On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

1.   Introduction

Let $Q = \Pi \times (0, T)$, where $\Pi = \{x\in\Bbb R^n: 0 < x_j < p_j, j = 1, \dotsc, n\}$ is a parallelepiped, ${p_j} > 0$, and $T > 0$. Consider the mappings ${S_j}:{\Bbb R^n} \to {\Bbb R^n}, 1 \le j \le n$ of the type ${S_j}x = \left({{x_1}, \dotsc, {x_{j - 1}}, {p_j} - {x_j}, {x_{j + 1}}, \dotsc, {x_n}} \right)$. Obviously, the mappings ${S_j}$ are involutions, i.e., $S_j^2 = I$, where $I$ is the identity mapping. Let us consider all possible products of mappings ${S_j}$, i.e., ${S_{ij}} = {S_i}{S_i}$, or ${S_{ijk}} = {S_i}{S_i}{S_k}, ...$. The total number of such mappings, taking into account the identity mapping ${S_0}x = x$, is equal to ${2^n}$. To number such mappings, we will use the binary number system, namely, if $0 \le i < {2^n}$ in the binary number system, the representation $i \equiv {({i_n} \ldots {i_1})_2} = {i_1} + 2{i_2} + \ldots + {2^{n - 1}}{i_n}$, where ${i_k} = 0, 1$, is valid. Therefore, introducing the vector ${\bf{i}} = ({i_1}, \ldots, {i_n})$, we can consider mappings of the type ${{\bf{S}}_{\bf{i}}} \equiv S_1^{{i_1}} \ldots S_n^{{i_n}}$ corresponding to the index $i$. Using these mappings, we introduce the operator

where ${a_0}, {a_1}, {a_2}, {a_3}, \ldots, {a_{{2^n} - 1}}$ is a set of real numbers, and ${\Delta ^2}$ is a biharmonic operator.

Let us consider the following problems in the domain $Q$.

Problem 1. Find a pair of such functions $\left\{{u(t, x), f(x)} \right\}$ that are smooth $u(t, x) \in C\left({\bar Q} \right)$, $f(x) \in C\left({\bar \Pi } \right)$, ${u_t}(t, x)$, ${L_x}u(t, x) \in C\left(Q \right)$, satisfy the equation

the boundary conditions

and the initial and final conditions

where $g(t)$, $\varphi(x)$, $\psi (x)$ are the given functions.

Problem 2. Find a pair of such functions $\left\{ {u(t, x), g(t)} \right\}$ that are smooth $u(t, x) \in C\left({\bar Q} \right)$, $g(t) \in C\left[{0, T}\right]$, ${u_t}(t, x), {L_x}u(t, x) \in C\left(Q \right)$ and satisfy conditions (1.1)–(1.3) and an additional condition

where ${x_0} \in \Pi$ is a given fixed point, $h(t), \varphi (x)$, and $f(x)$ are the given functions, and $h(0) = \varphi ({x_0})$.

Differential equations with involution are an important part of the general theory of functional differential equations. As it is known, the first works on differential equations with involution were written by Babbage ^[1] and continued by Carleman ^[2]. Various issues in the theory of differential equations with involution were studied in a series of papers written by Przeworska-Rolewicz ^[3,4,5] The case $n = 1$ is considered in several papers ^{[6,7,8,9,10,11]}, where the solvability of inverse problems on finding the righthand side of differential equations with involution is studied. Thus, in the work of Nasser Al-Salti et al. ^[7], in the rectangular domain $\Omega = \left\{ {(t, x):0 < t < T, - \pi < x < \pi } \right\}$ for the equation

inverse problems for determining a pair of functions $\left\{ {u(t, x), f(x)} \right\}$ have been studied. In this work, the authors state that redefined initial and final conditions

as well as one of the boundary conditions, Dirichlet, Neumann, periodic, or antiperiodic conditions, can be considered as additional conditions. The problem studied in the paper is solved using the method of separation of variables. In this case, a spectral problem arises with respect to the spatial variable for the equation with involution. For example, in the case of Dirichlet boundary conditions, we have

It is proven that the eigenfunctions of this problem are the functions

and the corresponding eigenvalues are the numbers

Similar studies in the case of two spatial variables for classical equations were carried out in ^{[12,13,14,15,16,17,18]}, as well as for equations with involution in ^[19,20]. The authors of this paper studied inverse problems for equations of parabolic type with an operator of the fourth and higher orders in the spatial variable (see, for example, ^{[21,22,23,24,25,26,27,28]}).

The methods used to solve the inverse problems in order to determine the right side of the equation depend on whether the function $f(x)$ or the function $g(t)$ is unknown. In the case when the function $f(x)$ is unknown, the problem under consideration is usually solved using the Fourier method, whereas in the case when the function $g(t)$ is unknown, the problem is reduced to the Volterra integral equation (see, for example, ^[18]).

As we have already noted, studies of direct and inverse problems for equations with involutive transformed arguments were mainly carried out for equations with one and two spatial variables. For equations with many variables, such problems are insufficiently studied. In this direction, we can only note the work of Kozhanov and Bzheumikhova ^[25].

When studying Problems 1 and 2, a spectral problem arises for the nonlocal operator $L_{x}$. In the one-dimensional case in ^[7], this problem is solved by finding a general solution to the studied equation and using the boundary conditions to determine the eigenfunctions and eigenvalues. In the two-dimensional case in ^[19], the corresponding spectral problem was solved by reducing it to four auxiliary problems.

In general, in the $n -$ dimensional case, $n \ge 3$, such spectral problems in a parallelepiped have not been considered. When solving this problem, to denote the summation index we used the notation in the binary number system. The transition to this system allowed us to construct eigenfunctions and eigenvalues of this problem explicitly and to prove the completeness of the systems of eigenfunctions in space ${L_2}\left(\Pi \right)$.

It should be also noted that the use of differential equations with involution when modeling a specific physical process of thermal diffusion in the case $n = 1$ is given in ^[7,9], and in ^[29,30] in the case $n = 2$, where equations with involutive transformations, which have applications in modeling of optical systems, are considered. In addition, the influence of nonlocality is graphically illustrated in ^[7] in the one-dimensional case.

2.   Eigenfunctions and eigenvalues of a nonlocal biharmonic operator

In this section, we study the eigenfunctions and eigenvalues of a Dirichlet type problem for a nonlocal biharmonic equation.

Consider the following boundary value problem.

Problem S. Find a function $v(x) \ne 0$ from the class $v(x) \in {C^2}\left({\bar \Pi } \right) \cap {C^4}\left(\Pi \right)$ that satisfies the equation

and boundary conditions

where $\lambda \in\Bbb R$.

If ${a_0} = 1$, $a_j = 0$, $j = 1, \ldots, 2^n-1$, then the problem coincides with the spectral problem with the Dirichlet condition for the classical biharmonic operator.

Note that in the case of a sphere, a similar spectral problem for the nonlocal Laplace operator was studied in ^[31]. Various boundary value problems for nonlocal harmonic and biharmonic equations are studied in ^[32,33,34].

Let us consider a set of functions

where ${\rm{ }}{\bf{k}} = ({k_1}{k_2}...{k_n}) \in {\Bbb N^n}$ and $C\left({n, \bf{p}} \right) = {2^{n/2}}\prod\limits_{j = 1}^n {\frac{1}{{\sqrt {{p_j}} }}}, \; {\bf{p}} = ({p_1}, \ldots, {p_n})$.

The following statement is proved in ^[35].

Lemma 2.1. The system of functions $\left\{ {{v_{\bf{k}}}(x):{\bf{k}} \in {\Bbb N^n}} \right\}$ is orthonormal and complete in space ${L_2}\left({(0, {p_1}) \times (0, {p_2}) \times... \times (0, {p_n})} \right)$.

Let us introduce the following numbers ${\varepsilon _{\bf{k}}} = \sum\limits_{i = 0}^{{2^n} - 1} {{{(- 1)}^{{\bf{i}} \cdot ({\bf{k}} + {\bf{e}})}}{a_i}}$, where ${\bf{k}} = ({k_1}, {k_2}, ..., {k_n})$, ${\bf{i}} = \left({{i_1}, \ldots, {i_n}} \right)$, ${\bf{i}} \cdot {\bf{k}} = {i_1}{k_1} +... + {i_n}{k_n}$, and ${\bf{e}} = (1, \ldots, 1)$. Note that components of the vector ${\bf{k}} = ({k_1}, {k_2}, ..., {k_n})$ can be calculated using mode 2 as in this case the ${\varepsilon_{\bf{k}}}$ value will not change. Taking this into account we can state that the equality ${\bf{k}} + {\bf{e}} = {{\bf{k}}^*}$ is valid, where the vector ${{\bf{k}}^*}$ is conjugate to ${\bf{k}}$. For example, ${\bf{k}} = (0, 1, 0, 1) \Rightarrow {{\bf{k}}^*} = (1, 0, 1, 0)$. Under these assumptions,

Theorem 2.1. Let the conditions ${\varepsilon _{\bf{k}}} \ne 0$ be satisfied for all ${\rm{ }}{\bf{k}} \in {\Bbb N^n}$, then the system of functions $\left\{ {{v_{\bf{k}}}(x):{\bf{k}} \in {\Bbb N^n}} \right\}$ is a system of eigenfunctions of Problem $\bf S$. The corresponding eigenvalues are determined by the equalities

Proof. It is obvious that the functions ${v_{\bf{k}}}(x)$, by their structure, satisfy conditions (2.2) ${v_{\bf{k}}}(x){|_{\partial \Pi }} = 0,$ and as

they also satisfy conditions (2.2). Let us check whether Eq (2.1) is fulfilled. Applying the Laplace operator to this function, we obtain

which means that

Hence, for any $m \in \left\{ {1, 2, ..., n} \right\}$, we get

Let $0 \le i \le {2^n} - 1$, which corresponds to vector ${\bf{i}} = \left({{i_1}, \ldots, {i_n}} \right)$. If ${i_m} = 1$, then

Obviously, this equality is also valid for the case ${i_m} = 0$. Thus, we get

In the general case, the following equality holds:

where ${\bf{k}} = ({k_1}, {k_2}, ..., {k_n})$, ${\bf{i}} = \left({{i_1}, \ldots, {i_n}} \right)$, ${\bf{i}} \cdot {\bf{k}} = {i_1}{k_1} +... + {i_n}{k_n}$ and ${\bf{e}} = (1, \ldots, 1)$.

Now, if we apply the operator ${L_x}$ to the function $v_{\bf{k}}$, then from the previous equalities it follows that

Thus, in addition to conditions (2.2), the function ${v_{\bf{k}}}(x)$ also satisfies the equality ${L_x}{v_{\bf{k}}}(x) = {\lambda _{\bf{k}}}{v_{\bf{k}}}(x)$, i.e., Eq (2.1). The theorem is proved.

Remark 2.1. If the condition ${\varepsilon _{{\bf{k}}\bmod 2}} > 0$ is satisfied for any ${\bf{k}} \in {\Bbb N^n}$, then all eigenvalues of Problem $\bf S$ are positive.

Example 2.1. Let $n = 2$, then Eq (2.1) takes the form

and the boundary conditions are written as

Eigenfunctions are specified in accordance with (2.3) as

and the corresponding eigenvalues are

where ${\varepsilon _{(m, k)}}$ is

More precisely, $\varepsilon_{(m, k)}$ can be written as

where $m, k = 1, 2, ...$.

3.   Study of convergence of Fourier series

Let

be the Fourier series expansion of the function $h(x)$ by the system $\left\{ {{v_{\bf{k}}}(x):{\bf{k}} \in {\Bbb N^n}} \right\}$, where

Further, we will use the symbol $C$ to denote an arbitrary constant whose value does not affect our conclusions.

Lemma 3.1. Let the function $h(x)$ be continuous in a closed domain $\bar \Pi$ and have continuous partial derivatives $\frac{{{\partial ^j}h(x)}}{{\partial {x_1}...\partial {x_j}}}$, $1 \le j \le n$ in $\bar \Pi$. If the conditions

are satisfied, then the number series $\sum\limits_{{\bf{k}} \in {\Bbb N^n}} {\left| {{h_{\bf{k}}}} \right|}$ converges.

Proof. If $h(x) \in C\left({\bar \Pi } \right)$ and the function $\frac{{\partial h(x)}}{{\partial {x_1}}}$ is continuous, then integrating the integral in (3.2) by parts over the variable ${x_1}$ and taking into account equality (3.3), we obtain

where

Applying this process to all $j \in \left\{ {2, 3, ..., n} \right\}$, we get

where

Using the Cauchy-Bunyakovsky inequality, we obtain

As the system $\left\{ {v_{{k_1}...{k_n}}^{1, 1, .., 1}({x_1}, {x_2}, ..., {x_n})} \right\}$ is orthogonal in space ${L_2}\left(\Pi \right)$ and $\frac{{{\partial ^n}h({x_1}, {x_2}, ..., {x_n})}}{{\partial {x_1}...\partial {x_n}}} \in {L_2}\left(\Pi \right)$, then due to Bessel's inequality, the series $\sum\limits_{{k_1} = 1}^\infty {...} \sum\limits_{{k_n} = 1}^\infty {{{\left| {h_{{k_1}...{k_n}}^{1, 1, .., 1}} \right|}^2}}$ converges. Moreover, the series

also converges. This implies the assertion of the lemma.

The following assertion is proved in a similar way.

Lemma 3.2. Let a function $h(x)$ belong to a class ${C^4}\left({\bar \Pi } \right)$ and have continuous partial derivatives of the form $\frac{{{\partial ^{n + 4}}h(x)}}{{\partial {x_1}...\partial x_j^5...\partial {x_n}}}$ for $j \in \left\{ {1, 2, ..., n} \right\}$ in $\bar \Pi$. If the functions $h(x), \frac{{{\partial ^2}h(x)}}{{\partial x_j^2}}$, and $\frac{{{\partial ^4}h(x)}}{{\partial x_j^4}}$ satisfy conditions (3.3), the number series $\sum\limits_{{k_1} = 1}^\infty {...} \sum\limits_{{k_j} = 1}^\infty {...} \sum\limits_{{k_n} = 1}^\infty {k_j^4\left| {{h_{{k_1}...{k_j}...{k_n}}}} \right|}$ converges.

If the functions $h(x), \frac{{{\partial ^2}h(x)}}{{\partial x_j^2}}$, and $\frac{{{\partial ^4}h(x)}}{{\partial x_j^4}}$ satisfy conditions (3.3), the number series

converges.

Proof. Let the functions $h(x)$ and $\frac{{{\partial ^2}h(x)}}{{\partial x_j^2}}, \frac{{{\partial ^4}h(x)}}{{\partial x_j^4}}$ satisfy conditions (3.3), then integrating the integral four times over the variable ${x_j}$ in the equality

we get

where

Using the Cauchy-Bunyakovsky inequality for $\sum\limits_{{k_1} = 1}^\infty {...} \sum\limits_{{k_n} = 1}^\infty {\left| {{h_{{k_1}...{k_n}}}} \right|}$, we obtain

As the system $\left\{ {v_{{k_1}...{k_n}}^{1, 1, .., 1}({x_1}, {x_2}, ..., {x_n})} \right\}$ is orthogonal in space ${L_2}\left(\Pi \right)$ and $\frac{{{\partial ^n}h({x_1}, {x_2}, ..., {x_n})}}{{\partial {x_1}...\partial {x_n}}} \in {L_2}\left(\Pi \right)$, then due to Bessel's inequality, the series $\sum\limits_{{k_1} = 1}^\infty {...} \sum\limits_{{k_n} = 1}^\infty {{{\left| {h_{{k_1}...{k_n}}^{1, 1, .., 1}} \right|}^2}}$ converges. This implies the statement of the lemma.

Corollary 3.1. Let the conditions of Lemma 3.2 be satisfied, then the number series

converges.

The proof of this assertion follows from the representation of eigenvalues in the form ${\lambda _{{k_1}{k_2}...{k_n}}} = {\varepsilon _{{k_1}{k_2}...{k_n}}}{\pi ^4}{\Big({\sum\limits_{j = 1}^n {\frac{{k_j^2}}{{p_j^2}}} } \Big)^2}$ and the assertion of Lemma 3.2.

4.   Uniqueness and existence of a solution to Problem 1

By definition, the solution to Problem 1 is expressed as functions $u(t, x), f(x)$. If such functions exist, they must be the elements of space ${L_2}\left(\Pi \right)$ and, therefore, they can be represented as series

where ${u_{\bf{k}}}(t) = {u_{{k_1}...{k_n}}}(t)$ and ${f_{\bf{k}}} = {f_{{k_1}...{k_n}}}$ are the coefficients to be determined.

Substituting (4.1) and (4.2) into Eq (1.1) and equating the coefficients for ${v_{\bf{k}}}(x)$, we obtain the following problem for the coefficients ${u_{\bf{k}}}(t)$:

The general solution to Eq (4.3) is the function

where ${C_{\bf{k}}}$ are arbitrary constants. Let us introduce the notation

then ${u_{\bf{k}}}(t)$ from (4.5) is represented in the form

Now, we will consider two cases separately: $g(t) = 1$ and $g(t) \ne 1$.

(Ⅰ) Suppose $g(t) = 1$, then

and ${u_{{\bf k}}}(t)$ are represented as

If we use condition (4.4), we obtain

then we find

The solution to problem (4.3), (4.4) is represented as

Hence, the solution to Problem 1 can be written as

By construction and also due to the properties of functions ${v_{{\bf k}}}(x)$, the function $u(t, x)$ from (4.8) formally satisfies conditions (1.1)–(1.4). Let us examine the smoothness of the functions $u(t, x)$ and $f(x)$.

Let the functions $\varphi (x)$ and $\psi (x)$ satisfy the conditions of Corollary 3.1, then the number series

converges. As ${\lambda _{{\bf k}}} = {\varepsilon _{{\bf k}}}\mu _{{\bf k}}^2$ and ${\varepsilon_{{\bf k}}} > 0$ does not tend to 0, the functions $\frac{1}{{1 - {e^{ - {\lambda _{{\bf k}}}T}}}}$ and $\frac{{{e^{ - {\lambda_{{\bf k}}}T}}}}{{1 - {e^{-{\lambda_{{\bf k}}}T}}}}$ are bounded. For the coefficients ${f_{{\bf k}}}$ from equality (4.6), we obtain

Based on this estimate and the uniform boundedness of the moduli of the eigenfunctions $\left| {{v_{{k_1}...{k_n}}}(x)} \right|$ from (2.3), we obtain absolute and uniform convergence of the functional series (4.9) in the closed domain $\bar \Pi$. Hence, the sum of this series, i.e., the function $f(x)$, is continuous in the domain $\bar \Pi$. For all $0 \le t \le T$, there are estimates

The series (4.8) satisfies the estimate

This means that this series converges absolutely and uniformly in a closed domain $\bar Q$ and, therefore the function $u(t, x)$, the sum of this series, belongs to the class $C({\bar Q})$.

Differentiating series (4.8) with respect to the variable $t$, we obtain

If $f(\lambda) = \lambda {e^{ - \lambda t}}$, then ${\max _{\lambda \ge 0}}f(\lambda) = f(1/t) = \frac{1}{t}{e^{ - 1}}$, which means that $\lambda {e^{ - \lambda t}} \le \frac{1}{{\delta e}}$ for $t \ge \delta$. Therefore, the series (4.10) satisfies the estimate

If the condition of Lemma 3.1 is satisfied, the last series converges, then for an arbitrary $\delta > 0$, series (4.10) converges absolutely and uniformly in the closed domain ${\bar Q_\delta } = [\delta \le t \le T] \times \bar \Pi \subset Q$. Hence, ${u_t}(t, x) \in C\left({{{\bar Q}_\delta }} \right)$. Thus, by virtue of arbitrariness of $\delta > 0$, it follows that ${u_t}(t, x) \in C\left(Q \right)$. The inclusion ${L_x}u(t, x) \in C\left(Q \right)$ is proved in a similar way. Thus, functions (4.8) and (4.9) for $g(t) = 1$ satisfy all the conditions of Problem 1.

If in Problem 1 homogeneous conditions (4.2) are specified, then for the coefficients ${f_{{k_1}...{k_n}}}$ in equality (4.6), we obtain that ${f_{{k_1}...{k_n}}} \equiv (f, {v_{{k_1}{k_2}...{k_1}}}) = 0$. Hence, the function $f(x)$ is orthogonal to all elements of the system $\left\{ {{v_{{\bf k}}}(x)} \right\}_{{\bf k}\in\Bbb N^n}$. Due to the completeness of this system and continuity of the function $f(x)$, we obtain $f(x) \equiv 0, x \in \bar \Pi$.

Similarly, for the coefficients ${u_{{\bf k}}}(t)$ from equality (4.5) we obtain ${u_{{\bf k}}}(t) \equiv (u, {v_{{\bf k}}}) = 0$. Hence, the equality $u(t, x) = 0$, $x \in \bar \Pi$ is valid for almost all $t \in [0, T]$. Due to the continuity of the function $u(t, x)$ in $\bar Q$, we obtain that $u(t, x) \equiv 0$, $(t, x) \in \bar Q$. This implies the uniqueness of the solution to Problem 1.

Thus, we proved the following assertion.

Theorem 4.1. Let the functions $\varphi (x)$ and $\psi (x)$ in Problem 1 satisfy the conditions of Corollary 3.1, $g(t) = 1$ and let the coefficients ${a_i}, \, i = 0, 1, ..., 2^n- 1$ be such that the conditions ${\varepsilon_{{\bf k}}} > 0$ are satisfied. The solution to the problem exists, is unique, and is represented in the form of series (4.8) and (4.9).

(Ⅱ) Let us study Problem 1 for the case $g(t) \ne 1$. If we search for a solution to the problem in the form of (4.1) and (4.2), then for unknown coefficients ${u_{{\bf k}}}(t)$, we obtain problem (4.3) and (4.4). Moreover, the general solution to Eq (4.3) is determined by equality (4.5). Substituting this function into condition (4.4), we have

Thus, if for all ${\bf k} \in\Bbb N^n$, the condition ${g_{{\bf k}}}(T) \ne 0$, then

and

If in Problem 1, as in the case of $g(t) = 1$, homogeneous conditions (4.2) are given, we obtain $f(x) \equiv 0$, $x \in \bar \Pi$ and $u(t, x) \equiv 0, (t, x) \in \bar Q$. Therefore, if the conditions ${g_{{\bf k}}}(T)\ne 0$, ${\bf k} \in\Bbb N^n$ are satisfied, the solution to Problem 1 is unique. If for some ${\bf m} = (m_1, m_2, ..., m_n)$ the equality ${g_{{\bf m}}}(T) = 0$ holds, then the homogeneous Problem 1 has a nonzero solution. Let us show that if this condition is satisfied, a pair of functions

will be a solution to homogeneous Problem 1, where ${f_{{\bf m}}}$ is an arbitrary constant. Indeed, applying the operators $\frac{\partial }{{\partial t}}$ and ${L_x}$ to the function $u(t, x)$, we get

Hence,

It is obvious that this function satisfies homogeneous conditions (1.2) and (1.3). Thus, we proved the following assertion.

Theorem 4.2. If a solution to Problem 1 exists, then it is unique if, and only if, the conditions ${g_{{\bf k}}}(T)\ne 0$ are satisfied for all ${\bf k}\in\Bbb N^n$.

Regarding the existence of a solution to Problem 1 in the case $g(t) \ne 1$, the following assertion is valid.

Theorem 4.3. Let in Problem 1 the functions $\varphi (x)$ and $\psi (x)$ satisfy the conditions of Corollary 3.1, and the coefficients ${a_i}$, $i = 0, 1, ..., 2^n- 1$ are such that the conditions $\varepsilon_{{\bf k}} > 0$ are satisfied. A solution to the problem exists and is represented in the form of series (4.8) and (4.9), where the coefficients $f_{{\bf k}}$ and $u_{{\bf k}}$ are respectively determined by equalities (4.11) and (4.12).

Proof. If $g(t) \in C[0, T], \left| {g(t)} \right| \ge {g_0} \equiv C$, then according to the mean value theorem there is a point $\xi \in [0, T]$ such that

This gives us the following lower estimate:

Using inequality (4.13) for ${f_{{\bf k}}}$ from equality (4.11), we get

Similarly from equality (4.12), we obtain for ${u_{{\bf k}}}(t)$,

As the function $g(t)$ is continuous on the interval $[0, T]$, it follows that

Hence, $\left| {\frac{{{g_{{\bf k}}}(t)}}{{{g_{{\bf k}}}(T)}}} \right| \le C$. Therefore, the estimate

is valid for ${u_{{\bf k}}}(t)$. The estimate

is proved in a similar way. From these estimates and the convergence of the series

absolute and uniform convergence of series (4.8) and (4.9) follows. For the sums of these series we get $f(x) \in C\left({\bar \Pi } \right)$ and $u(t, x) \in C\left({\bar Q} \right)$.

The absolute and uniform convergence of the series

in an arbitrary closed domain $\bar Q_\delta \subset Q$ and $\delta > 0$ is proved similar to the case $g(t) = 1$. Therefore, the inclusions $\frac{{\partial u(t, x)}}{{\partial t}}$, $Lu(t, x) \in C\left(Q \right)$ are valid. The theorem is proved.

5.   Uniqueness and existence of a solution to Problem 2

The main assertion regarding Problem 2 is the following theorem.

Theorem 5.1. Let ${\varepsilon _{{\bf k}}} > 0$, the function $\varphi (x)$, satisfy the conditions of Corollary 3.1, then if $f({x_0}) \ne 0$, $h(t) \in {C^1}[0, T]$, $h(0) = \varphi ({x_0})$, the solution to Problem 2 exists and is unique.

Proof. If we assume that the function $g(t)$ is known, then the solution to Problem 2 can be represented as

where

Let us suppose that $x = x_0$ in (5.1), then

where

Denote $r(t) = h(t) - {\varphi _0}(t)$ and

For the function $g(t)$, we obtain the following Volterra integral equation of the first kind

Lemma 5.1. If the function $\varphi (x)$ satisfies the conditions of Corollary 3.1, then the function ${\varphi_0}(t)$ from (5.2) is continuous and has a continuous derivative on the interval $[0, T]$.

Proof. Let us differentiate the series (5.2)

As for the points of the domain $\bar Q$, we have the estimate $\left| {{e^{-{\lambda _{{\bf k}}}t}}{v_{{\bf k}}}(x_0)} \right| \le C$, then

If the condition of Corollary 3.1 is satisfied, the last number series converges, then series (5.2) converges absolutely and uniformly. Therefore, the sum of this series represents a continuous function, i.e., ${\varphi _0}(t) \in {C^1}[0, T]$. The lemma is proved.

Let us study the properties of the kernel $K(t, \tau)$. Differentiating series (5.3) with respect to $t$, we get

For series (5.3) and (5.5), we obtain the estimates

If the function $f(x)$ satisfies the conditions of Corollary 3.1, then the number series

converges. The series (5.3) and (5.5) converge absolutely and uniformly in the closed domain $[0, T] \times [0, T]$. Therefore, the functions $K(t, \tau)$ and ${K_t}(t, \tau)$ are continuous in this domain. Further, differentiating equality (5.4), we obtain

As

the equality (5.6) is an integral Volterra equation of the second kind with a continuous kernel and a continuous righthand side. According to the general theory, such an equation has a unique solution $g(t)$ from the class $C[0, T]$.

If we substitute this function into equality (5.1), then the pair of functions $\left\{ {u(t, x), g(t)} \right\}$ satisfies all the conditions of Problem 2. The smoothness of the function and the uniqueness of the solution are proved as in the case of Problem 1. The theorem is proved.

6.   Conclusions

In this paper, the solvability of some inverse problems for a nonlocal analogue of the fourth-order parabolic equation is studied. The nonlocal operator is introduced using involutive mappings. Unlike previous works of the authors, in this paper the problems are studied in the $n$-dimensional case. The considered problems are solved by applying the Fourier method and reducing them to the Volterra integral equation. In this case, a spectral problem arises for the nonlocal analogue of the biharmonic operator. The eigenfunctions and eigenvalues for this problem are found explicitly and the completeness of the system of eigenfunctions is proved. Solutions to the main problems are constructed in the form of series using a system of eigenfunctions. Further, it is planned to continue the study of inverse problems for high order differential equations with involutions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research has been funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant No. AP19677926).

Conflict of interest

The authors declare no conflicts of interest in this paper.