Inverse kernel determination problem for a class of pseudo-parabolic integro-differential equations

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Introduction

There are numerous cases where practical applications lead to challenges in determining the coefficients, the right-hand side of the differential equation, and the kernel of integrodifferential equations. Such problems are referred to as inverse problems of mathematical physics.

Inverse problems currently represent a rapidly developing branch of modern mathematics. Various inverse problems for second-order hyperbolic and parabolic equations, as well as first-order systems, are discussed in the monographs [1–5] (see also the extensive bibliographies therein). The recently published monograph [6] investigates a new class of inverse problems involving the determination of the convolution kernel in second-order hyperbolic integrodifferential equations.

Water filtration in double-porosity media, moisture transfer in soil, and similar natural phenomena often lead to boundary value problems involving pseudo-parabolic equations (see, e.g., [7, 8]). When such processes occur in viscoelastic media, Volterra operators — representing the convolution of a time-dependent viscosity function with a solution operator (typically elliptic) — are incorporated into the right-hand side of the pseudo-differential equations.

The study of inverse problems for pseudo-parabolic equations began in the 1980s. The first significant result, obtained in [9], addressed the inverse identification of an unknown source function. Among recent works, we highlight [10], where the author examined an inverse problem of recovering a space-dependent source coefficient in a third-order pseudo-parabolic equation under a final over-determination condition (see also references therein).

To the best of our knowledge, the problem of determining the convolution kernel in an integrodifferential pseudo-parabolic equation remains unexplored. However, a series of works [11–20] has investigated inverse problems involving convolution kernel determination for linear parabolic integrodifferential equations. These studies established local existence and global uniqueness theorems, as well as stability estimates for the solutions.

In this study, we employ the Fourier method, integral inequalities, and the fixed-point principle to prove the existence and uniqueness of a solution to the inverse problem of determining the kernel of a multidimensional third-order integrodifferential pseudo-parabolic equation. The problem is supplemented with an additional condition specified at a fixed point for the solution of the first boundary value problem. 

Consider the following nonhomogeneous pseudo-parabolic integrodifferential equation:
\[ \begin{equation}
u_{t} - \Delta u_t - \Delta u = (k * \Delta u)(x,t) + f(x,t), \quad (x,t) \in D,
\end{equation} \tag{1} \]
where $D = \Omega \times (0, T]$, $T > 0$, and $\Omega \subset \mathbb{R}^N$ is a bounded domain with a sufficiently smooth boundary $\partial \Omega$. Here, $\Delta$ denotes the Laplacian, $k(t)$ is the convolution kernel representing the "memory effect" (or viscosity function), $f(x,t)$ is a source function, and $(k * \Delta u)(x,t)$ denotes the Laplace convolution:
\[ \begin{equation*}
(k * u)(x,t) := \int_0^t k(t-s) u(x,s) \, ds.
\end{equation*} \]
In the domain $D$, we study the following problem for Eq. (1): Find a function $u(x,t)$ satisfying (1) with the initial condition
\[ \begin{equation}
u(x,0) = \varphi(x), \quad x \in \Omega,
\end{equation} \tag{2} \]
and the boundary condition
\[ \begin{equation}
u = 0 \quad \text{on} \quad \partial\Omega \times (0,T),
\end{equation} \tag{3} \]
where $f(x,t)$ and $\varphi(x)$ are given functions. This problem is commonly referred to as the direct (forward) problem.

A function $u(x,t)$ is called a classical solution to problem (1)–(3) if it satisfies the following conditions:

1. $u(x,t)$ is continuous in $\overline{D}$ along with all derivatives appearing in Eq. (1);
2. all given conditions are satisfied in the classical sense.

Based on this direct problem, we now consider the following inverse problem.

Inverse problem. Determine the kernel $k(t)$, $t > 0$, appearing in equation (1), given that the solution of the direct problem satisfies the additional condition
\[ \begin{equation}
u(x_0,t) = h(t), \quad x_0 \in \Omega,\ t \in [0,T],
\end{equation} \tag{4} \]
where $x_0 \in \Omega$ is a fixed point and $h(t)$ is a given sufficiently smooth function.

1. Investigation of the Direct Problem

This section studies problem (1)–(3). We prove the existence and uniqueness of a classical solution to problem (1)–(3).

1.1. Uniqueness of the Solution

The following uniqueness result holds for (1)–(3).

Theorem 1. If problem (1)–(3) has a solution, then this solution is unique.

Proof. Applying the method of separation of variables, we seek a solution to (1)–(3) in the form
\[ \begin{equation}
u(x,t) = U(t)X(x).
\end{equation} \tag{5} \]
Substituting (5) into (1) with
\[ \begin{equation*}
\int_0^t k(t-\tau)\Delta u(x,\tau)d\tau + f(x,t) = 0,
\end{equation*} \]
we require that $X(x) \not\equiv 0$ satisfies the spectral problem
\[ \begin{equation*}
\begin{cases}
\Delta X + \lambda X = 0, & \text{in } \Omega, \\
X = 0, & \text{on } \partial\Omega.
\end{cases}
\end{equation*} \]

It is well-known that the operator $-\Delta$ has only positive real and simple eigenvalues $\lambda_m$, which when properly ordered satisfy
$0 < \lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lim\limits_{m\to\infty}\lambda_m = +\infty$. We denote by $X_m$ the eigenfunction corresponding to $\lambda_m$, normalized such that $\|X_m\|^2_{L^2(\Omega)} = (X_m,X_m) = 1$, where $({}\cdot{},{}\cdot{})$ denotes the inner scalar product in the Hilbert space $L^2(\Omega)$.

Let $u(x,t)$ be a solution to problem (1)–(3). Consider the scalar product
\[ \begin{equation}
u_m(t) = \bigl(u({}\cdot{},t), X_m \bigr)_{L^2(\Omega)}.
\end{equation} \tag{6} \]

From (6) and using equation (1), we obtain
\[ \begin{equation}
u_m'(t) + \lambda_m u_m'(t) + \lambda_m u_m(t) = -\lambda_m (k * u_m)(t) + f_m(t),
\end{equation} \tag{7} \]
where $f_m(t) = (f,X_m)$, $m = 1,2,\dots$. The initial condition (2) yields
\[ \begin{equation}
\varphi_m := u_m(0) = (\varphi, X_m )_{L^2(\Omega)}, \quad m = 1,2,\dots.
\end{equation} \tag{8} \]

One can verify that problem (7), (8) has a unique solution $u_m(t) \in C^1[0,T]$ given by
\[ \begin{equation}
u_m(t) = \chi_m(t)\varphi_m + \frac{1}{1+\lambda_m}(\chi_m * f_m)(t) - \frac{\lambda_m}{1+\lambda_m}\bigl(\chi_m * (k * u_m)\bigr)(t),
\end{equation} \tag{9} \]
where \(\chi_m(t) = \exp\bigl\{-\tfrac{\lambda_m}{1+\lambda_m}t\bigr\}\).

This implies the uniqueness of the solution to problem (1)–(3), since for ${\varphi(x) \equiv 0}$ and $f(x,t) \equiv 0$, we obtain $\varphi_m \equiv 0$ and $f_m(t) \equiv 0$. From (9) it follows that $u_m(t) \equiv 0$. By (6), this is equivalent to
\[ \begin{equation*}
\bigl(u({}\cdot{},t), X_m \bigr)_{L^2(\Omega)} = 0.
\end{equation*} \]

Since the system $\{X_m\}$ is complete in $L^2(\Omega)$, we have $u(x,t) = 0$ almost everywhere in $\Omega$ for all $t \in [0,T]$. As $u(x,t)$ is continuous on $\overline{D}$, we conclude that $u(x,t) \equiv 0$ on $\overline{D}$. This completes the proof of uniqueness for problem (1)–(3). $\square$

1.2. Existence of the Classical Solution

This subsection establishes the existence of a solution to problem (1)–(3).

Under appropriate conditions on the functions $\varphi(x)$ and $f(x,t)$, we prove that the function
\[ \begin{equation}
u(x,t) = \sum\limits_{m=1}^{\infty} u_m(t) X_m(x)
\end{equation} \tag{10} \]
represents a solution to problem (1)–(3).

Lemma 1. The following estimates hold for all $m = 1,2,\ldots$:
\[ \begin{equation}
|u_m(t)| \leqslant \max\{1,T\} \bigl[ |\varphi_m| + \|f_m\|_{0} \bigr] e^{\|k\|_{0} T^2/2}, \quad t \in [0,T],
\end{equation} \tag{11} \]
\[ \begin{multline}
|u'_m(t)| \leqslant \|f_m\|_0 +{}\\
{}+ \max\{1,T\}(1 + \|k\|_0 T) \bigl[ |\varphi_m| + \|f_m\|_{0} \bigr] e^{\|k\|_{0} T^2/2}, \quad t \in [0,T],
\end{multline} \tag{12} \]
where $\|k\|_0 = \max\limits_{t\in[0,T]} |k(t)|$.

Proof. From (9), we estimate $u_m(t)$ as follows:
\[ \begin{equation*}
|u_m(t)| \leqslant |\varphi_m| + t|f_m(t)| + \|k\|_0 \int _0^t (t-s) |u_m(s)|\, ds.
\end{equation*} \]
Applying Gronwall's lemma for all $t \in [0,T]$, we obtain (11). Furthermore, from (7) and (11), we derive (12). This completes the proof of the lemma. $\square$

Assume the following regularity conditions:
\[ \begin{equation}
\begin{cases}
\varphi(x) \in H^{\left[\frac{n}{2}\right]+3}(\Omega), \quad f(x,t) \in C([0,T]; H^{\left[\frac{n}{2}\right]+3}(\Omega)), \\
\varphi = \Delta\varphi = \dots = \Delta^{\left[\frac{n+2}{4}\right]} \varphi \in H_0^1(\Omega), \\
f({}\cdot{},t)=\Delta f({}\cdot{},t)=\cdots=\Delta^{\left[\frac{n+2}{4}\right]}f({}\cdot{},t)\in
H_0^1(\Omega),\quad t\in[0,T].
\end{cases}
\end{equation} \tag{A1} \]

By the Cauchy–Schwarz inequality and Lemma 1 in [17], the series (10) converges uniformly on $\overline{D}$ in view of (11):
\[ \begin{multline*}
\sum\limits_{m=1}^{\infty} |u_m(t) X_m(x)| \leqslant
C_1 \biggl( \sum\limits_{m=1}^{\infty} \frac{X_m^2(x)}{\lambda_m^{\left[\frac{n}{2}\right]+1}}
\sum\limits_{m=1}^{\infty} \varphi_m^2 \lambda_m^{\left[\frac{n}{2}\right]+1} \biggr)^{1/2} + {}
\\
{} + C_2 \biggl( \sum\limits_{m=1}^{\infty} \frac{X_m^2(x)}{\lambda_m^{\left[\frac{n}{2}\right]+1}} \sum\limits_{m=1}^{\infty} \|f_m\|_0^2 \lambda_m^{\left[\frac{n}{2}\right]+1} \biggr)^{1/2} \leqslant{}
\\
{}
\leqslant \widetilde{C}_1 \int_{\Omega} \bigl(\Delta^{[\frac{n}{2}]+1} \varphi\bigr)^2 \,dx +
\widetilde{C}_2 \int_{\Omega} \bigl(\Delta^{[\frac{n}{2}]+1} \|f(x,{}\cdot{})\|\bigr)^2 dx.
\end{multline*} \]

Differentiating the series in (10) term-wise, we obtain:
\[ \begin{equation}
u_t = \sum\limits_{m=1}^{\infty} u'_m(t) X_m(x),
\end{equation} \tag{13} \]
\[ \begin{equation}
u_{x_i x_i} = \sum\limits_{m=1}^{\infty} u_m(t) \frac{\partial^2 X_m(x)}{\partial x_i^2}, \quad i = 1,2,\ldots,n,
\end{equation} \tag{14} \]
\[ \begin{equation}
u_{x_i x_i t} = \sum\limits_{m=1}^{\infty} u'_m(t) \frac{\partial^2 X_m(x)}{\partial x_i^2}, \quad i = 1,2,\ldots,n.
\end{equation} \tag{15} \]

Obviously, that if either series (14) or (15) converges uniformly, then series (13) also converges uniformly.

For series (14), using (11) and (A1), and applying the Cauchy–Schwarz inequality for $(x,t) \in D$ and $i = 1,2,\dots,n$, we have:
\[ \begin{equation*}
\biggl| \sum\limits_{m=1}^{\infty} u_m(t) \frac{\partial^2 X_m(x)}{\partial x_i^2} \biggr|
\leqslant C_3 \biggl[ \int_{\Omega} \bigl(\Delta^{[\frac{n}{2}]+3} \varphi\bigr)^2 \,dx +
\int_{\Omega} \bigl(\Delta^{[\frac{n}{2}]+3} \|f(x,{}\cdot{})\|\bigr)^2 dx \biggr].
\end{equation*} \]

Consequently, the series (14), as well as (13) and (15), converge uniformly in $\overline{D}$.

These results lead to the following theorem.

Theorem 2. Let $\varphi(x)$ and $f(x,t)$ satisfy condition (A1), and let $k(t) \in C[0, T]$. Then problem (1)–(3) admits a classical solution $u \in C(\overline{D}) \cap C_{x,t}^{2,1}(D)$ defined by the series (10).

1.3. A Priori Estimates

This subsection establishes estimates for the solution and its first derivative in the direct problem (1)–(3). In the following section, we will prove that problem (1)–(4) has a unique solution for any $T > 0$. For this purpose, we employ weighted norms: for each $\sigma \geqslant 0$, we define the Bielecki norm
\[ \begin{equation*}
\|k\|_{\sigma} = \max\limits_{t\in[0,T]} \bigl(e^{-\sigma t}|k(t)|\bigr).
\end{equation*} \]

Remark. The weighted norm eliminates restrictions on the maximum value of $T$. In contrast, using the standard supremum norm would require $T$ to be smaller than some finite quantity depending on the problem's data.

The space $C_{\sigma}[0,T] := (C[0,T], \|\cdot\|_{\sigma})$ forms a Banach space, and the norms $\|{}\cdot{}\|_{\sigma}$ and $\|{}\cdot{}\|_0$ are equivalent. Moreover, the convolution operator is both commutative and invariant under multiplication by $e^{-\sigma t}$:
\[ \begin{equation*}
(f * g)(t) = (g * f)(t), \quad \text{and} \quad
e^{-\sigma t}(f * g)(t) = \bigl(e^{-\sigma t}f(t)\bigr) * \bigl(e^{-\sigma t}g(t)\bigr).
\end{equation*} \]
Additionally, we have the estimate
\[ \begin{equation}
\|f * g\|_{\sigma} \leqslant \frac{1}{\sigma} \|f\|_0 \|g\|_{\sigma}, \quad \sigma > 0
\end{equation} \tag{16} \]
(see [16]).

Let $\widetilde{u}_m$ denote the solution of (7), (8) with coefficients $\widetilde{\varphi}_m$, $\widetilde{f}_m$, and $\widetilde{k}$. From (9), we estimate the difference $u_m - \widetilde{u}_m$ in the Bielecki norm:
\[ \begin{multline}
e^{-\sigma t}|u_m(t) - \widetilde{u}_m(t)| \leqslant |\varphi_m - \widetilde{\varphi}_m| + t\|f_m - \widetilde{f}_m\|_0 +{}\\
{}+ t^2\|u_m\|_0\|k - \widetilde{k}\|_{\sigma} + t\|\widetilde{k}\|_{\sigma} \int_0^t e^{-\sigma s}|u_m - \widetilde{u}_m|(s) \,ds.
\end{multline} \]

Applying Gronwall's lemma for all $t \in [0,T]$ and $m \in \mathbb{N}$ yields:
\[ \begin{equation}
\|u_m - \widetilde{u}_m\|_{\sigma} \leqslant
\bigl( |\varphi_m - \widetilde{\varphi}_m| + T\|f_m - \widetilde{f}_m\|_{0} + T^2\|k - \widetilde{k}\|_{\sigma}\|u_m\|_{0} \bigr) e^{T^2\|\widetilde{k}\|_{\sigma}}.
\end{equation} \tag{17} \]

Theorem 2 established that problem (1)–(3) possesses a unique classical solution in $\overline{D}$. Consequently, for all $t \geqslant 0$, $u_t$ belongs to $C(\overline{D})$, and the difference of its Fourier coefficients satisfies:
\[ \begin{multline}
\|u_m' - \widetilde{u}_m'\|_{\sigma} \leqslant \|u_m - \widetilde{u}_m\|_{\sigma} +
T\|u_m\|_0\|k - \widetilde{k}\|_{\sigma} +{}
\\
{}+ T\|\widetilde{k}\|_{\sigma}\|u_m - \widetilde{u}_m\|_{\sigma} + \|f_m - \widetilde{f}_m\|_0.
\end{multline} \tag{18} \]

2. The Existence and Uniqueness Theorem for the Inverse Problem

This section investigates the inverse problem of determining the functions $u(x,t)$ and $k(t)$ from relations (1)–(4). We employ the contraction mapping principle to solve this problem.

Let
\[ \begin{equation*}
\mu = \biggl(\sum\limits_{m=1}^{\infty}\frac{\lambda_m}{1+\lambda_m}\varphi_m X_m(x_0)\biggr)^{-1} \neq 0.
\end{equation*} \tag{A2} \]
Under condition (A1), the numerical series
\[ \begin{equation*}
\sum_{m=1}^{\infty}\frac{\lambda_m}{1+\lambda_m}\varphi_m X_m(x_0)
\end{equation*} \]
converges.

Substituting $x = x_0$ into (10) and using (4), we obtain
\[ \begin{equation}
h(t) = \sum\limits_{m=1}^{\infty} u_m(t) X_m(x_0), \quad t \in [0,T].
\end{equation} \tag{19} \]

Replacing $u_m(t)$ in (19) with the right-hand side of (9) and differentiating twice yields the integral equation for $k(t)$:
\[ \begin{multline}
k(t) = k_0(t) + \mu\sum\limits_{m=1}^{\infty}\Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^2 (k * u_m)(t) X_m(x_0) -{}
\\
{}- \mu\sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^3 \bigl(\chi_m * (k * u_m)(\tau)\bigr)(t) X_m(x_0) - {}
\\
{}- \mu\sum\limits_{m=1}^{\infty} \frac{\lambda_m}{1+\lambda_m} (k * u_m')(t) X_m(x_0),
\end{multline} \tag{20} \]
where
\[ \begin{multline*}
k_0(t) = -\mu h''(t) + \mu\sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^2 \chi_m(t) \varphi_m X_m(x_0) -{}
\\
{}- \mu\sum\limits_{m=1}^{\infty}\frac{\lambda_m}{(1+\lambda_m)^2} f_m(t) X_m(x_0) + \mu\sum\limits_{m=1}^{\infty}\frac{1}{1+\lambda_m} f_m'(t) X_m(x_0) +{} \\
{}+ \mu\sum\limits_{m=1}^{\infty}\frac{\lambda_m^2}{(1+\lambda_m)^3} (\chi_m * f_m)(t) X_m(x_0).
\end{multline*} \]

Assume the following regularity conditions:
\[ \begin{equation*}
\begin{cases}
h \in C^2[0,T], \quad f \in C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega)) \cap C^1([0,T]; H^{\left[\frac{n}{2}\right]-1}(\Omega)), \\
\varphi(x_0) = h(0), \\
f({}\cdot{},t)=\Delta f({}\cdot{},t)=\cdots=
\Delta^{\left[\frac{n+4}{4}\right]}f({}\cdot{},t)\in H_0^1(\Omega),\quad t\in[0,T].
\end{cases}
\end{equation*} \tag{A3} \]

Equation (20) can be expressed as the fixed-point equation
\[ \begin{equation}
k = Ak
\end{equation} \tag{21} \]
for the operator $A$ defined by
\[ \begin{multline*}
Ak(t) = k_0(t) + \mu\sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^2 (k * u_m)(t) X_m(x_0) -{}\\
{}- \mu\sum\limits_{m=1}^{\infty}\Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^3 \bigl(\chi_m * (k * u_m)(\tau)\bigr)(t) X_m(x_0) -{} \\
{}- \mu\sum\limits_{m=1}^{\infty} \frac{\lambda_m}{1+\lambda_m} (k * u_m')(t) X_m(x_0).
\end{multline*} \]

To establish that $A$ has a fixed point, we first demonstrate that $A$ maps a closed convex set into itself in the space $C[0,T]$ equipped with the Bielecki norm.

Lemma 2. Under conditions (A1)–(A3), there exists $\sigma_0 > 0$ such that for all $\sigma \geqslant \sigma_0$, there exists $R > 0$ for which the closed convex ball
\[ \begin{equation*}
K = \{k \in C[0,T] : \|Ak - k_0\|_{\sigma} \leqslant R\}
\end{equation*} \]
is invariant under $A$, i.e., $A(K) \subset K$.

Proof. For any $k \in C[0,T]$, $t \in [0,T]$, and $\sigma > 0$, estimate (16) yields:
\[ \begin{multline}
\|Ak - k_0\|_{\sigma} \leqslant
\max\limits_{t \in [0,T]} \biggl| \mu\sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^2 e^{-\sigma t} (k * u_m)(t) X_m(x_0) \biggr| +{}
\\
{}+ \max\limits_{t \in [0,T]} \biggl| \mu\sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^3 e^{-\sigma t} \bigl(\chi_m * (k * u_m)(\tau)\bigr)(t) X_m(x_0) \biggr| +{}
\\
{}+ \max\limits_{t \in [0,T]} \biggl| \mu \sum\limits_{m=1}^{\infty} \frac{\lambda_m}{1+\lambda_m} e^{-\sigma t} (k * u_m')(t) X_m(x_0) \biggr| \leqslant{}
\\
{}\leqslant \frac{\|k\|_{\sigma}}{\sigma} |\mu| \sum\limits_{m=1}^{\infty} \|u_m\|_0 |X_m(x_0)| + \frac{\|k\|_{\sigma}}{\sigma} |\mu| \sum\limits_{m=1}^{\infty} \|\chi_m * u_m'\|_0 |X_m(x_0)| \\
+ \frac{\|k\|_{\sigma}}{\sigma} |\mu| \sum\limits_{m=1}^{\infty} \|u_m'\|_0 |X_m(x_0)| := I_1 + I_2 + I_3.
\end{multline} \tag{22} \]

For $k \in K$, we have
\[ \begin{equation}
\|k\|_{\sigma} \leqslant \|k_0\|_0 + R := R_0,
\end{equation} \tag{23} \]
since $\|\cdot\|_{\sigma} \leqslant \|\cdot\|_0$.

Applying Lemma 1 and (A1) to $I_1$ with (23) gives:
\[ \begin{multline}
I_1 \leqslant \frac{R_0}{\sigma} \max\{1,T\} e^{\|k\|_0 T^2/2} |\mu| \sum\limits_{m=1}^{\infty} \bigl(|\varphi_m| + \|f_m\|_0\bigr) |X_m(x_0)| \leqslant{} \\
{}
\leqslant \frac{R_0}{\sigma} \max\{1,T\} e^{\|k\|_0 T^2/2} |\mu|
\Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)} + \|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega))}\Bigr) := \frac{\widetilde{\sigma}_1}{\sigma}.
\end{multline} \tag{24} \]

Similarly, for $I_2$:
\[ \begin{multline}
I_2 \leqslant \frac{R_0}{\sigma} T |\mu| \sum\limits_{m=1}^{\infty} \|u_m'\|_0 |X_m(x_0)| \leqslant \frac{R_0}{\sigma} T |\mu| \sum\limits_{m=1}^{\infty} \|f_m\|_0 |X_m(x_0)| +{}
\\
{}+ \frac{R_0}{\sigma} \max\{1,T\} (1 + R_0 T) T e^{\|k\|_0 T^2/2} |\mu| \sum\limits_{m=1}^{\infty}
\bigl(|\varphi_m| + \|f_m\|_0\bigr) |X_m(x_0)| \leqslant{}
\\
\leqslant \frac{R_0}{\sigma} T |\mu| \|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega))} + \frac{R_0}{\sigma} \max\{1,T\} (1 + R_0 T) \times{} \\
{}\times T e^{\|k\|_0 T^2/2} |\mu| \Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)} + \|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega))}\Bigr) := \frac{\widetilde{\sigma}_2}{\sigma}.
\end{multline} \tag{25} \]

For $I_3$, we obtain:
\[ \begin{multline}
I_3 \leqslant \frac{R_0}{\sigma} |\mu| \|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega))} + \frac{R_0}{\sigma} \max\{1,T\} (1 + R_0 T) \times{} \\
{}\times e^{\|k\|_0 T^2/2} |\mu| \Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)} + \|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega))}\Bigr) := \frac{\widetilde{\sigma}_3}{\sigma}.
\end{multline} \tag{26} \]

Combining (24)–(26) for (22) yields
\[ \begin{equation*}
\|Ak - k_0\|_{\sigma} \leqslant \frac{\widetilde{\sigma}_0}{\sigma},
\end{equation*} \]
where $\widetilde{\sigma}_0 := \widetilde{\sigma}_1 + \widetilde{\sigma}_2 + \widetilde{\sigma}_3$. Choosing $\sigma \geqslant \sigma_0 := (1/R)\widetilde{\sigma}_0$ ensures $A(K) \subset K$. $\square$

Lemma 3. Under the same conditions as in Lemma 2, the family $(A(k))_{k\in K}$ is contractive, i.e., there exists $q\in[0,1)$ such that
\[ \begin{equation*}
\|Ak-A\tilde{k}\|_{\sigma} \leqslant q\|k-\widetilde{k}\|_{\sigma}
\end{equation*} \]
for all $k$, $\widetilde{k}\in K$.

Proof. From the commutative and invariant properties of the convolution operator, we have
\[ \begin{equation*}
e^{-\sigma t}v_1*v_2(t) - e^{-\sigma t}\widetilde{v}_1*\widetilde{v}_2(t) =
e^{-\sigma t}(v_1-\widetilde{v}_1)*v_2(t) + e^{-\sigma t}\widetilde{v}_1*(v_2-\widetilde{v}_2)(t)
\end{equation*} \]
and
\[ \begin{equation*}
\|v_1*v_2(t) - \widetilde{v}_1*\widetilde{v}_2(t)\|_{\sigma} \leqslant
\frac{1}{\sigma}\bigl(\|v_1-\widetilde{v}_1\|_{\sigma}\|v_2\|_0 + \|\widetilde{v}_1\|_0\|v_2-\widetilde{v}_2\|_{\sigma}\bigr).
\end{equation*} \]

For any $k$, $\widetilde{k}\in K$, we estimate
\[ \begin{multline}
\|Ak - A\widetilde{k}\|_{\sigma} \leqslant
\frac{|\mu|}{\sigma} \sum\limits_{m=1}^{\infty}\Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^2
\bigl(\|k-\widetilde{k}\|_{\sigma}\|u_m\|_0 + \|\widetilde{k}\|_0\|u_m-\widetilde{u}_m\|_{\sigma}\bigr)|X_m(x_0)| +{}
\\
{}+ \frac{|\mu|}{\sigma} \sum\limits_{m=1}^{\infty} \Bigl(\frac{\lambda_m}{1+\lambda_m}\Bigr)^3
\bigl(\|k-\widetilde{k}\|_{\sigma}\|\chi_m*u_m\|_0 + \|\widetilde{k}\|_0\|\chi_m*(u_m-\widetilde{u}_m)\|_{\sigma}\bigr)|X_m(x_0)| +{}\\
{}+ \frac{|\mu|}{\sigma} \sum\limits_{m=1}^{\infty}\frac{\lambda_m}{1+\lambda_m}
\bigl(\|k-\widetilde{k}\|_{\sigma}\|u'_m\|_0 + \|\widetilde{k}\|_0\|u'_m-\widetilde{u}'_m\|_{\sigma}\bigr)|X_m(x_0)|
:= \hat{I}_1 + \hat{I}_2 + \hat{I}_3.
\end{multline} \tag{27} \]

We now estimate each term in (27). Using Lemma 1, (A1), and (17) for $\hat{I}_1$, we obtain
\[ \begin{multline*}
\hat{I}_1\leqslant \frac{|\mu|}{\sigma}\bigl(1+T^2e^{T^2R_0}\|\tilde{k}\|_0\bigr) \|k-\widetilde{k}\|_{\sigma}\sum\limits_{m=1}^{\infty}\|u_m\|_0|X_m(x_0)| \leqslant{}
\\
{}
\leqslant\frac{|\mu|}{\sigma}\bigl(1+T^2e^{T^2R_0}\|\tilde{k}\|_0\bigr)
\max\{1,T\}e^{\|k\|_0T^2/2}\|k-\widetilde{k}\|_{\sigma}
\sum\limits_{m=1}^{\infty} \bigl(|\varphi_m|+\|f_m\|_0 \bigr)|X_m(x_0)| \leqslant{}
\\
{}
\leqslant \frac{|\mu|}{\sigma}\bigl(1+T^2e^{T^2R_0}\|\tilde{k}\|_0\bigr)
\max\{1,T\}e^{\|k\|_0T^2/2}\|k-\tilde{k}\|_{\sigma} \times {}
\\
{}\times
\Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)}+\|f\|_{C([0,T];H^{\left[\frac{n}{2}\right]+1}(\Omega)}\Bigr):= \frac{\hat{\sigma}_1}{\sigma}\|k-\widetilde{k}\|_{\sigma}.
\end{multline*} \]

For $\hat{I}_2$, we have
\[ \begin{multline*}
\hat{I}_2 \leqslant \frac{|\mu|}{\sigma}\Bigl(T + T^2 e^{T^2 R_0}\frac{\|\widetilde{k}\|_0}{\sigma}\Bigr) \|k-\widetilde{k}\|_{\sigma} \sum\limits_{m=1}^{\infty} \|u_m\|_0 |X_m(x_0)| \leqslant {}
\\
\leqslant \frac{|\mu|}{\sigma}\Bigl(T + T^2 e^{T^2 R_0}\frac{\|\widetilde{k}\|_0}{\sigma}\Bigr)
\max\{1, T\} e^{\|k\|_0 T^2/2} \|k-\widetilde{k}\|_{\sigma} \times{}
\\
{}\times \Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)} + \|f\|_{C([0,T];H^{\left[\frac{n}{2}\right]+1}(\Omega))}\Bigr) := \frac{\hat{\sigma}_2}{\sigma} \|k-\widetilde{k}\|_{\sigma}.
\end{multline*} \]

Similarly, for $\hat{I}_3$, Lemma 1 yields
\[ \begin{multline*}
\hat{I}_3 \leqslant \frac{|\mu|}{\sigma}\|k-\widetilde{k}\|_{\sigma}\sum\limits_{m=1}^{\infty}\|u_m'\|_0|X_m(x_0)| +{}
\\
{}+ \frac{|\mu|}{\sigma} \bigl(T+T^2e^{T^2R_0}+T^3e^{T^2R_0}\bigr)\|\tilde{k}\|_0\|k-\widetilde{k}\|_{\sigma}
\sum\limits_{m=1}^{\infty}\|u_m\|_0|X_m(x_0)| \leqslant{}
\\
{}
\leqslant \frac{|\mu|}{\sigma}\|k-\widetilde{k}\|_{\sigma}\sum\limits_{m=1}^{\infty}\|f_m\|_0|X_m(x_0)|
+{} \hspace{6cm}
\\
{}
+\frac{|\mu|}{\sigma} \|k-\widetilde{k}\|_{\sigma} \max\{1,T\} \bigl(1+\|k\|_0T\bigr) e^{\|k\|_0T^2/2}
\sum\limits_{m=1}^{\infty} \bigl(|\varphi_m|+\|f_m\|_0\bigr)|X_m(x_0)|+{}
\\
{}+\frac{|\mu|}{\sigma}\bigl(T+T^2e^{T^2R_0}+T^3e^{T^2R_0}\bigr) \max\{1, T\} e^{\|k\|_0T^2/2}\|\tilde{k}\|_0
\|k-\widetilde{k}\|_{\sigma} \times{}
\\
\hspace{6cm}
{}\times\sum\limits_{m=1}^{\infty} \bigl(|\varphi_m|+\|f_m\|_0\bigr)|X_m(x_0)|\leqslant {}
\\
{}\leqslant \frac{|\mu|}{\sigma}\|f\|_{C([0,T]; H^{\left[\frac{n}{2}\right]+1}(\Omega)}\|k-\widetilde{k}\|_{\sigma}
+\frac{|\mu|}{\sigma}\max\{1, T\} \bigl(1+\|k\|_0T \bigr) e^{\|k\|_0T^2/2} \times{}
\\
\hspace{4cm}
{}\times
\Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)} +
\|f\|_{C([0,T];H^{\left[\frac{n}{2}\right]+1}(\Omega)}\Bigr)\|k-\widetilde{k}\|_{\sigma} +{}
\\
{}
+\frac{|\mu|}{\sigma}\bigl(T+T^2e^{T^2R_0}+T^3e^{T^2R_0}\bigr) \max\{1,T\}e^{\|k\|_0T^2/2} \|\tilde{k}\|_0
\times{} \hspace{3cm}
\\
{}\times
\Bigl(\|\varphi\|_{H^{\left[\frac{n}{2}\right]+1}(\Omega)}+\|f\|_{C([0,T];H^{\left[\frac{n}{2}\right]+1}(\Omega)}\Bigr)\|k-\widetilde{k}\|_{\sigma}
:=\frac{\hat{\sigma}_3}{\sigma}\|k-\widetilde{k}\|_{\sigma}.
\end{multline*} \]
Choosing $q := {\hat{\sigma}_0}/{\sigma} < 1$, where $\hat{\sigma}_0 := \max\{\hat{\sigma}_1, \hat{\sigma}_2, \hat{\sigma}_3\}$, establishes that $A$ is a contraction on $K$, completing the proof. $\square$

By the Banach fixed-point theorem, equation (21) has a unique solution for any $T > 0$, yielding:

Theorem 3. Under assumptions (A1)–(A3), for any $T > 0$, problem (1)–(4) admits a unique solution.

Conclusion

This study has established the existence and uniqueness of a solution to the inverse problem of determining the kernel of a multidimensional third-order integrodifferential pseudo-parabolic equation. Our approach combines the Fourier method, integral inequalities, and the fixed-point principle, with the solution specified by an additional condition at a fixed point for the first boundary value problem.

All results presented in this article remain valid when the Laplacian operator $\Delta$ in (1) is replaced by a more general self-adjoint differential operator $L$ defined in the domain $\Omega$. This operator takes the form:
\[ \begin{equation*}
L = \sum\limits_{i, j=1}^n \frac{\partial}{\partial x_i} \Bigl[a_{ij}(x)\frac{\partial}{\partial x_j}\Bigr] - c(x),
\end{equation*} \]
where the coefficients satisfy:

• symmetry: $a_{ij}(x) = a_{ji}(x)$ for all $i$, $j$;
• uniform ellipticity: $\sum_{i, j=1}^n a_{ij}(x)\xi_i\xi_j \geqslant \alpha\sum_{i=1}^n\xi_i^2$ with $\alpha = \text{const} > 0$;
• non-negativity: $c(x) \geqslant 0$ in $\Omega$.

We additionally assume the coefficients $a_{ij}(x)$ and $c(x)$ satisfy appropriate smoothness conditions (see [17] for details).

Competing interests. The authors declare no conflict of interest regarding the authorship and publication of this article.
Authors’ contributions and responsibilities. All authors contributed equally to the development of the article’s concept and the writing of the manuscript. The authors take full responsibility for submitting the final version of the manuscript for publication. The final version of the manuscript was approved by all authors.
Funding. The research was conducted without external funding.

About the authors

Durdimurod K. Durdiev

Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University

Email: d.durdiev@mathinst.uz
ORCID iD: 0000-0002-6054-2827
http://www.mathnet.ru/person29112

Dr. Phys. & Math. Sci., Professor; Head of Branch¹; Professor, Dept. of Differential Equations²

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11

Hilola B. Elmuradova

Bukhara State University

Email: helmuradova@mail.ru
ORCID iD: 0000-0003-4306-2589
https://www.mathnet.ru/person228134

Teacher; PhD Student; Dept. of Differential Equations²

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11

Askar A. Rahmonov

Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University

Author for correspondence.
Email: araxmonov@mail.ru
ORCID iD: 0000-0002-7641-9698
https://www.mathnet.ru/person67047

Cand. Phys. & Math. Sci., Associate Professor; Senior Researcher¹; Associate Professor, Dept. of Differential Equations²

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11

References

Supplementary files