On Solvability of Some Inverse Problems for a Pseudoparabolic Equation with Multiple Involution

Abstract

In this paper, solvability of some inverse problems for a nonlocal analog of a pseudoparabolic equation is studied. The nonlocal analog of a pseudoparabolic equation is formed using transformations that have the involution property. Two types of inverse problems are considered. In the first problem, in addition to the solution, a function in the right-hand side of the equation depending on the spatial variable is determined. In the second problem, a function depending on the time variable is found. The first problem is solved using the Fourier method, and the second problem is solved by reducing to the integral Volterra equation.

1. Introduction

In recent years, considerable attention from researchers in the field of differential equations has been directed toward equations with transformed arguments. Within this class, differential equations involving transformed involute arguments play a particularly significant role. It should be noted that a mapping $S:R^n\to R^n$ is called an involution if it satisfies the condition $S^{2}x=x$ for all $x\in R^n$.

Plenty of works [1,2,3,4,5,6,7,8,9,10] are devoted to the study of the solvability of direct and inverse problems for involutionary analogs of classical equations of mathematical physics.

Direct and inverse problems were also studied for non-classical equations of the type as follows.

$${\displaystyle \frac{\partial }{\partial t}}\left[u(t,x)-Lu(t,x)\right]-Lu(t,x)=f,$$

where L is a linear differential operator of even order. As noted in [11], equations of this type belong to the class of pseudoparabolic equations.

Mathematical models of various processes described by pseudoparabolic equations are considered in [12,13]. In particular, in [14], it is indicated that under certain assumptions, pseudoparabolic equations in the one-dimensional case describe the filtration of liquid in porous media, and the corresponding solution is interpreted as soil moisture.

In [15,16,17,18,19,20,21,22,23,24,25,26,27,28], various initial-boundary value problems with local and nonlocal conditions were investigated for pseudoparabolic equations. In [29,30,31,32,33,34], numerical methods for solving such problems were considered.

Inverse problems for pseudoparabolic equations were first studied by Rundell in [26], where the problems of identification of an unknown source function f were considered. In this work, theorems on the global existence and uniqueness of solutions are proved in the cases where the function f depends only on one variable: either x or the time variable t.

In recent papers [35,36,37,38], in the one-dimensional case, the questions of solvability of some direct and inverse problems for pseudoparabolic equations with involution have been investigated. In particular, in the work of D. Serikbaev [38], in the domain $\mathsf{\Omega }=\left\{(t,x):0<t<T,-\pi <x<\pi \right\}$, the pseudoparabolic equation with involution is as follows:

$$D_t^{\alpha }\left[u(t,x)-u_{xx}(t,x)+\epsilon u_{xx}(t,Sx)\right]-u_{xx}(t,x)+\epsilon u_{xx}(t,Sx)=f(t,x),$$

(1)

where $Sx=\pi -x$, $0<\alpha \le 1$ and $D_t^{\alpha }$ is a fractional derivative in the Caputo sense that is studied.

In this paper, the author considers the initial-boundary value problems with the conditions

$$u(0,x)=\varphi \left(x\right),-\pi \le x\le \pi ,$$

(2)

$$u(t,-\pi )=u(t,\pi )=0.$$

(3)

For this case, classical and generalized solutions of the direct problem (1)–(3) are found. The inverse problem, for which, in addition to the solution $u(t,x)$, when the right-hand side is determined depends only on the spatial variable, i.e., when $f(t,x)\equiv f\left(x\right)$, is also studied.

When studying problems (1)–(3) with respect to the spatial variable, a spectral problem arises for a one-dimensional equation with involution, as follows:

$$-y^{″}\left(x\right)+\epsilon y^{″}(\pi -x)=\lambda y\left(x\right),0<x<\pi ,y\left(0\right)=y\left(\pi \right)=0.$$

(4)

Solutions to the problems are sought in the form of Fourier series for the system of eigenfunctions of problem (4). Theorems on the existence and uniqueness of solutions to these problems are proved.

Later, in [35,37], the authors used numerical methods to solve inverse problems for Equation (1) and to determine the right-hand side of the equation depending on the spatial variable. We should also note [36], where the boundary control problems were studied for Equation (1).

It should be pointed out that the initial boundary value problems for pseudoparabolic equations with involution have been studied only in the one-dimensional case. Therefore, it is important to study differential equations of type (1) in the general n-dimensional case, where $n\ge 1$. In this paper, we study a pseudoparabolic equation with involution in a multidimensional domain, with involution in all spatial variables. In [39,40], a parabolic equation with involution was also considered. However, in [39], unlike the present paper, the problems were studied only in a rectangular domain. When studying our problems using the Fourier method, a spectral problem for the operator $L_a$ (see: Section 2) arises, which has not previously been studied in the context of problems for pseudo-differential equations with involution. The spectral problem for this operator is also a new element and it was previously considered separately only in [41]. In this work, two statements of inverse problems are considered. In the first problem, the right-hand side depends only on this and such a problem for pseudo-differential equations with involution was previously studied only in the one-dimensional case in [35,36,37,38]. The overdetermination condition in the second problem differs from the conditions set in previous works [39,40]. It is shown that in this formulation, the problem is well-posed. This paper analyzes the applicability of the Fourier method in the case of a high-order equations, namely, pseudo-differential equations with involution, and shows correctness of the considered problems. As we have already mentioned, such equations were previously studied only in the one-dimensional case.

2. Statement of Problems

Let $Q=\left\{(t,x):0<t<T,x\in \mathsf{\Omega }\right\}$ be a cylindrical domain, where $\mathsf{\Omega }=\left\{x\in R^n:\left|x\right|<1\right\}$ is a unit ball and $\partial \mathsf{\Omega }$ is a unit sphere. Let also $S_1,S_2,\dots ,S_l$ be a set of real symmetric commutative matrices $S_i{S}_j=S_j{S}_i$ such that $S_j^2=I$. As an example, we can consider the mapping matrix $S_{1}x=\left(-x_1,x_2,\dots ,x_n\right)$.

The product of such mappings is also an involution. The total number of all possible products of such mappings, taking into account the identity mapping $S_{0}x=x$, is equal to $2^l$. Let i be the summation index from the range $\left\{0,1,\dots ,2^{l-1}\right\}$ and $i={\left(i_l{i}_{l-1}\dots i_1\right)}_2\equiv 2^0{i}_1+\dots +2^{l-2}{i}_{l-1}+2^{l-1}{i}_l$ be its representation in the binary system of calculus, where $i_k$ takes one of the values either 0 or 1. Then, for any index $i\in \left\{0,1,\dots ,2^l-1\right\}$, we can consider mappings of the type $S_1^{i_1}\dots S_{l-1}^{i_{l-1}}{S}_l^{i_l}x$. Using these mappings, we can introduce the following operator:

$$L_{a}v\left(x\right)=\sum _{i=0}^{2^l-1}{a}_i\mathsf{\Delta }v\left(S_1^{i_1}\dots S_l^{i_l}x\right),$$

where $a_i$ is some set of real numbers, $\mathsf{\Delta }$ is the Laplace operator. Let us denote the operator $L_a$ a nonlocal analog of the Laplace operator.

Let us consider the following equation:

$${\displaystyle \frac{\partial }{\partial t}}\left(u(t,x)-L_{a}u(t,x)\right)-L_{a}u(t,x)=F(t,x),(t,x)\in Q$$

(5)

in the domain Q.

In this paper, the following inverse problems for Equation (5) are studied.

Problem 1.

Let $F(t,x)=f\left(x\right)$. Find a pair of functions $\left\{u(t,x),f\left(x\right)\right\}$ satisfying Equation (5), the initial condition

$$u(0,x)=\varphi \left(x\right),x\in \overline{\mathsf{\Omega }},$$

(6)

the boundary condition is following

$$u(t,x)=0,0<t<T,x\in \partial \mathsf{\Omega },$$

(7)

and the redefinition condition is as follows:

$$u(T,x)=\psi \left(x\right),x\in \overline{\mathsf{\Omega }},$$

(8)

where $\varphi \left(x\right),\psi \left(x\right)$ are given functions.

Problem 2.

Let $F(t,x)=g\left(t\right)f\left(x\right)$. Find a pair of functions $\left\{u(t,x),g\left(t\right)\right\}$ satisfying Equation (5), conditions (6), (7) and the additional condition

$$u(t,x_0)-L_{a}u(t,x_0)=h\left(t\right),0\le t\le T,$$

(9)

where $x_0$ is an arbitrary point in the domain Ω, $f\left(x\right)$ and $h\left(t\right)$ are given functions.

A regular solution to problem 1 is a pair of smooth functions $\left\{u(t,x),f\left(x\right)\right\}$: $f\left(x\right)\in C\left(\overline{\mathsf{\Omega }}\right)$; the function $u(t,x)$ and all its derivatives in Equation (5) are continuous in a closed domain $\overline{Q}$ and satisfy conditions (5)–(7) in the classical sense.

3. Information on the Spectral Problem and Convergence of Fourier Series

In this section, we will present some well-known statements about the convergence of Fourier series in the system of eigenfunctions of the following spectral problem:

$$L_{a}v\left(x\right)+\lambda v\left(x\right)=0,x\in \mathsf{\Omega },$$

(10)

$$v\left(x\right)=0,x\in \partial \mathsf{\Omega }.$$

(11)

Let us assume that $w_k\left(x\right)$ and ${\eta }_k,k=1,2,\dots$, respectively, are eigenfunctions and eigenvalues of the classical Dirichlet problem

$$\mathsf{\Delta }w\left(x\right)+\eta w\left(x\right)=0,x\in \mathsf{\Omega },w\left(x\right)=0,x\in \partial \mathsf{\Omega }.$$

(12)

Let $p,q\in \left\{0,1,\dots ,2^l-1\right\},$ $p={\left(p_l\dots p_2{p}_1\right)}_2,q={\left(q_l\dots q_2{q}_1\right)}_2$ be a binary representation of numbers p and q, and $p\otimes q=p_1{q}_1+p_2{q}_2+\dots +p_l{q}_l$. Let us introduce the following functions

$$v_{k,p}\left(x\right)={\displaystyle \frac{1}{2^l}}\sum _{q=0}^{2^l-1}{(-1)}^{p\otimes q}{w}_k\left(S_l^{q_l}\dots S_1^{q_1}x\right).$$

On the other hand, for a system of functions $v_{k,p}\left(x\right)$, the following assertion was proved [39].

Lemma 1.

The elements of the system ${\left\{v_{k,p}\left(x\right)\right\}}_{k=1}^{\infty },0\le p\le 2^l-1$ are eigenfunctions of the problem (10) and (11). The corresponding eigenvalues are determined by the following equality:

$${\lambda }_{k,p}={\eta }_k\sum _{i=0}^{2^l-1}{(-1)}^{p\otimes i}{a}_i\equiv {\eta }_k{\epsilon }_p,{\epsilon }_p=\sum _{i=0}^{2^l-1}{(-1)}^{p\otimes i}{a}_i.$$

Moreover, ${\left\{v_{k,p}\left(x\right)\right\}}_{k=0}^{\infty },0\le p\le 2^l-1$ is a complete and orthonormal system in the space $L_2\left(\mathsf{\Omega }\right)$.

The proof of Lemma 1 is presented in [39] (Theorems 6 and 7).

Let us express the eigenfunctions $v_{k,p}\left(x\right)$ and eigenvalues ${\lambda }_{k,p}$ as follows:

$$v_{2^{l}k-p}\left(x\right)={\displaystyle \frac{1}{2^l}}\sum _{q=0}^{2^l-1}{(-1)}^{p\otimes q}{w}_{2^{l}k-p}\left(S_l^{q_l}\dots S_1^{q_1}x\right),$$

$${\lambda }_{2^{l}k-p}={\epsilon }_p{\eta }_{2^{l}k-p},k=1,2,\dots ,p=0,1,\dots ,2^l-1.$$

Subsequently, let us assume that the inequalities ${\epsilon }_p={\displaystyle \sum _{i=0}^{2^l-1}}{(-1)}^{p\otimes i}{a}_i>0$ are valid for all $p\in \left\{0,1,\dots ,2^l-1\right\}$.

Further, the symbol C will be used to denote arbitrary constants, distinguished when necessary using various subscripts, and the symbol $\left[{\displaystyle \frac{m}{n}}\right]$ will denote the integer part of the number ${\displaystyle \frac{m}{n}}$.

Let us present some assertions about eigenfunctions $w_k\left(x\right)$ and eigenvalues ${\eta }_k$ proved by V.A. Ilyin [42].

Lemma 2

([42] Lemma 1). For the system ${\left\{w_k\left(x\right)\right\}}_{k=1}^{\infty }$, the following statements are valid:

(1) series ${\displaystyle \sum _{k=1}^{\infty }}{\eta }_k^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{w}_k^2\left(x\right)$ converge uniformly in a closed domain $\overline{\mathsf{\Omega }}$;

(2) series ${\displaystyle \sum _{k=1}^{\infty }}{\eta }_k^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+2\right)}{\left[{\displaystyle \frac{\partial w_k\left(x\right)}{\partial x_i}}\right]}^2$ and ${\displaystyle \sum _{k=1}^{\infty }}{\eta }_k^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left[{\displaystyle \frac{{\partial }^2{w}_k\left(x\right)}{\partial x_i\partial x_j}}\right]}^2$ converge uniformly in an arbitrary closed subdomain ${\overline{\mathsf{\Omega }}}_0$ located strictly inside Ω.

Lemma 3

([42], Lemma 5). Let the function $g\left(x\right)$ satisfy the conditions

(1) $g\left(x\right)\in C^r\left(\overline{\mathsf{\Omega }}\right),{\displaystyle \frac{{\partial }^{r+1}g\left(x\right)}{\partial x_1^{r_1}\dots \partial x_n^{r_n}}}\in L_2\left(\mathsf{\Omega }\right),r_1+\dots +r_n=r+1,r\ge 1$,

(2) ${\left.g\left(x\right)\right|}_{\partial \mathsf{\Omega }}={\left.\mathsf{\Delta }g\left(x\right)\right|}_{\partial \mathsf{\Omega }}=\dots ={\left.{\mathsf{\Delta }}^{\left[{\displaystyle \frac{r}{2}}\right]}g\left(x\right)\right|}_{\partial \mathsf{\Omega }}=0$.

Then the number series ${\displaystyle \sum _{k=1}^{\infty }}{g}_k^2{\eta }_k^{r+1}$ converges, where $g_k=\left(g,w_k\right)$ and ${\mathsf{\Delta }}^{\left[r/2\right]}$ is the polyharmonic operator of order $\left[r/2\right]$.

As far as ${\left\{v_{2^{n}k-p}\left(x\right)\right\}}_{k=1}^{\infty },p=0,1,\dots ,2^l-1$ are eigenfunctions of problem (10) and (11) and form a complete orthonormal system in $L_2\left(\mathsf{\Omega }\right)$, then Lemma 2 implies the following assertions.

Corollary 1.

For the system ${\left\{v_{2^{n}k-p}\left(x\right)\right\}}_{k=1}^{\infty },p=0,1,\dots ,2^l-1$, the following statements are valid:

(1) series ${\displaystyle \sum _{p=0}^{2^l-1}}{\displaystyle \sum _{k=1}^{\infty }}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{v}_{2^{l}k-p}^2\left(x\right)$ converge uniformly in a closed domain $\overline{\mathsf{\Omega }};$

(2) series ${\displaystyle \sum _{p=0}^{2^l-1}}{\displaystyle \sum _{k=1}^{\infty }}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+2\right)}{\left[{\displaystyle \frac{\partial v_{2^{l}k-p}\left(x\right)}{\partial x_i}}\right]}^2$ and ${\displaystyle \sum _{p=0}^{2^l-1}}{\displaystyle \sum _{k=1}^{\infty }}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left[{\displaystyle \frac{{\partial }^2{v}_{2^{l}k-p}\left(x\right)}{\partial x_i\partial x_j}}\right]}^2$ converge uniformly in an arbitrary strictly closed subdomain ${\overline{\mathsf{\Omega }}}_0$ located inside Ω.

By assumption $\left({\displaystyle \sum _{i=0}^{2^l-1}}{(-1)}^{(p\otimes i)}{a}_i\right)>0$, there exist constants $C_1$ and $C_2$ such that the estimates hold

$$C_1{\eta }_{2^{l}k-p}\le {\lambda }_{2^{l}k-p}\le C_2{\eta }_{2^{l}k-p}.$$

(13)

Hence, we obtain the following assertion.

Corollary 2.

For the system ${\left\{v_{2^{l}k-p}\left(x\right)\right\}}_{k=1}^{\infty },p=0,1,\dots ,2^l-1$, the following statements are valid:

(1) series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\lambda }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{v}_{2^{l}k-p}^2\left(x\right)$ converge uniformly in a closed domain $\overline{\mathsf{\Omega }};$

(2) series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\lambda }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+2\right)}{\left[{\displaystyle \frac{\partial v_{2^{l}k-p}\left(x\right)}{\partial x_i}}\right]}^2$ and ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\lambda }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left[{\displaystyle \frac{{\partial }^2{v}_{2^{l}k-p}\left(x\right)}{\partial x_i\partial x_j}}\right]}^2$ converge uniformly in an arbitrary strictly closed subdomain ${\overline{\mathsf{\Omega }}}_0$ of the domain Ω.

Corollary 3.

Let the function $g\left(x\right)$ satisfy the conditions of Lemma 3 with $r\ge 1.$ Then, the number series ${\displaystyle \sum _{p=0}^{2^l-1}}{\displaystyle \sum _{k=1}^{\infty }}{\left|g_{2^{l}k-p}\right|}^2{\eta }_{2^{l}k-p}^{r+1}$ converges.

4. Uniqueness and Existence of a Solution to Problem 1

In Problem 1, the system of functions $v_{2^{l}k-p}\left(x\right)$ is an orthonormal basis of the space $L_2\left(\mathsf{\Omega }\right)$. Therefore, we can look for a solution to Problem 1 in the form

$$f\left(x\right)=\sum _{k=1}^{\infty }{f}_{2^{l}k-p}{v}_{2^{l}k-p}\left(x\right),$$

(14)

$$u(t,x)=\sum _{k=1}^{\infty }{u}_{2^{l}k-p}\left(t\right)v_{2^{l}k-p}\left(x\right),$$

(15)

where $f_{2^{l}k-p}$ and $u_{2^{l}k-p}\left(t\right)$ are the coefficients to be determined.

Let us assume that the functions $\varphi \left(x\right)$ and $\psi \left(x\right)$ can be expanded in a Fourier series with respect to the system $v_{k_1\dots k_n}\left(x\right)$:

$$\varphi \left(x\right)=\sum _{k=1}^{\infty }{\varphi }_{2^{l}k-p}{v}_{2^{l}k-p}\left(x\right),\psi \left(x\right)=\sum _{k=1}^{\infty }{\psi }_{2^{l}k-p}{v}_{2^{l}k-p}\left(x\right),$$

where ${\varphi }_{2^{l}k-p}=\left(\varphi ,v_{2^{l}k-p}\right)$ and ${\psi }_{2^{l}k-p}=\left(\psi ,v_{2^{l}k-p}\right)$.

Substituting the functions $f\left(x\right)$ and $u(t,x)$ into Equation (5), we obtain

$${\displaystyle \frac{\partial }{\partial t}}u(t,x)-{\displaystyle \frac{\partial }{\partial t}}{L}_{a}u(t,x)-L_{a}u(t,x)-f\left(x\right)=$$

$$=\sum _{k=1}^{\infty }\left[u_{2^{l}k-p}^{\prime }\left(t\right)+{\lambda }_{2^{l}k-p}{u}_{2^{l}k-p}^{\prime }\left(t\right)+{\lambda }_{2^{l}k-p}{u}_{2^{l}k-p}\left(t\right)-f_{2^{l}k-p}\right]v_{2^{l}k-p}\left(x\right).$$

From conditions (6) and (7) it follows that

$$0=u(0,x)-\varphi \left(x\right)=\sum _{k=1}^{\infty }\left[u_{2^{l}k-p}\left(0\right)-{\varphi }_{2^{l}k-p}\right]v_{2^{l}k-p}\left(x\right),$$

$$0=u(T,x)-\psi \left(x\right)=\sum _{k=1}^{\infty }\left[u_{2^{l}k-p}\left(T\right)-{\psi }_{2^{l}k-p}\right]v_{2^{l}k-p}\left(x\right).$$

For the coefficients $u_{2^{l}k-p}\left(t\right)$, we obtain the following problem:

$$u_{k_1\dots k_n}^{\prime }\left(t\right)+{\mu }_{2^{l}k-p}{u}_{2^{l}k-p}\left(t\right)={\displaystyle \frac{f_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}},0<t<T,$$

(16)

$$u_{2^{l}k-p}\left(0\right)={\varphi }_{2^{l}k-p},u_{2^{l}k-p}\left(T\right)={\psi }_{2^{l}k-p},$$

(17)

where ${\mu }_{2^{l}k-p}={\displaystyle \frac{{\lambda }_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}}$.

The solution to Equation (16) that satisfies the first condition in (17) is written as

$$u_{2^{l}k-p}\left(t\right)={\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}+{\displaystyle \frac{f_{2^{l}k-p}}{{\lambda }_{2^{l}k-p}}}\left(1-e^{-{\mu }_{2^{l}k-p}t}\right).$$

(18)

If we use the second condition of equality (17), we get

$${\psi }_{2^{l}k-p}=u(T,x)={\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}T}+{\displaystyle \frac{f_{2^{l}k-p}}{{\lambda }_{2^{l}k-p}}}\left(1-e^{-{\mu }_{2^{l}k-p}T}\right).$$

As $1-e^{-{\mu }_{2^{l}k-p}T}\ne 0$, then for $f_{2^{l}k-p}$, we obtain

$$f_{2^{l}k-p}={\lambda }_{2^{l}k-p}{\displaystyle \frac{{\psi }_{2^{l}k-p}-e^{-{\mu }_{2^{l}k-p}T}{\varphi }_{2^{l}k-p}}{1-e^{-{\mu }_{2^{l}k-p}T}}}.$$

(19)

Substituting the value of $f_{2^{l}k-p}$ from (19) into the right-hand side of equality (18) to solve problems (16) and (17), we obtain the following representation:

$$u_{2^{l}k-p}\left(t\right)={\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}t}-e^{-{\mu }_{2^{l}k-p}T}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\varphi }_{2^{l}k-p}+{\displaystyle \frac{1-e^{-{\mu }_{2^{l}k-p}t}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\psi }_{2^{l}k-p}.$$

Then, the function $u_{2^{l}k-p}^{\prime }\left(t\right)$ will get the following form:

$$u_{2^{l}k-p}^{\prime }\left(t\right)={\mu }_{2^{l}k-p}\left({\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}t}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\psi }_{2^{l}k-p}-{\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}t}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\varphi }_{2^{l}k-p}\right).$$

Further, as $0<{\mu }_{2^{l}k-p}\le 1,\delta =\underset{k,p}{min}\left[1-e^{-{\mu }_{2^{l}k-p}T}\right]>0$ and functions

$${\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}T}}{1-e^{-{\mu }_{2^{l}k-p}T}}},{\displaystyle \frac{1}{1-e^{-{\mu }_{2^{l}k-p}T}}},{\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}t}-e^{-{\mu }_{2^{l}k-p}T}}{1-e^{-{\mu }_{2^{l}k-p}T}}},{\displaystyle \frac{1-e^{-{\mu }_{2^{l}k-p}t}}{1-e^{-{\mu }_{2^{l}k-p}T}}}$$

are limited, we obtain the following estimates:

$$\left|f_{2^{l}k-p}\right|\le C_1\left(\left|{\varphi }_{2^{l}k-p}\right|+\left|{\psi }_{2^{l}k-p}\right|\right)\left|v_{2^{l}k-p}\left(x\right)\right|,$$

(20)

$$\left\{\begin{array}{c}\left|u_{2^{l}k-p}\left(t\right)\right|\le C_2\left(\left|{\varphi }_{2^{l}k-p}\right|+\left|{\psi }_{2^{l}k-p}\right|\right)\left|v_{2^{l}k-p}\left(x\right)\right|,\hfill \\ \left|u_{2^{l}k-p}^{\prime }\left(t\right)\right|\le C_3\left(\left|{\varphi }_{2^{l}k-p}\right|+\left|{\psi }_{2^{l}k-p}\right|\right)\left|v_{2^{l}k-p}\left(x\right)\right|.\hfill \end{array}\right.$$

(21)

Let us formulate the main assertion regarding Problem 1.

Theorem 1.

Let the functions $\varphi \left(x\right)$ and $\psi \left(x\right)$ satisfy the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$. Then, the solution to Problem 1 exists, is unique, and can be represented as

$$f\left(x\right)=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left[{\displaystyle \frac{1}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\psi }_{2^{l}k-p}-{\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}T}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\varphi }_{2^{l}k-p}\right]v_{2^{l}k-p}\left(x\right),$$

(22)

$$u(t,x)=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left[{\displaystyle \frac{e^{-{\mu }_{2^{l}k-p}t}-e^{-{\mu }_{2^{l}k-p}T}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\varphi }_{2^{l}k-p}+{\displaystyle \frac{1-e^{-{\mu }_{2^{l}k-p}t}}{1-e^{-{\mu }_{2^{l}k-p}T}}}{\psi }_{2^{l}k-p}\right]v_{2^{l}k-p}\left(x\right).$$

(23)

Proof. 

Apparently, functions $f\left(x\right)$ and $u(t,x)$ represented in the form (22) and (23) formally satisfy all the conditions of Problem 1. Let us show that these functions have the smoothness of a regular solution. Using estimates (20) and (21), we obtain

$$\left|f\left(x\right)\right|\le C_4\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left[\left|{\psi }_{2^{l}k-p}\right|+\left|{\varphi }_{2^{l}k-p}\right|\right]\left|v_{2^{l}k-p}\left(x\right)\right|,$$

(24)

$$\left|u(t,x)\right|\le C_5\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left[\left|{\psi }_{2^{l}k-p}\right|+\left|{\varphi }_{2^{l}k-p}\right|\right]\left|v_{2^{l}k-p}\left(x\right)\right|,$$

(25)

$$\left|u_t(t,x)\right|\le C_6\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left[\left|{\psi }_{2^{l}k-p}\right|+\left|{\varphi }_{2^{l}k-p}\right|\right]\left|v_{2^{l}k-p}\left(x\right)\right|,$$

(26)

$$\left|L_{a}u(t,x)\right|\le C_7\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left[\left|{\psi }_{2^{l}k-p}\right|+\left|{\varphi }_{2^{l}k-p}\right|\right]\left|v_{2^{l}k-p}\left(x\right)\right|$$

(27)

$$\left|{\displaystyle \frac{\partial }{\partial t}}{L}_{a}u(t,x)\right|\le C_8\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left[\left|{\psi }_{2^{l}k-p}\right|+\left|{\varphi }_{2^{l}k-p}\right|\right]\left|v_{2^{l}k-p}\left(x\right)\right|.$$

(28)

Let us study the convergence of series

$$\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|,\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|.$$

(29)

Using estimate (17) for the second series in (29), we obtain

$$\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\lambda }_{2^{l}k-p}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|\le C\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|$$

$$=C_9\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}\sqrt{{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\sqrt{{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|$$

$$=C_{10}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\sqrt{{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}}\sqrt{{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|.$$

Then, applying the Cauchy–Bunyakovsky inequality, we have

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\sqrt{{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}}\sqrt{{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|\\ \hfill \le \sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\psi }_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x\right)\right|}^2}.\end{array}$$

We obtain a similar estimate for the first series in (29):

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|{\psi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x\right)\right|=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|{\psi }_{2^{l}k-p}\right|\sqrt{{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\sqrt{{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\left|v_{2^{l}k-p}\left(x\right)\right|\le \end{array}$$

$$\begin{array}{c}\hfill \le \sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|{\psi }_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x\right)\right|}^2}.\end{array}$$

By the assertion of Corollary 1, the series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x\right)\right|}^2$ converges absolutely and uniformly in the closed domain $\overline{\mathsf{\Omega }}$. In addition, if the function $\psi \left(x\right)$ satisfies the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$, then the numerical series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\psi }_{2^{l}k-p}\right|}^2$ and ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|{\psi }_{2^{l}k-p}\right|}^2$ converge. Similarly, if the function $\varphi \left(x\right)$ satisfies the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$, then the numerical series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\varphi }_{2^{l}k-p}\right|}^2$ and ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{p=0}^{2^l-1}}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|{\varphi }_{2^{l}k-p}\right|}^2$ also converge. □

Thus, if the condition of the theorem is satisfied, all series on the right-hand side of inequalities (24)–(28) converge uniformly in the domain $\overline{\mathsf{\Omega }}$. Then, the series representing the functions $u(t,x),u_t(t,x),L_{a}u(t,x)$ and ${\displaystyle \frac{\partial }{\partial t}}{L}_{a}u(t,x)$ converge absolutely and uniformly in the closed domain $\overline{Q}$. Therefore, the function $u(t,x)$ is a regular solution of Problem 1.

Now we will consider the uniqueness of the solution. Let a pair of functions $\left\{u(t,x),f\left(x\right)\right\}$ satisfy Equation (5) and homogeneous conditions (6) and (8). Let us consider the functions

$$\begin{array}{c}\hfill u_{2^{l}k-p}\left(t\right)=\underset{\mathsf{\Omega }}{\int }u(t,x)v_{2^{l}k-p}\left(x\right)dx,k\ge 1,p=0,1,\dots ,2^l-1.\end{array}$$

For these functions, we obtain Equation (16) with the homogeneous condition (17). In this case, the solution to Equation (16) satisfying the condition $u_{2^{l}k-p}\left(0\right)=0$ is the function

$$u_{2^{l}k-p}\left(t\right)={\displaystyle \frac{f_{2^{l}k-p}}{{\lambda }_{2^{l}k-p}}}\left(1-e^{-{\mu }_{2^{l}k-p}t}\right).$$

(30)

Using the homogeneous condition $u_{2^{l}k-p}\left(T\right)=0$ for the function $u_{k_1\dots k_n}\left(t\right)$, we obtain

$$0=u_{k_1\dots k_n}\left(T\right)={\displaystyle \frac{f_{2^{l}k-p}}{{\lambda }_{2^{l}k-p}}}\left(1-e^{-{\mu }_{2^{l}k-p}T}\right).$$

As far as $1-e^{-{\mu }_{2^{l}k-p}T}\ne 0$, then $f_{2^{l}k-p}=0$. Therefore, in equality (30), coefficients $u_{2^{l}k-p}\left(t\right)\equiv 0$ for all $k\ge 1,p=0,1,\dots ,2^l-1$. Thus, we have shown that the functions $f\left(x\right)$ and $u(t,x)$ are orthogonal to all elements of the system ${\left\{v_{2^{l}k-p}\left(x\right)\right\}}_{k=1}^{\infty },p=0,1,\dots ,2^l-1$. As far as this system is complete and the functions $f\left(x\right)$ and $u(t,x)$ are continuous, we obtain $f\left(x\right)\equiv 0,x\in \overline{\mathsf{\Pi }}$ and $u(t,x)\equiv 0,(t,x)\in \overline{Q}$. This implies the uniqueness of the solution to Problem 1. The theorem is proved.

5. Uniqueness and Existence of a Solution to Problem 2

Let us present the main assertion regarding Problem 2.

Theorem 2.

Let the function $f\left(x\right)$ satisfy the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]$, and the function $\varphi \left(x\right)$ with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$. Then, if $f\left(x_0\right)\ne 0,$$h\left(t\right)\in C^1[0,T]$ and $h\left(0\right)=\varphi \left(x_0\right)-L_a\varphi \left(x_0\right)$, then a solution to Problem 2 exists and is unique.

Proof. 

If we represent the desired function $u(t,x)$ in the form (15), then in this case, for $u_{k_1\dots k_n}\left(t\right)$, we obtain the following problem:

$$u_{2^{l}k-p}^{\prime }\left(t\right)+{\mu }_{2^{l}k-p}{u}_{2^{l}k-p}\left(t\right)={\displaystyle \frac{f_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}}g\left(\tau \right),0<t<T,$$

(31)

$$u_{2^{l}k-p}\left(0\right)={\varphi }_{2^{l}k-p},$$

(32)

where

$$f_{2^{l}k-p}=\underset{\mathsf{\Omega }}{\int }f\left(x\right)v_{2^{l}k-p}\left(x\right)dx.$$

The solution to problems (31)–(32) is the function as follows:

$$\begin{array}{c}\hfill u_{2^{l}k-p}\left(t\right)={\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}+{\displaystyle \frac{f_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}}\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}g\left(\tau \right)d\tau .\end{array}$$

Therefore, the function $u(t,x)$ is represented as

$$\begin{array}{c}\hfill u(t,x)=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}·v_{2^{l}k-p}\left(x\right)\end{array}$$

$$+\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\displaystyle \frac{f_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}}\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}g\left(\tau \right)d\tau ·v_{2^{l}k-p}\left(x\right).$$

(33)

In addition, we find the function $u_{2^{l}k-p}^{\prime }\left(t\right)$ from the following equality:

$$\begin{array}{c}\hfill u_{2^{l}k-p}^{\prime }\left(t\right)=-{\mu }_{2^{l}k-p}{\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}+\end{array}$$

$$\begin{array}{c}\hfill +{\displaystyle \frac{f_{2^{l}k-p}}{1+{\lambda }_{2^{l}k-p}}}\left[g\left(t\right)-{\mu }_{2^{l}k-p_n}\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}g\left(\tau \right)d\tau \right].\end{array}$$

It should be noted that if the function $g\left(t\right)$ exists and $g\left(t\right)\in C\left[0,T\right]$, then

$$\begin{array}{c}\hfill \left|\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}g\left(\tau \right)d\tau \right|\le {∥g∥}_{C[0,T]}\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}d\tau \\ \hfill ={\displaystyle \frac{C_{11}}{{\mu }_{2^{l}k-p}}}\left({\left.-e^{-{\mu }_{2^{l}k-p}(t-\tau )}\right|}_0^t\right)={\displaystyle \frac{C_{11}}{{\mu }_{2^{l}k-p}}}\left(1-e^{-{\mu }_{2^{l}k-p}t}\right),\end{array}$$

where $\left|\right|g|{|}_{C[0,T]}=\underset{0\le t\le T}{max}\left|g\left(t\right)\right|.$ □

Hence, due to the inequalities ${\mu }_{2^{l}k-p}\le 1,{\displaystyle \frac{1}{{\mu }_{2^{l}k-p}}}\le C,{\displaystyle \frac{1}{{\lambda }_{2^{l}k-p}}}\le {\displaystyle \frac{C}{{\eta }_{2^{l}k-p}}}$ for functions $u_{2^{l}k-p}\left(t\right)$ and $u_{2^{l}k-p}^{\prime }\left(t\right)$, we obtain the following estimates:

$$\left|u_{2^{l}k-p}\left(t\right)\right|\le C\left(\left|{\varphi }_{2^{l}k-p}\right|+{\displaystyle \frac{1}{{\eta }_{2^{l}k-p}}}\left|f_{2^{l}k-p}\right|\right),$$

$${\lambda }_{2^{l}k-p}\left|u_{2^{l}k-p}\left(t\right)\right|\le C_{12}\left[{\eta }_{2^{l}k-p}\left|{\varphi }_{2^{l}k-p}\right|+\left|f_{2^{l}k-p}\right|\right],$$

$$\left|u_{2^{l}k-p}^{\prime }\left(t\right)\right|\le C_{13}\left(\left|{\varphi }_{2^{l}k-p}\right|+{\displaystyle \frac{1}{{\lambda }_{2^{l}k-p}}}\left|f_{2^{l}k-p}\right|\right).$$

This means that the following estimates are valid for the function $u(t,x)$ in (33):

$$\left|u(t,x)\right|\le C_{14}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left(\left|{\varphi }_{2^{l}k-p}\right|+{\displaystyle \frac{1}{{\eta }_{2^{l}k-p}}}\left|f_{2^{l}k-p}\right|\right),$$

(34)

$$\left|u_t(t,x)\right|\le C_{15}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left(\left|{\varphi }_{2^{l}k-p}\right|+{\displaystyle \frac{1}{{\eta }_{2^{l}k-p}}}\left|f_{2^{l}k-p}\right|\right),$$

(35)

$$\left|L_{a}u(t,x)\right|\le C_{16}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}\left(\left|{\varphi }_{2^{l}k-p}\right|+\left|f_{2^{l}k-p}\right|\right)+C_{17}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|f_{2^{l}k-p}\right|,$$

(36)

$$\left|{\displaystyle \frac{\partial }{\partial t}}{L}_{a}u(t,x)\right|\le C_{18}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}\left(\left|{\varphi }_{2^{l}k-p}\right|+\left|f_{2^{l}k-p}\right|\right)+C_{19}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|f_{2^{l}k-p}\right|.$$

(37)

These estimates will be used in the study of the smoothness of the function $u(t,x)$ and its derivatives.

Hereby, let us find the function $g\left(t\right)$. Using condition (9), we obtain

$$\begin{array}{c}\hfill h\left(t\right)=u(t,x)-{\left.L_{a}u(t,x)\right|}_{x=x_0}=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}·\left(1+{\lambda }_{2^{l}k-p}\right)v_{2^{l}k-p}\left(x_0\right)+\end{array}$$

$$+\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{f}_{2^{l}k-p}\underset{0}{\overset{t}{\int }}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}g\left(\tau \right)d\tau ·v_{2^{l}k-p}\left(x_0\right).$$

(38)

It should be noted that for the product of ${\lambda }_{2^{l}k-p}$ and ${\varphi }_{2^{l}k-p}$ we can write the equality

$$\begin{array}{c}\hfill {\lambda }_{2^{l}k-p}·{\varphi }_{2^{l}k-p}={\lambda }_{2^{l}k-p}\underset{\mathsf{\Omega }}{\int }\varphi \left(x\right)v_{2^{l}k-p}\left(x\right)dx=\underset{\mathsf{\Omega }}{\int }\varphi \left(x\right){\lambda }_{2^{l}k-p}{v}_{2^{l}k-p}\left(x\right)dx=\end{array}$$

$$\begin{array}{c}\hfill =-\underset{\mathsf{\Omega }}{\int }\varphi \left(x\right)L_a{v}_{2^{l}k-p}\left(x\right)dx=\underset{\mathsf{\Omega }}{\int }{L}_a\varphi \left(x\right)·v_{2^{l}k-p}\left(x\right)dx={\left(L_a\varphi \right)}_{2^{l}k-p},\end{array}$$

where ${\left(L_a\varphi \right)}_{2^{l}k-p}$ is the Fourier coefficient of the function $L_a\varphi \left(x\right)$. Then, the following equality holds:

$$\begin{array}{c}\hfill {\left.\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}·\left(1+{\lambda }_{2^{l}k-p}\right)v_{2^{l}k-p}\left(x_0\right)\right|}_{t=0}\\ \hfill =\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\varphi }_{2^{l}k-p}·v_{2^{l}k-p}\left(x_0\right)+\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\left(L_a\varphi \right)}_{2^{l}k-p}·v_{2^{l}k-p}\left(x_0\right)=\end{array}$$

$$=\varphi \left(x_0\right)+L_a\varphi \left(x_0\right).$$

This means that the matching conditions must be satisfied:

$$h\left(0\right)=\varphi \left(x_0\right)-L_a\varphi \left(x_0\right).$$

Let us introduce the following notations:

$$\mathsf{\Phi }\left(t\right)=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left(1+{\lambda }_{2^{l}k-p}\right){\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}·v_{2^{l}k-p}\left(x_0\right),$$

(39)

$$K(t,\tau )=\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{f}_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}·v_{2^{l}k-p}\left(x_0\right)$$

(40)

and $\mathsf{\Psi }\left(t\right)=h\left(t\right)-\mathsf{\Phi }\left(t\right)$.

Then, equality (38) can be rewritten as follows:

$$\underset{0}{\overset{t}{\int }}K(t,\tau )g\left(\tau \right)d\tau =\mathsf{\Psi }\left(t\right).$$

(41)

Thus, for the function $g\left(t\right)$, we have obtained the Volterra integral equation of the first kind. Let us consider this equation.

First, we need prove the following assertion.

Lemma 4.

If the function $\varphi \left(x\right)$ satisfies the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$, then the function $\mathsf{\Phi }\left(t\right)$ from (39) is continuous and has a continuous derivative on the interval $[0,T]$.

Proof. 

Differentiating the series (39), we obtain the following:

$${\mathsf{\Phi }}^{\prime }\left(t\right)=-\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left(1+{\lambda }_{2^{l}k-p}\right){\mu }_{2^{l}k-p}{\varphi }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}t}·v_{2^{l}k-p}\left(x_0\right).$$

(42)

As the function $e^{-{\mu }_{2^{l}k-p}t}$ is limited and ${\mu }_{2^{l}k-p}={\lambda }_{2^{l}k-p}{\left(1+{\lambda }_{2^{l}k-p}\right)}^{-1}$, the following estimates are valid:

$$\begin{array}{c}\hfill \left|\mathsf{\Phi }\left(t\right)\right|\le C_{20}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}\left|{\varphi }_{2^{l}k-p}\right|·\left|v_{2^{l}k-p}\left(x_0\right)\right|\\ \hfill =C_{20}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\sqrt{{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}}\sqrt{{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}}\left|{\varphi }_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x_0\right)\right|\le \end{array}$$

$$\begin{array}{c}\hfill \le \sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\varphi }_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x_0\right)\right|}^2},\end{array}$$

$$\begin{array}{c}\hfill \left|{\mathsf{\Phi }}^{\prime }\left(t\right)\right|\le \sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\varphi }_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x_0\right)\right|}^2}.\end{array}$$

By assumption, the function $\varphi \left(x\right)$ satisfies the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]+2$. Then, the following numerical series

$$\begin{array}{c}\hfill \sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+3\right)}{\left|{\varphi }_{2^{l}k-p}\right|}^2,\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x_0\right)\right|}^2\end{array}$$

converge. It means that the series (39) and (42) converge absolutely and uniformly on the segment $[0,T]$. Hence, the sums of these series represent a continuous function, i.e., $\mathsf{\Phi }\left(t\right),{\mathsf{\Phi }}^{\prime }\left(t\right)\in C[0,T]$. The lemma is proved.

A similar assertion is also valid for the function $K(t,\tau )$. □

Lemma 5.

If the function $f\left(x\right)$ satisfies the conditions of Lemma 3 with the exponent $r=\left[{\displaystyle \frac{n}{2}}\right]$, then the function $K(t,\tau )$ from (40) is continuous and has a continuous derivative on the set $[0,T]\times [0,T]$.

Proof. 

Differentiating the series (40) with respect to t, we get

$$K_t(t,\tau )=-\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{f}_{2^{l}k-p}{\mu }_{2^{l}k-p}{e}^{-{\mu }_{2^{l}k-p}(t-\tau )}·v_{2^{l}k-p}\left(x_0\right).$$

(43)

Then, for the series (40) and (43), we obtain the following estimates:

$$\begin{array}{c}\hfill \left|K(t,\tau )\right|\le C_{21}\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}\left|f_{2^{l}k-p}\right|\left|v_{2^{l}k-p}\left(x_0\right)\right|\\ \hfill \le C_{22}\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|f_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\le \sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x_0\right)\right|}^2},\end{array}$$

$$\begin{array}{c}\hfill \left|K_t(t,\tau )\right|\le C_{23}\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|f_{2^{l}k-p}\right|}^2}·\sqrt{\sum _{k=1}^{\infty }\sum _{p=0}^{2^l-1}{\eta }_{2^{l}k-p}^{-\left(\left[{\displaystyle \frac{n}{2}}\right]+1\right)}{\left|v_{2^{l}k-p}\left(x_0\right)\right|}^2}.\end{array}$$

Hence, the sums of these series, i.e., the functions $K(t,\tau )$ and $K_t(t,\tau )$, are continuous in this domain. The lemma is proved. □

Differentiating equality (41), we obtain

$$K(t,t)g\left(t\right)+\underset{0}{\overset{t}{\int }}{K}_t(t,\tau )g\left(\tau \right)d\tau ={\mathsf{\Psi }}^{\prime }\left(t\right).$$

(44)

As far as $K(t,t)={\displaystyle \sum _{k_1=1}^{\infty }}{\displaystyle \sum _{k_n=1}^{\infty }}{f}_{k_1\dots k_n}{v}_{k_1\dots k_n}\left(x_0\right)=f\left(x_0\right)\ne 0$, then equality (44) turns out to be a Volterra integral equation of the second kind with a continuous kernel and a continuous right-hand side. According to the general theory, such an equation has a unique solution $g\left(t\right)$ from the class $C[0,T]$.

If we substitute this function into equality (33), then the pair of functions $\left\{u(t,x),g\left(t\right)\right\}$ satisfies all the conditions of Problem 2. The smoothness of the function $u(t,x)$ and its derivatives follows from estimates (34)–(37), and the uniqueness of the solution is studied as in the case of Problem 1. The theorem is proved.

6. Conclusions

In this paper, inverse initial-boundary value problems are considered for a pseudoparabolic equation with involution. Two types of inverse problems are considered in the paper. For the first type of the inverse problems, in addition to the solution itself, the right-hand side of the equation, depending on the spatial variable, is determined. For the second type of problem, a function depending on the time variable is found. The specific feature of this research is that these problems are studied in the general n-dimensional case. To solve the first problem, the Fourier method is used. The second problem is solved by reducing it to the Volterra integral equation. The proposed method can be applied to pseudoparabolic equations of high order and to inverse problems with nonlocal conditions in the time variable. These problems will be the subject of further research.