/(2\alpha_i)+1/(n+1)-1/2)} \|u\|_{W_0^{2,2k+1}(D_T)}^{\alpha_i} \\ &\qquad =\beta_1^{\alpha_i}T^{\alpha_i (1/(2\alpha_i)+1/(n+1)-1/2)} \|u\|_0^{\alpha_i} \quad\forall\, u \in W_0^{2,2k+1}(D_T), \qquad i=1,\dots,n, \end{aligned} \end{equation} \tag{3.11} $$
and in a similar way
$$ \begin{equation} \begin{aligned} \, \notag &\biggl[\int_{D_T}|u_t|^{2\alpha_{n+1}}\,dx\,dt \biggr]^{1/2} \\ &\qquad \leqslant \beta_1^{\alpha_{n+1}}T^{\alpha_{n+1} (1/(2\alpha_{n+1})+1/(n+1)-1/2)} \|u\|_0^{\alpha_{n+1}} \quad\forall\, u \in W_0^{2,2k+1}(D_T). \end{aligned} \end{equation} \tag{3.12} $$

Below, instead of condition (1.12) on $\alpha_0$, for the simplicity of presentation we make the more restrictive assumption

$$ \begin{equation} 1<\alpha_0<\frac{n+1}{n-1}, \end{equation} \tag{3.13} $$
which is similar to conditions (1.13) imposed on the other exponents $\alpha_i$, $i=1,\dots,n+1$.

By (3.13), similarly to (3.11) and (3.12) we have

$$ \begin{equation} \biggl[\int_{D_T}|u|^{2\alpha_0} \,dx\,dt \biggr]^{1/2} \leqslant \beta_1^{\alpha_0}T^{\alpha_0(1/(2 \alpha_0)+1/(n+1)-1/2)} \|u\|_0^{\alpha_0} \quad\forall\, u \in W_0^{2,2k+1}(D_T). \end{equation} \tag{3.14} $$

Note that by (1.13) and (3.13)

$$ \begin{equation} \gamma_i=\alpha_i \biggl(\frac{1}{2\alpha_i}+\frac{1}{n+1} -\frac{1}{2} \biggr) > 0, \qquad i=0,\dots,n+1. \end{equation} \tag{3.15} $$

Below we present an analogue of the first inequality in (2.5) which we require below. By (1.5), (1.6), (2.3), (2.8) and (2.10) we have

$$ \begin{equation} \begin{aligned} \, \notag &\|u\|_0^2\leqslant \int_{D_T}\biggl[c_0(\Delta u)^2+2c \biggl((\Delta u)^2+\biggl(\frac {\partial^2 u}{\partial t^2}\biggr)^2\biggr)+\sum_{i=1}^{2k+1}\biggl(\frac{\partial^i u}{\partial t^i}\biggr)^2\biggr]\,dx\,dt \\ \notag &\leqslant \int_{D_T}\biggl[(c_0\,{+}{\kern1pt}2 c)(\Delta u)^2{+}\,2c T^{4k-2} \biggl(\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \biggr)^2{+}\, \biggl\{\sum_{i=1}^{2k+1} T^{4k-2i+2}\!\biggr\} \biggl(\frac{\partial^{2k+1} u}{\partial t^{2k+1}}\biggr)^2 \biggr] \,dx\,dt \\ &\leqslant \lambda^2(T) \int_{D_T}\biggl[\biggl(\frac{\partial^{2k+1} u}{\partial t^{2k+1}} \biggr)^2+(\Delta u)^2 \biggr] \,dx\,dt = \lambda^2(T)\|u\|_1^2 \quad\forall\, u \in W_0^{2,2k+1}(D_T), \end{aligned} \end{equation} \tag{3.16} $$
where
$$ \begin{equation} \lambda(T)= \begin{cases} (c_0+4c+2k+1)^{1/2}, & T \leqslant 1, \\ (c_0+4c+2k+1)^{1/2}T^{2k}, & T >1. \end{cases} \end{equation} \tag{3.17} $$

Now, taking (1.11), (3.3), (3.4) and (3.11)–(3.17) into account we find an estimate for the quantity $\|K u\|_{W_0^{2,2k+1}(D_T)}=\|K u\|_1$ in (3.6):

$$ \begin{equation} \begin{aligned} \, \notag &\|K u\|_{W_0^{2,2k+1}(D_T)} =\|K u\|_1 \leqslant \|L_0^{-1}\|_{L_2(D_T) \to W_0^{2,2k+1}(D_T)} \|N u-F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1}\|N u\|_{L_2(D_T)}+c_1^{-1}\|F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1} \biggl[\int_{D_T} \biggl(M+M_0|u|^{\alpha_0}+\sum_{i=1}^n M_i |u_{x_i}|^{\alpha_i} + M_{n+1}|u_t|^{\alpha_{n+1}} \biggr)^2\,dx\,dt\biggr]^{1/2} \\ \notag &\qquad+c_1^{-1}\|F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1} \biggl[\int_{D_T} (n+3) \bigl(M^2+M_0^2 |u|^{2 \alpha_0} +\sum_{i=1}^n M_i^2 |u_{x_i}|^{2\alpha_i}+M_{n+1}^2|u_t|^{2\alpha_{n+1}} \biggr) \,dx\,dt \biggr]^{1/2} \\ \notag &\qquad+c_1^{-1}\|F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1}(n+3)^{1/2} \biggl[\biggl(\int_{D_T} M^2 \,dx\,dt \biggr)^{1/2}+ \biggl(\int_{D_T} M_0^2 |u|^{2 \alpha_0} \,dx\,dt \biggr)^{1/2} \\ \notag &\qquad+\sum_{i=1}^n \biggl(\int_{D_T} M_i^2 |u_{x_i}|^{2\alpha_i} \,dx\,dt \biggr)^{1/2}+\biggl(\int_{D_T} M_{n+1}^2|u_t|^{2\alpha_{n+1}} \,dx\,dt \biggr)^{1/2} \biggr] \\ \notag &\qquad+ c_1^{-1}\|F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1}(n+3)^{1/2} \biggl[\biggl(M^2\operatorname{mes}D_T \biggr)^{1/2}+ \sum_{i=0}^{n+1} M_i \beta_1^{\alpha_i} T^{\gamma_i} \|u\|_0^{\alpha_i} \biggr] + c_1^{-1}\|F\|_{L_2(D_T)} \\ \notag &\leqslant c_1^{-1}(n+3)^{1/2} \sum_{i=0}^{n+1} M_i \beta_1^{\alpha_i} T^{\gamma_i} \lambda^{\alpha_i}(T) \|u\|_1^{\alpha_i}+c_1^{-1}(n+3)^{1/2} (M^2 \operatorname{mes} D_T)^{1/2} \\ \notag &\qquad+ c_1^{-1}\|F\|_{L_2(D_T)} \\ &= \sum_{i=0}^{n+1} \widetilde{a}_i(T)\|u\|_1^{\alpha_i}+b(T) \quad\forall\, u \in W_0^{2,2k+1}(D_T). \end{aligned} \end{equation} \tag{3.18} $$
Here
$$ \begin{equation} \widetilde{a}_i(T)=c_1^{-1}(n+3)^{1/2} M_i \beta_1^{\alpha_i} T^{\gamma_i} \lambda^{\alpha_i}(T), \qquad i=0,\dots,n+1, \end{equation} \tag{3.19} $$
$$ \begin{equation} b(T)=c_1^{-1}(n+3)^{1/2} (M^2 \operatorname{mes} \Omega)^{1/2} T^{1/2} +c_1^{-1}\|F\|_{L_2(D_T)}; \end{equation} \tag{3.20} $$
in addition, in the derivation of (3.18) we used the following inequalities:
$$ \begin{equation*} \biggl(\sum_{i=1}^m k_i\biggr)^2 \leqslant m \sum_{i=1}^m k_i^2, \qquad \biggl(\sum_{i=1}^m k_i^2\biggr)^{1/2} \leqslant \sum_{i=1}^m |k_i|. \end{equation*} \notag $$

We simplify the expression on the right in (3.18). Since $\alpha_i >1$, $i=0,\dots,n+1$, for $\|u\|_1 \leqslant 1$ we have $\|u\|_1^{\alpha_i} \leqslant 1$, while for $\|u\|_1 >1$ we have $\|u\|_1^{\alpha_i} \leqslant \|u\|_1^\alpha$, where

$$ \begin{equation} \alpha=\max_{0 \leqslant i \leqslant n+1}\alpha_i >1. \end{equation} \tag{3.21} $$
Bearing this in mind, from (3.18), by (3.19) and (3.20) we obtain
$$ \begin{equation} \|K u\|_{W_0^{2,2k+1}(D_T)} \leqslant a_1(T) \|u\|_1^\alpha+b_1(T) \quad\forall\, u \in W_0^{2,2k+1}(D_T), \end{equation} \tag{3.22} $$
where
$$ \begin{equation} a_1(T) =\sum_{i=0}^{n+1}\widetilde{a}_i(T), \qquad b_1(T)=\sum_{i=0}^{n+1}\widetilde{a}_i(T)+b(T), \end{equation} \tag{3.23} $$
and the $\widetilde{a}_i(T)$, $i=0,1,\dots,n+1$, and $b(T)$ are defined by (3.19) and (3.20), respectively.

§ 4. Existence and absence of solutions to (1.1)–(1.3)

Below, under the assumption that

$$ \begin{equation} F \colon D_\infty \to \mathbb{R}^N, \qquad F|_{D_T} \in L_2(D_T) \quad\forall\, T > 0, \end{equation} \tag{4.1} $$
where $D_\infty := \Omega \times (0, \infty)$, and under certain assumptions about the nonlinear vector-valued function $f$ we show that there exists a positive number $T_0=T_0(F)$ such that for $0<T<T_0$ problem (1.1)–(1.3) has at least one generalized solution $u \in W_0^{2,2k+1}(D_T)$ in the domain $D_T$ in the sense of Definition 1.1, whereas on the other hand, for large $T$ this problem can turn out unsolvable in $D_T$. We also distinguish a class of nonlinear vector functions $f$ such that for each $F$ satisfying (4.1) problem (1.1)–(1.3) is solvable in the domain $D_T$ for each $T>0$.

In accordance with (3.22), consider the following algebraic equation:

$$ \begin{equation} a_1 z^{\alpha}+b_1=z \end{equation} \tag{4.2} $$
with respect to the unknown positive number $z$, where $a_1=a_1(T)$ and $b_1=b_1(T)$ are as in (3.23).

For $T>0$ it is obvious from (3.19), (3.20) and (3.23) that $a_1>0$ and ${b_1 > 0}$. A simple analysis similar to the analysis for $\alpha =3$ in [17], Ch. VIII, § 35.4, Example 2, shows that

(1) for $0<b_1<b_0$, where

$$ \begin{equation} b_0=[\alpha^{-1/(\alpha-1)}-\alpha^{-\alpha/(\alpha-1)}]a_1^{-1/(\alpha-1)}, \end{equation} \tag{4.3} $$
equation (4.2) has two positive roots $z_1$ and $z_2$. For $b_1=b_0$ these roots coincide, and we have the single positive root
$$ \begin{equation*} z_1=z_2=z_0=(\alpha a_1)^{-1/(\alpha-1)}; \end{equation*} \notag $$

(2) if $b_1>b_0$, then (4.2) has no nonnegative roots.

Note that for $0<b_1<b_0$ we have the inequalities

$$ \begin{equation*} z_1<z_0=(\alpha a_1)^{-1/(\alpha-1)} <z_2. \end{equation*} \notag $$

By (3.17), (3.19), (3.20), (3.23) and (4.3) the condition $b_1<b_0$ is equivalent to

$$ \begin{equation} \begin{aligned} \, \notag g(T) &:= a_1^{\alpha/(\alpha-1)}(T)+a_1^{1/(\alpha-1)}(T)\bigl[ c_1^{-1}(n+3)^{1/2} (M^2 \operatorname{mes}\Omega)^{1/2}T^{1/2} \\ &\qquad+c_1^{-1}\|F\|_{L_2(D_T)} \bigr]<\alpha^{-1/(\alpha-1)}-\alpha^{-\alpha/(\alpha-1)}. \end{aligned} \end{equation} \tag{4.4} $$

Since Lebesgue integral is absolutely continuous, by (4.1) we have

$$ \begin{equation*} \lim_{T \to 0}\|F\|_{L_2(D_T)} =0, \end{equation*} \notag $$
so that it follows from (3.15), (3.19), (3.23) and (4.4) that
$$ \begin{equation} \lim_{T \to 0}g(T)=0. \end{equation} \tag{4.5} $$

On the other hand, since $\alpha > 1$, the right-hand side of (4.4) is positive. Hence by (4.5) there exists a positive number $T_0=T_0(F)$ such that we have $b_1< b_0$ if

$$ \begin{equation} 0<T<T_0(F). \end{equation} \tag{4.6} $$
Thus, if $T$ satisfies (4.6), then the operator
$$ \begin{equation*} K\colon W_0^{2,2k+1}(D_T) \to W_0^{2,2k+1}(D_T) \end{equation*} \notag $$
acting by formula (3.6) takes the ball
$$ \begin{equation*} B(0,z_2):=\bigl\{u \in W_0^{2,2k+1}(D_T)\colon \|u\|_{W_0^{2,2k+1}(D_T)} \leqslant z_2 \bigr\} \end{equation*} \notag $$
to itself, where $z_2=z_2(T)$ is the greatest positive root of equation (4.2). In fact, if $u \in B(0,z_2)$, then by (3.22) and (4.2) we have
$$ \begin{equation*} \|K u\|_{W_0^{2,2k+1}(D_T)} \leqslant a_1 \|u\|_1^\alpha+b_1 \leqslant a_1 z_2^\alpha+b_1=z_2. \end{equation*} \notag $$
So, in view of the fact that $K$ is a compact continuous operator which takes the closed convex ball $B(0,z_2) \subset W_0^{2,2k+1}(D_T)$ to itself, by Schauder’s theorem ([17], Ch. VIII, 35.3, the theorem (Schauder’s principle)) equation (3.6) has at least one solution $u$ in the space $W_0^{2,2k+1}(D_T)$, which incidentally is a weak generalized solution of problem (1.1)–(1.3) in $W_0^{2,2k+1}(D_T)$ in the sense of Definition 1.1.

Thus, leaning on the above assumptions on the domain $\Omega$, nonlinear vector function $f$ and the right-hand side $F$ of equation (1.1), the above arguments ensure the following result.

Theorem 4.1. Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ with boundary $\partial \Omega$ of the class $C^2$, let the matrix $A=A(x,t)$, $(x,t) \in \Gamma$, satisfy condition (2.2), the nonlinear vector function $f$ satisfy conditions (1.10)–(1.13) and (3.13), and the vector function $F$ satisfy (4.1). Then there exists a positive number $T_0=T_0(F)$ such that for $0<T<T_0$ problem (1.1)–(1.3) has at least one weak generalized solution $ u \in W_0^{2,2k+1}(D_T)$ in the sense of Definition 1.1.

Now we present an example of a nonlinear vector function $f=f(u)$ for which problem (1.1)–(1.3) need not be solvable. Consider the following condition imposed on $f$: there exist numbers $l_1,\dots,l_N$, $\sum_{i=1}^{N}|l_i| \neq 0$, such that

$$ \begin{equation} \sum_{i=1}^{N}l_i f_i (u) \leqslant-\biggl|\sum_{i=1}^{N}l_i u_i\biggr|^\beta \quad\forall\, u \in \mathbb{R}^N,\qquad 1<\beta=\mathrm{const}<\frac{n+1}{n-1}. \end{equation} \tag{4.7} $$

For simplicity we assume below that $\Omega\colon |x|<1$.

Theorem 4.2. Let $f$ be a vector function satisfying conditions (1.10)–(1.13) and (4.7). Let $F^0=(F_1^0,\dots,F_N^0) \in L_2(D_T)$, $G\!=\!\sum_{i=1}^{N}l_i F_i^0 \!\geqslant\! 0$ and ${\|G\|_{L_2(D_T)} \!\neq\! 0}$. Then there exists a positive number $\mu_0=\mu_0 (G,\beta) $ such that for $\mu > \mu_0 $ problem (1.1)–(1.3) cannot have a weak generalized solution in the space $W_0^{2,2k+1}(D_T)$ for $F=\mu F^0$.

Proof. Assume that the hypotheses of the theorem hold and problem (1.1)–(1.3) has a solution $u \in W_0^{2,2k+1}(D_T)$ for each fixed $\mu > 0$. By Definition 1.1 the vector function $u$ satisfies equality (1.16) for each vector function $\varphi \in W_0^{2,2k+1}(D_T)$. Below we use the method of test functions (see [4]). As such a test vector function we can take $\varphi=(l_1 \varphi_0, l_2 \varphi_0,\dots,l_N \varphi_0)$, where $\varphi_0$ is a scalar function satisfying the conditions
$$ \begin{equation} \varphi_0 \in C_0^{4,4k+2}(\overline{D}_T), \qquad \varphi_0|_{\Gamma}= \frac {\partial \varphi_0}{\partial \nu} \bigg|_{\Gamma}=0 \quad \text{and}\quad \varphi_0|_{D_T}>0, \end{equation} \tag{4.8} $$
where the space $C_0^{4,4k+2}(\overline{D}_T)$ was defined in (1.4).

Integration by parts in (1.16) taking (4.8) into account and using the notation $v=\sum_{i=1}^{N}l_i u_i$ yields

$$ \begin{equation*} -\int_{D_T}\biggl[\sum_{i=1}^{N}l_i f_i(u) \biggr] \varphi_0 \,dx\,dt =\int_{D_T} v L_0 \varphi_0 \,dx\,dt -\mu \int_{D_T} G \varphi_0 \,dx\,dt, \end{equation*} \notag $$
which in view of (4.7) implies that
$$ \begin{equation} \int_{D_T} |v|^\beta \varphi_0 \,dx\,dt \leqslant \int_{D_T} v L_0 \varphi_0 \,dx\,dt -\mu \int_{D_T} G \varphi_0 \,dx\,dt, \end{equation} \tag{4.9} $$
where $L_0 := \partial^{2(2k+1)}/\partial t^{2(2k+1)}-\Delta^2$.

If in Young’s inequality with parameter $\varepsilon > 0$,

$$ \begin{equation*} ab \leqslant \frac{\varepsilon}{\beta}a^\beta +\frac{1}{{\beta'\varepsilon^{\beta'-1}}}b^{\beta'}, \qquad a,b \geqslant 0, \qquad \beta'=\frac{\beta}{{\beta-1}}, \end{equation*} \notag $$
we take $a=|v|\varphi_0^{1/\beta}$ and $b=|L_0 \varphi_0|/\varphi_0^{1/\beta}$, then, as $\beta'/\beta=\beta'-1$, we have
$$ \begin{equation} |v L_0 \varphi_0 |=|v|\varphi_0^{1/\beta} \frac{|L_0 \varphi_0 |} {\varphi_0^{1/\beta}} \leqslant \frac{\varepsilon}{\beta}|v|^\beta \varphi_0+\frac{1} {\beta'\varepsilon^{\beta'-1}}\, \frac{|L_0 \varphi_0 |^{\beta'}} {\varphi_0^{\beta'-1}}. \end{equation} \tag{4.10} $$
From (4.9) and (4.10) we obtain
$$ \begin{equation*} \biggl(1-\frac{\varepsilon}{\beta}\biggr) \int_{D_T}|v|^\beta \varphi_0 \,dx\,dt \leqslant \frac{1}{\beta'\varepsilon^{\beta'-1}} \int_{D_T}\frac{|L_0 \varphi_0 |^{\beta'}}{\varphi_0^{\beta'-1}}\,dx\,dt -\mu \int_{D_T} {G \varphi_0 \,dx\,dt}, \end{equation*} \notag $$
so that for $\varepsilon<\beta$
$$ \begin{equation} \int_{D_T} {|v|^\beta \varphi_0 \,dx\,dt} \leqslant \frac{\beta}{(\beta-\varepsilon)\beta'\varepsilon^{\beta'-1}} \int_{D_T} \frac{|L_0 \varphi_0 |^{\beta'}}{\varphi_0^{\beta'-1}}\,dx\,dt -\frac{\beta \mu}{\beta-\varepsilon} \int_{D_T}G \varphi_0 \,dx\,dt. \end{equation} \tag{4.11} $$

In view of the equalities

$$ \begin{equation*} \beta'=\frac{\beta}{\beta-1}, \qquad \beta=\frac{\beta'}{\beta'-1}\quad\text{and} \quad \min_{0<\varepsilon<\beta} \frac{\beta}{(\beta-\varepsilon)\beta'\varepsilon^{\beta'-1}}=1 \end{equation*} \notag $$
(the minimum is attained at $\varepsilon=1 $) it follows from (4.11) that
$$ \begin{equation} \int_{D_T}|v|^\beta \varphi_0 \,dx\,dt \leqslant \int_{D_T}\frac{|L_0 \varphi_0 |^{\beta'}}{\varphi_0^{\beta'-1}}\,dx\,dt -\beta' \mu \int_{D_T}G \varphi_0 \,dx\,dt. \end{equation} \tag{4.12} $$

It is easy to show that there exists a function $\varphi_0$ that, apart from (4.8), also satisfies the following condition:

$$ \begin{equation} \kappa_0=\int_{D_T}\frac{|L_0 \varphi_0 |^{\beta'}}{\varphi_0^{\beta'-1}}\,dx\,dt <+\infty. \end{equation} \tag{4.13} $$
In fact, it is easy to verify that for sufficiently large $m$ the function
$$ \begin{equation*} \varphi_0 (x,t)=[(1-|x|^2) t (T-t)]^m \end{equation*} \notag $$
satisfies (4.8) and (4.13).

Since $G \in L_2(D_T)$, $G|_{D_T} \geqslant 0$ and $\|G\|_{L_2(D_T)} \neq 0$ and since $\varphi_0 |_{D_T} >0$, we have

$$ \begin{equation} 0<\kappa_1=\int_{D_T} G \varphi_0 \,dx\,dt <+\infty. \end{equation} \tag{4.14} $$
Let $g(\mu)$ denote the right-hand side of (4.12), which is a linear function with respect to $\mu $. From (4.13) and (4.14) it is easy to see that
$$ \begin{equation} g(\mu)<0 \quad \text{for}\ \mu > \mu_0\quad\text{and} \quad g(\mu) > 0 \quad \text{for}\ \mu<\mu_0, \end{equation} \tag{4.15} $$
where
$$ \begin{equation*} g(\mu)=\kappa_0-\beta' \mu \kappa_1, \qquad \mu_0=\frac{\kappa_0}{\beta' \kappa_1} > 0. \end{equation*} \notag $$

By (4.15), for $\mu > \mu_0$ the right-hand side of (4.12) is negative, while the left-hand side is nonnegative. This contradiction proves Theorem 4.2.

Remark 4.1. Note that in Theorem 4.2 we assumed for simplicity that $\Omega\colon |x|<1$. However, this result also holds in the more general case when $\Omega$ is a convex domain with sufficiently smooth boundary $\partial \Omega$. Our assumption was connected with the construction of the test function $\varphi_0$ satisfying (4.8) and (4.13), in accordance with the formula

$$ \begin{equation} \varphi(x,t)=[(1-|x|^2)t(T-t)]^m, \end{equation} \tag{4.16} $$
where $m$ is positive and sufficiently large. If the boundary $\partial \Omega$ of the convex domain $\Omega$ is defined by an equation $ \omega (x)=0$, where $\nabla_x \omega |_{\partial \Omega} \neq 0$, $\omega|_\Omega >0$ and $\omega \in C^4(R^n)$, then instead of the test function defined by (4.16) we must take
$$ \begin{equation*} \varphi(x,t)=[\omega(x)t(T-t)]^m, \end{equation*} \notag $$
where $m$ is positive and sufficiently large. In this case Theorem 4.2 remains valid.

Remark 4.2. In the proof of Theorem 4.2 we can replace (4.7) by the more general condition

$$ \begin{equation} \sum_{i=1}^N l_if_i(u) \leqslant-d_0\biggl|\sum_{i=1}^N l_i u_i\biggr|^\beta \quad \forall\, u \in \mathbb{R}^N, \quad 1<\beta=\mathrm{const}<\frac{n+1}{n-1}, \end{equation} \tag{4.17} $$
where $d_0=\mathrm{const} >0$. In fact, the case (4.17) reduces to (4.7) if from $l_i$ we go over to $\widetilde{l}_i$ by the formula $l_i=\lambda \widetilde{l}_i$, where $\lambda=d_0^{1/(1-\beta)}$. As a result, we obtain (4.7) with $\widetilde{l}_i$ in place of $l_i$. Now we describe a class of vector functions $f$ satisfying (4.17):
$$ \begin{equation} f_i(u_1,\dots,u_N)=\sum_{j=1}^N a_{ij}|u_j|^{\beta_{ij}}+b_i, \qquad i=1,\dots,N, \end{equation} \tag{4.18} $$
where the constants $a_{ij}$, $\beta_{ij}$ and $b_i$ satisfy the inequalities
$$ \begin{equation} a_{ij}>0, \qquad 1<\beta_{ij}<\frac{n+1}{n-1} \quad\text{and} \quad \sum_{i=1}^N b_i >0, \qquad i,j= 1,\dots,N. \end{equation} \tag{4.19} $$
In this case we must take $l_1=\dots =l_N=-1$ in (4.17). In fact, in view of (4.19) we can fix constants $\alpha_0$ and $\beta$ such that
$$ \begin{equation} 0<a_0 \leqslant \min_{i,j} {a_{ij}}, \qquad \sum_{i=1}^N b_i- a_0N^2 \geqslant 0\quad\text{and} \quad 1<\beta <\beta_{ij}, \qquad i,j=1,\dots,N. \end{equation} \tag{4.20} $$
It is easy to verify that $|s|^{\beta_{ij}} \geqslant |s|^\beta -1$ for $ s \in (-\infty, \infty)$. Using the well-known inequality (see [18], Ch. IV, § 2.144, Example 1)
$$ \begin{equation*} \sum_{i=1}^N |y_i|^\beta > N^{1-\beta}\biggl|\sum_{i=1}^N y_i\biggr|^\beta \quad \forall\, y=(y_1,\dots,y_N) \in \mathbb{R}^N, \qquad \beta=\mathrm{const} >1, \end{equation*} \notag $$
in view of (4.18) and (4.19) we obtain
$$ \begin{equation} \begin{aligned} \, \notag \sum_{i=1}^N f_i(u_1,\dots,u_N) &\geqslant a_0 \sum_{i,j=1}^N |u_j|^{\beta_{ij}}+ \sum_{i=1}^N b_i \geqslant a_0 \sum_{i,j=1}^N (|u_j|^{\beta}-1)+\sum_{i=1}^N b_i \\ \notag &\geqslant a_0 N \sum_{j=1}^N |u_j|^{\beta}-a_0 N^2+\sum_{i=1}^N b_i \\ &\geqslant a_0 N^{2-\beta}\biggl|\sum_{j=1}^N u_j\biggr|^{\beta} +\sum_{i=1}^N b_i-a_0 N^2 \geqslant a_0 N^{2-\beta}\biggl|\sum_{j=1}^N u_j\biggr|^{\beta}. \end{aligned} \end{equation} \tag{4.21} $$

From (4.21) we conclude that if conditions (4.18) and (4.19) are satisfied, then (4.17) holds for $l_1=\dots =l_N=-1$ and $d_0=a_0 N^{2-\beta}$.

Remark 4.3. It follows from Theorems 4.1 and 4.2 that for a vector function $F$ satisfying (4.1) problem (1.1)–(1.3) is always solvable for sufficiently small $T>0$, although it is not necessarily solvable for large $T$. Below we distinguish a class of nonlinear vector functions $f$ such that for any vector function $F$ satisfying (4.1) problem (1.1)–(1.3) has at least one solution for each fixed $T>0$.

Consider the following condition on $f=(f_1,\dots,f_N)(s_0,s_1,\dots,s_{n+1})$:

$$ \begin{equation} \lim_{|s|\to \infty} \sup \frac{s_0 \cdot f(s_0,s_1,\dots,s_{n+1})}{|s|^2} \leqslant 0, \end{equation} \tag{4.22} $$
where $s_i \in \mathbb{R}^N$, $i=0,\dots,n+1$; $s_0 \cdot f$ is the standard scalar product in Euclidean space $\mathbb{R}^N$, $s=(s_0,s_1,\dots,s_{n+1}) \in \mathbb{R}^{(n+2)N}$ and $|s|^2=\sum_{i=0}^{n+1}|s_i|^2$.

Here is one class of vector functions $f=(f_1,\dots,f_N)$ satisfying (4.22):

$$ \begin{equation*} \begin{gathered} \, f_i=f_i^0(s_0,s_1,\dots,s_{n+1}) |s_{0i}|^{\beta_i}\operatorname{sign} s_{0i}, \\ f_i^0(s_0,s_1,\dots,s_{n+1}) \leqslant 0, \qquad \beta_i =\mathrm{const} >0, \\ \forall\, s=(s_0,s_1,\dots,s_{n+1}) \in \mathbb {R}^N, \qquad s_0=(s_{01},\dots,s_{0N}), \\ f_i^0 \in C(\mathbb{R}^N), \qquad i=1,\dots,N. \end{gathered} \end{equation*} \notag $$

Theorem 4.3. Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ with boundary $\partial \Omega$ in the class $C^2$, let $A=A(x,t)$, $(x,t) \in \overline{\Gamma}$, be a matrix satisfying (2.2) and assume that a nonlinear vector function $f=f(s_0,s_1,\dots,s_{n+1})$ satisfies conditions (1.10)–(1.13) and (4.22), while a vector function $F$ satisfies (4.1). Then for each fixed $T>0$ problem (1.1)–(1.3) has at least one weak generalized solution $u \in W_0^{2,2k+1}(D_T)$ in the sense of Definition 1.1.

Proof. We begin with an a priori estimate for a solution $u \in W_0^{2,2k+1}(D_T)$ of problem (1.1)–(1.3). Since $f \in C(\mathbb{R}^{(n+2)N})$, by (4.22), for each $\varepsilon > 0$ there exists $\widetilde{C}_\varepsilon \geqslant 0$ such that
$$ \begin{equation} s_0 \cdot f(s_0,s_1,\dots,s_{n+1}) \leqslant \widetilde{C}_\varepsilon +\varepsilon |s|^2 \quad\forall\, (s_0,s_1,\dots,s_{n+1}) \in \mathbb{R}^{(n+2)N}. \end{equation} \tag{4.23} $$

Setting $\varphi=u \in W_0^{2,2k+1}(D_T)$ in (1.16) and taking (4.23) and (2.4) into account, for each $\varepsilon >0$ we obtain

$$ \begin{equation} \begin{aligned} \, \notag &\|u\|_1^2 =\int_{D_T} u f(u, \nabla u) \,dx\,dt-\int_{D_T} {Fu\,dx\,dt} \\ \notag &\leqslant \widetilde{C}_\varepsilon \operatorname{mes} D_T\,{+}\,\varepsilon \int_{D_T} \biggl[u^2{+}\,\sum_{i=1}^{n} \biggl(\frac{\partial u}{\partial x_i} \biggr)^2{+}\,\biggl(\frac{\partial u}{\partial t} \biggr)^2 \biggr] \,dx\,dt\,{+}\int_{D_T}\!\biggl(\frac{1}{4\varepsilon} F^2\,{+}\,\varepsilon u^2 \biggr)\,dx\,dt \\ &= \widetilde{C}_\varepsilon \operatorname{mes}D_T+2\varepsilon \int_{D_T} \biggl[u^2+\sum_{i=1}^{n} \biggl(\frac{\partial u}{\partial x_i} \biggr)^2+\biggl(\frac{\partial u}{\partial t} \biggr)^2 \biggr] \,dx\,dt+\frac{1}{4\varepsilon} \|F\|_{L_2(D_T)}^2. \end{aligned} \end{equation} \tag{4.24} $$

It follows from (4.24) in view of (1.5), (2.1) and (2.5) that

$$ \begin{equation*} c_1^2\|u\|_0^2 \leqslant \|u\|_1^2 \leqslant \frac{{1}}{{4\varepsilon}}\|F\|_{L_2(D_T)}^2+ \widetilde{C}_\varepsilon \operatorname{mes}D_T+2 \varepsilon \|u\|_0^2, \end{equation*} \notag $$
so that for $\varepsilon=\frac{1}{{4}} c_1^2$ we obtain
$$ \begin{equation} \|u\|_0^2 \leqslant 2 c_1^{-4} \|F\|_{L_2 ({D_T})}^2 +2 c_1^{-2} \widetilde{C}_\varepsilon \operatorname{mes}D_T. \end{equation} \tag{4.25} $$

Inequality (4.25) yields the following a priori estimate for a solution $u \in W_0^{2,2k+1}(D_T)$ of problem (1.1)–(1.3):

$$ \begin{equation} \|u\|_0=\|u\|_{W_0^{2,2k+1}(D_T)} \leqslant c_3 \|F\|_{L_2 ({D_T})}+c_4, \end{equation} \tag{4.26} $$
where the constants $c_3=(2c_1^{-4})^{1/2}$ and $c_4=(2c_1^{-2} \widetilde{C}_\varepsilon \operatorname{mes} D_T)^{1/2}$ are independent of $u$ and $F$ and $ \varepsilon=\frac{1}{4}c_1^2$.

By Remark 3.1 problem (1.1)–(1.3) is equivalent to the functional equation (3.6), where the operator $K$ in the Hilbert space $W_0^{2,2k+1}(D_T)$ is continuous and compact. On the other hand the a priori estimate (4.26) for a solution of the equation $u=Ku$ in (3.6) also holds for a solution of equation $u=\tau K u$ with the parameter ${\tau \in [0,1]}$, for the same constants $c_3$ and $c_4$ as in (4.26). Hence by the Leray–Schauder fixed point theorem (see [17], Ch. VIII, § 35.5, Theorem 3) equation (3.6), and therefore also problem (1.1)–(1.3), has at least one weak generalized solution in the space $W_0^{2,2k+1}(D_T)$ in the sense of Definition 1.1, which completes the proof of Theorem 4.3.

§ 5. Uniqueness of the solution of problem (1.1)–(1.3)

Theorem 5.1. Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ with boundary $\partial \Omega$ of class $C^2$, let the matrix $A=A(x,t)$, $(x,t) \in \overline{\Gamma}$, satisfy (2.2), the nonlinear vector function $f=f(u)$, which only depends on $u \in \mathbb{R}^N$, satisfy conditions (1.10)–(1.12), and also let

$$ \begin{equation} (f(u)-f(v)) \cdot (u-v) \leqslant 0 \quad\forall\, u, v \in \mathbb{R}^N. \end{equation} \tag{5.1} $$
Then for each $F=(F_1,\dots,F_N) \in L_2(D_T)$ problem (1.1)–(1.3) can have at most one weak generalized solution $u=(u_1,\dots,u_N)$ in the space $W_0^{2,2k+1}(D_T)$.

Proof. Let $F \in L_2 (D_T)$, and let $u^1$ and $u^2$ be two weak generalized solutions of problem (1.1)–(1.3) in the space $W_0^{2,2k+1}(D_T)$, so that by (1.16) we have the equalities
$$ \begin{equation} \begin{aligned} \, \notag &\int_{D_T} \biggl[\frac{\partial^{2k+1} u^i}{\partial t^{2k+1}} \, \frac{\partial^{2k+1} \varphi}{\partial t^{2k+1}}+\Delta u^i \Delta \varphi \biggr]\,dx\,dt +\int_{\Gamma} A \, \frac{\partial u^i}{\partial \nu} \, \frac{\partial \varphi}{\partial \nu}\,ds \\ &\qquad\qquad - \int_{D_T} f(u^i) \varphi \,dx\,dt =- \int_{D_T} F \varphi \,dx\,dt \quad\forall\, \varphi \in W_0^{2,2k+1}(D_T), \qquad i=1,2. \end{aligned} \end{equation} \tag{5.2} $$

From (5.2), for the difference $v=u^2-u^1$ we obtain

$$ \begin{equation} \begin{aligned} \, \notag &\int_{D_T} \biggl[\frac{\partial^{2k+1} v}{\partial t^{2k+1}} \, \frac{\partial^{2k+1} \varphi}{\partial t^{2k+1}}+\Delta v \Delta \varphi \biggr]\,dx\,dt+\int_{\Gamma}A \, \frac{\partial v}{\partial \nu} \, \frac{\partial \varphi}{\partial \nu}\,ds \\ &\qquad =\int_{D_T} \bigl(f(u^2)-f(u^1) \bigr) \varphi \,dx\,dt \quad\forall\, \varphi \in W_0^{2,2k+1}(D_T). \end{aligned} \end{equation} \tag{5.3} $$

Setting $\varphi=v \in W_0^{2,2k+1}(D_T)$ in (5.3) and taking (2.4) into account we see that

$$ \begin{equation} \|v\|_1^2=\int_{D_T}\bigl(f(u^2)-f(u^1)\bigr)(u^2-u^1) \,dx\,dt. \end{equation} \tag{5.4} $$

It follows from (5.4) in view of (2.5) and (5.1) that

$$ \begin{equation*} c_1 \|v\|_0 \leqslant \|v\|_1 \leqslant 0, \end{equation*} \notag $$
which means that $v=0$, that is, $u^2=u^1$. Theorem 5.1 is proved.

Theorems 4.3 and 5.1 yield the following result.

Theorem 5.2. Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ with boundary $\partial \Omega$ of class $C^2$, let the matrix $A=A(x,t)$, $(x,t) \in \overline{\Gamma}$, satisfy (2.2), and let the nonlinear vector function $f=f(u)$, which only depends on $u \in \mathbb{R}^N$, satisfy conditions (1.10)–(1.12), (5.1) and also

$$ \begin{equation*} \lim_{s_0 \to \infty} \sup \frac{s_0 \cdot f(s_0)}{|s_0|^2} \leqslant 0. \end{equation*} \notag $$
Then for each $F \in L_2(D_T)$ problem (1.1)–(1.3) has a unique weak generalized solution $u \in W_0^{2,2k+1}(D_T)$ in the sense of Definition 1.1.

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Citation: S. S. Kharibegashvili, B. G. Midodashvili, “On the solvability of the boundary value problem for one class of nonlinear systems of high-order partial differential equations”, Sb. Math., 215:6 (2024), 841–860
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\jour Sb. Math.
\yr 2024
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