The second initial-boundary value problem with integral displacement for second-order hyperbolic and parabolic equations

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\Section[n]{Introduction} The aim of this paper is to obtain new results on the solvability of spatially nonlocal boundary value problems with integral conditions for second-order hyperbolic and parabolic equations. The direction in the theory of differential equations associated with the study of the solvability of nonlocal problems with boundary conditions of the integral form, apparently, originates from the works of~[\citen{Cann}], [\citen{Kamsh}], published in 1963 and 1964, respectively. An important role in the development of this direction was played by the work [\citen{Ion}], published in 1977. Among the subsequent works (in general, very numerous), we will single out the works closest to the methods of this article [\citen{Bou}]--[\citen{PNS}], [\citen{Pul}]. Note that most of the works devoted to the study of the solvability of boundary value problems with boundary conditions of the integral form were related to the one-dimensional case with respect to spatial variables. In the work~[\citen{KgP}]

\Section{Solvability of non-local problems I and II}

Let's define the function $N_1 (x,y)$ as the solution of the problem
\begin{equation}\label{repin:eq6}
\Delta_xN_1(x,y) - N_1(x,y) = 0
\end{equation}
\begin{equation}\label{repin:eq7}
\frac{\partial N_1(x,y)}{\partial\nu} = N(x,y) \quad\mbox{\it at}\quad{x\in\Gamma, y\in\Omega}
\end{equation}
(the $y$ variable in this problem is a parameter).
\begin{newthm}{Theorem~1} Let the condition be satisfied
$c(x,t) \in C(\overline{Q});$
$N(x,y) \in C^3(\overline{\Omega} \times \overline{\Omega}).$
Then for any function $f(x,t)$ such that $f (x,t) \in L_2 (Q)$, $f_t(x,t) \in L_2 (Q)$, the non-local problem I has a solution $u (x,t)$ such that
$u(x,t) \in L_\infty (0,T;W^2_2(\Omega)), \quad u_t(x,t) \in L_\infty (0,T;W^1_2(\Omega)),$
$u_{tt}(x,t) \in L_\infty (0,T;L_2(\Omega)).$
\end{newthm}
\begin{proof}
Let $M$ be an operator defined on the space $L_2(\Omega)$ and matching the function $v (x)$ to the function $(Mv) (x)$:
$(Mv)(x) = v(x) - \int_{\Omega} N_1(x,y) v (y) dy.$

There are two possible cases: 1) the number 1 is not an eigenvalue of the operator $M$; 2) the number 1 is an eigenvalue of the operator $M$. In the first case, the operator $I-M$ will be a continuously invertible operator from $L_2(\Omega)$ in $L_2(\Omega)$ , and thus for the non-local problem I, all the working conditions of~[\citen{KgP}] will be met. Therefore, this problem will have a solution that belongs to the class required in the theorem.

Let case 2 now hold).

As you know [\citen{Tren}, гл.VI, \S 24], the spectrum of the operator $M$ consists of at most a countable set of eigenvalues, and this set can have only the number 0 as its limit point. Therefore, there exists a number $ \ epsilon_0$ such that $\epsilon_0 \in (0,1)$, and there are no eigenvalues of the operator $M$ in the interval $(1-\epsilon_0,1)$.

For the number $ \ epsilon$ from the interval $(1-\epsilon_0, 1)$, consider the problem: Find a function $u(x,t)$ that is in the cylinder $Q$ the solution of the equation~\eqref{repin:eq1} and such that the conditions \eqref{eq:ulg are satisfied for it:GeneralModel} and \eqref{repin:eq4}, as well as the condition
\begin{equation}\label{repin:eq8}
(1-\epsilon)\frac{\partial u(x,y)}{\partial\nu} - \int_{\Omega} N(x,y)u(y,t)dy\bigl |_{(x,t)\in S} = 0.
\end{equation}

The $(1-\epsilon) operator corresponds to this problem)I - M$~([\citen{KgP}]). Since the number $1- \ epsilon$ is not an eigenvalue of the operator $M$, then, according to~[\citen{KgP}], non-local problem \eqref{repin:eq1}, \eqref{eq:ulg:GeneralModel}, \eqref{repin:eq4}, \eqref{repin:eq8} has a solution $u^\epsilon(x,t)$ such that $u^\epsilon \in L_\infty(0,T; W^2_2(\Omega))$, $u^\epsilon_t \in L_\infty(0,T; W^1_2(\Omega))$, $u^\epsilon_{tt} \in L_\infty(0,T; L_2(\Omega))$. We show that for the family $\{u^\epsilon(x,t)\}$, there are a priori estimates sufficient to implement the limit transition procedure.

Consider the equality
$\int_{0}^{t}\int_{\Omega}L_1u^\epsilon u_\tau^\epsilon dxd\tau = \int_{0}^{t}\int_{\Omega}fu^\epsilon_\tau dxd\tau.$

By integrating parts by parts and using the conditions \eqref{eq:ulg:GeneralModel}, \eqref{repin:eq4}, \eqref{repin:eq8} it is not difficult to convert this equality to the form
\begin{equation}\label{repin:eq9}
\frac{1}{2}\int_{\Omega}[u^\epsilon_t(x,t)]^2dx + \frac{1}{2}\sum_{i=1}^{n}\int_{\Omega}[u^\epsilon_{x_i}(x,t)]^2dx =
\end{equation}
$=\frac{1}{1-\epsilon}\int_{0}^{t}\int_{\Gamma}(\int_{\Omega}N(x,y)u^\epsilon_\tau(y,\tau)dy)u^\epsilon(x,\tau) dsd\tau-$
$-\frac{1}{1-\epsilon}\int_{\Gamma}(\int_{\Omega}N(x,y)u^\epsilon(y,t)dy)u^\epsilon(x,t)ds-\int_{0}^{t}\int_{\Omega}cu^\epsilon u^\epsilon_\tau dxd\tau + \int_{0}^{t}\int_{\Omega}fu^\epsilon_\tau dxd\tau.$

For the function $u^\epsilon(x,t)$, there is an inequality
\begin{equation}\label{repin:eq10}
\int_{\Gamma}[u^\epsilon(x,t)]^2ds \leq \delta\sum_{i=1}^{n}\int_{\Omega}[u^\epsilon_{x_i}(x,t)]^2dx + c(\delta)\int_{\Omega}[u^\epsilon(x,t)]^2dx,
\end{equation}
\begin{equation}\label{repin:eq11}
\int_{\Omega}[u^\epsilon(x,t)]^2ds \leq T\int_{0}^{t}\int_{\Omega}[u^\epsilon_{\tau}(x,\tau)]^2dxd\tau.
\end{equation}

In the first of these inequalities, $ \ delta$ is an arbitrary positive number, and this inequality itself is a consequence of the embedding theorem~[\citen{Lad}, ch.II, \S 2]. The second inequality is proved elementary using the Newton-Leibniz formula.

Using the inequalities \eqref{repin:eq10} and \eqref{repin:eq11}, selecting the number $ \ delta$ small, applying Young's inequality, and finally using Gronwall's lemma, we get that the consequence of the equality \eqref{repin:eq9} will be an a priori estimate
\begin{equation}\label{repin:eq12}
\int_{\Omega}[u^\epsilon_t(x,t)]^2dx + \sum_{i=0}^{n}\int_{\Omega}[u^\epsilon_{x_i}(x,t)]^2dx \leq R_1\int_{Q}f^2dxdt,
\end{equation}
the constant $R_1$ in which is defined only by the functions $c(x,t)$ and $N (x,t)$, as well as by the number $T$ and the domain $\Omega$.

In the next step, consider the equality
$\int_{0}^{t}\int_{\Omega}(L_1 u^\epsilon)_\tau u^\epsilon_{\tau\tau}dxdt = \int_{0}^{t}\int_{\Omega} f_\tau u^\epsilon_{\tau\tau}dxd\tau.$

In fact, repeating the previous reasoning, but using the estimate \eqref{repin:eq12} and taking into account that the equality $u^ \ epsilon_{tt}(x,0) = f (x,0)$ holds, we get that for the functions $u^\epsilon(x,t)$ the second a priori estimate holds
\begin{equation}\label{repin:eq13}
\int_{\Omega}[u^\epsilon_{tt}(x,t)]^2dx + \sum_{i=2}^{n}\int_{\Omega}[u^\epsilon_{x_it}(x,t)]^2dx \leq R_2\int_{Q}(f^2 + f^2_t) dxdt,
\end{equation}
with a constant $R_2$, defined only by the functions $c(x,t)$ and $N (x,t)$, as well as the number $T$ and the domain $\Omega$.

Using the inequalities \eqref{repin:eq12} and \eqref{repin:eq13}, it is not difficult to show that for the function $u^\epsilon(x,t)$, the third a priori bound holds
\begin{equation}\label{repin:eq14}
\int_{\Omega}[\Delta u^\epsilon(x,t)]^2dx \leq R_3\int_{Q}(f^2 + f^2_t) dxdt,
\end{equation}
the constant $R_3$ in which is again defined only by the functions $c(x,t)$ and $N (x,t)$, the number $T$, and the domain $\Omega$.

Define a function $w^\epsilon(x,t)$:
$w^\epsilon(x,t) = (1 - \epsilon)u^\epsilon(x,t) - \int_{\Omega} N_1(x,y)u^\epsilon(y,t)dy.$

There are equalities
$L_1w^\epsilon(x,t) = (1 - \epsilon)f - \int_{\Omega} N(x,y)u^\epsilon_{tt}(y,t)dy +$
$+ \int_{\Omega}[\Delta_xN(x,y)-c(x,t)N(x,y)]u^\epsilon(y,t)dy,$
$\frac{\partial w^\epsilon(x,t)}{\partial\nu}\big |_{(x,t)\in S}.$

Multiplying the first of these equalities by the function $w^ \ epsilon_t(x,t) - \Delta w^\epsilon_t(x,t)$, integrating by celindr with a variable height (variable upper limit) using the second equality and inequalities
\eqref{repin:eq12}-\eqref{repin:eq14}, we get that for the functions $w^\epsilon(x,t)$ there are estimates similar to the estimates obtained above for the functions $u^\epsilon(x, t)$, but with other constants $R'_1$, $R'_2$, $R'_3$ on the right side.

From the proved estimates for the function $w^\epsilon (x,t)$, as well as from the inversion of the derivative $ \ frac{\partial w^\epsilon(x,t)}{\partial\nu}$ to zero on the boundary and the second main inequality for the elleptic operator~[\citen{Lad}, ch. III, \S 8], it follows that for the function $w^\epsilon(x,t)$, inclusions are performed
$w^\epsilon(x,t) \in L_\infty (0,T;W^2_2(\Omega))$, $w^\epsilon_t(x,t) \in L_\infty (0,T;W^1_2(\Omega)),$
$w^\epsilon_{tt}(x,t) \in L_\infty (0,T;L_2(\Omega))$, and the norms of the functions $w^\epsilon(x,t)$, $w^\epsilon_t(x,t)$, $w^\epsilon_{tt} (x,t)$ in the corresponding spaces will be bounded uniformly by $\epsilon$. But then the functions $u^\epsilon(x,t)$, $u^\epsilon_t(x,t)$, $u^\epsilon_{tt}(x,t)$ will also have the same inclusions, and the norms of these functions in the spaces $L_\infty (0, T;W^2_2(\Omega))$, $L_\infty (0,T;W^1_2(\Omega))$, $L_\infty (0,T;L_2(\Omega))$ will be bounded evenly by $\epsilon$.

The established estimates and inclusions allow for the standard procedure of the limit transition.

Select the sequence $\{\epsilon_m\}^\infty_{m=1}$ so that $\epsilon_m\in(1-\epsilon_0,1)$, $\epsilon_m \to 0$ is executed when $m \to \infty$.
Let $u_m(x,t)$ be the solution to \ eqref{repin:eq1}, \eqref{eq:ulg:GeneralModel}, \eqref{repin:eq4}, \eqref{repin:eq8} in the case of $\epsilon = \epsilon_m$.
The families $\{u_m(x,t)\}^\infty_{m=1}$, $\{u_{mt}(x, t)\}^\infty_{m=1}$, $\{u_{mtt}(x, t)\}^\infty_{m=1}$ are uniformly bounded by $m$ in the spaces $L_\infty (0, T;W^2_2(\Omega))$, $L_\infty (0,T;W^1_2(\Omega)),$ $ L_\infty (0,T;L_2(\Omega))$ respectively. These estimates and theorems attachments ~[\citen{Sobol}-\citen{Tri}] follows that there exists a sequence $\{m_k\}^\infty_{k=1}$
natural numbers and the function $u(x,t)$ such that for $k \to \infty$ is the place of convergence
$$u_{m_k}(x,t) \to u(x,t) \quad\mbox{\it is weak in space}\quad W^2_2(Q),$$
$$u_{m_k}(x,t) \to u(x,t) \quad\mbox{\it is strongly in space}\quad W^1_2(Q),$$
$$u_{m_kx_i}(x,t) \to u_{x_i}(x,t) \quad\mbox{\it is strongly in space}\quad L_2(S), i=1,.., n,$$
$$\epsilon_{m_k}\frac{\partial u_{m_k}(x,t)} {\partial\nu} \to 0 \quad\mbox{\it is strongly in space}\quad L_2 (S).$$

Obviously, the limit function $u(x,t)$ will be the solution from the space $W^2_2 (Q)$ of the non-local problem I. Moreover,for the function $u(x, t)$, the estimates of \eqref{repin:eq12}-\eqref{repin:eq14} will be preserved, as well as the estimates of the derivatives of $u_{x_ix_j}(x, t)$ in the space $ L_\infty (0, T;L_2(\Omega))$. This means that the function $u(x,t)$ will be the solution of the non-local problem I from the required class.
\end{proof}

\begin{newthm}{Theorem~2} Let the condition be satisfied
$c(x,t) \in C(\overline{Q});$
$N(x,y) \in C^2(\overline{\Omega} \times \overline{\Omega}).$
Then for any function $f(x,t)$ such that $f (x,t) \in L_2 (Q)$ the non-local problem II has a solution $u (x,t)$ such that
$u(x,t) \in L_2 (0,T;W^2_2(\Omega)), \quad u_t(x,t) \in L_2(Q).$
\end{newthm}

\begin{proof}
Let's turn to the $M$operator again. If the number 1 is not an eigenvalue of the operator $M$, then the non-local problem II has a solution $u (x,t)$ belonging to the required class - this is not difficult to prove using the methods of~[\citen{KgP}]. If the number 1 is the eigenvalue of the operator $M$, then you can again use the regularization method. The a priori estimates necessary for the implementation of the limit transition procedure are not difficult to obtain by analyzing the equality sequence
$\int_{0}^{t}\int_{\Omega}L_2u^\epsilon u^\epsilon dxd\tau = \int_{0}^{t}\int_{\Omega}fu^\epsilon dxd\tau,$
$\int_{0}^{t}\int_{\Omega}L_2u^\epsilon u^\epsilon_\tau dxd\tau = \int_{0}^{t}\int_{\Omega}fu^\epsilon_\tau dxd\tau$
($u^\epsilon(x,t)$ - solution of the equation~\eqref{repin:eq5} under the regularized condition~\eqref{repin:eq6}, as well as the condition~\eqref{eq:ulg:GeneralModel}). The embedding theorems allow us to use the obtained estimates to select a sequence from the family $\{u^ \ epsilon(x,t)\}$ that converges to the solution of the non-local problem II.
\end{proof}

\Section{Comments and additions}

1. For the solution of non-local problems I and II, the uniqueness property holds when the conditions of Theorem 1 and 2 are met. This property is not difficult to prove by analyzing the equalities
$<$
\int_{0}^{t}\int_{\Omega}L_1u u_\tau dxd\tau = \int_{0}^{t}\int_{\Omega}fu^\epsilon dxd\tau,

$<$
\int_{0}^{t}\int_{\Omega}u dxd L_2u\tau = \int_{0}^{t}\int_{\Omega}fu^\epsilon_\dxd tau\tau

($u(x,t)$ is a solution of a nonlocal task I or II of the theorems of existence of classes). And, note, the reversibility or irreversibility of the $M$ operator does not play any role.

2. In non-local problems I or II, the Laplace operator can be replaced by a more general linear elliptic operator of the second order. Further, the boundary conditions~\eqref{repin:eq2} of non-local problems I or II can be replaced by the condition of the third boundary value problem
$\frac{\partial u(x,t)}{\partial\nu} + \sigma(x) u(x,t) - \int_{\Omega} N(x,y)u(y,t) dy\bigg |_{(x,t) \in S} = 0.$

3. By combining the method of this paper and the work~[\citen{KgP}], we can obtain new conditions for the solvability of integral analogues of the second or third boundary value problems for some other classes of differential equations - for example, for the equation
$u_{ttt} + \ alpha u_{tt} - \beta\Delta u_t - \Delta u = f$
simulating high-density sound waves~[\citen{Sh}] (here $\alpha$ and $\beta$ are positive constants).

About the authors

Alexander I. Kozhanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences; Samara State Technical University

Email: kozhanov@math.nsc.ru
ORCID iD: 0000-0003-4376-4003
SPIN-code: 9132-3234
Scopus Author ID: 55892833300
ResearcherId: R-5686-2016
http://www.mathnet.ru/person18220

Dr. Phys. & Math. Sci., Professor; Chief Researcher; Lab. of Differential and Difference Equations¹; Professor; Dept. of Higher Mathematics²

4, Acad. Koptyug pr., Novosibirsk, 630090, Russian Federation; 244, Molodogvardeyskaya st., Samara, 443100, Russian Federation

Alexandra V. Dyuzheva

Samara State Technical University

Author for correspondence.
Email: duzhevaalexandra@yandex.ru
ORCID iD: 0000-0002-3284-5302
Scopus Author ID: 57221800436
http://www.mathnet.ru/person53016

Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Higher Mathematics

244, Molodogvardeyskaya st., Samara, 443100, Russian Federation

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