Well-Posedness of Problems for the Heat Equation with a Fractional-Loaded Term and Memory

Abstract

We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, the problem is reformulated as a Volterra-type integral equation. Under appropriate regularity assumptions, we establish existence and uniqueness of classical solutions. Furthermore, we address an inverse problem aimed at simultaneously recovering the memory kernel and the solution. Using a differentiability-based approach, we derive a stable and well-posed formulation that enables the identification of memory effects in fractional heat models.

1. Introduction

Modeling heat conduction and diffusion processes in complex physical systems requires taking into account memory and nonlocal effects that are not captured adequately by classical heat equations. The presence of hereditary effects, temporal nonlocality, and heterogeneity in the medium motivate the use of fractional calculus and convolution-type integral terms within parabolic partial differential equations. These models are essential for accurately describing anomalous diffusion in porous and viscoelastic media, biological tissues, and advanced materials with memory, which are relevant in materials science, geophysics, and thermodynamics [1,2,3,4,5]. Recent studies emphasize the necessity of including fractional and memory-dependent terms to capture experimentally observed nonlocal heat transport phenomena in micro- and nano-scale systems [6,7,8,9].

The objective of this paper is to study a time fractional heat equation with memory and loading, to establish its well-posedness, and to address an associated inverse problem for identifying the unknown memory kernel. The motivation arises from the need to accurately describe anomalous heat conduction and diffusion processes in media with hereditary and nonlocal effects.

Within this framework, we study a time–fractional heat equation with loading in ${\mathbb{R}}^{n+1}$, which involves a Riemann–Liouville type fractional integral operator together with a memory contribution expressed via convolution. The kernel $\kappa (x,t)$ is assumed to depend on n spatial variables and time, and is taken in the separable form

$$\kappa (x_1,x_2,...x_n,t)=p\left(x_n\right)q(x_1,x_2,...x_{n-1},t),x\in {\mathbb{R}}^n,t>0,$$

which enables the memory effect to be analyzed through convolutional representations and facilitates inverse parameter recovery.

We begin by constructing the fundamental solution of the related parabolic problem with variable coefficients, relying on Fourier transform methods and paramet rix techniques [10,11,12]. Provided that the initial data are given by $u(x,0)=\phi \left(x\right)$ on $\mathbb{R},$ the corresponding differential equation can be reformulated in an integral framework and represented as a Volterra-type relation.

Assuming appropriate regularity and growth conditions, we prove the existence and the uniqueness of solutions in the classical sense in anisotropic Holder spaces $H^{l,l}({\mathbb{R}}^n\times (0,T))$ following the frameworks developed in [13,14,15]. These results ensure that the proposed model is well-posed for practical applications and allow further theoretical analysis and numerical implementation.

Furthermore, we consider an inverse problem aimed at simultaneously identifying the unknown kernel function $q(x_1,x_2,...x_{n-1},t)$ together with the state variable $u(x,t)$ of the loaded heat equation, based on supplementary integral or boundary measurements. Through techniques relying on differentiability arguments, this inverse setting can be transformed into an auxiliary formulation that guarantees well-posedness for the pair $(u,q)$. Using tools from operator theory and integral equations [2,16,17,18,19,20], we establish conditions ensuring stable identification, contributing to the theoretical foundation of recovering memory kernels in fractional parabolic models.

The paper is organized in the following way. Section 2 provides the required preliminaries, formulates the Cauchy problem, and develops the corresponding funda mental solution, after which the formulation is recast into a Volterra-type integral representation. Section 3 addresses the issues of existence and uniqueness of solutions under suitable assumptions. Section 4 is devoted to the inverse problem, where both the memory kernel and the solution are to be determined, and stability together with uniqueness results are established. Finally, Section 5 presents concluding remarks and points to possible applications in the study of anomalous heat transfer and viscoelastic media with hereditary effects.

To rigorously justify the inclusion of memory and nonlocal effects within heat conduction and anomalous diffusion models, various theoretical frameworks and numerical analyses have been developed. Classical foundations on parabolic equations with variable coefficients and fundamental solutions are presented in [10,11], while the analysis of fractional models for anomalous diffusion in complex media is thoroughly discussed in [3,4,5]. Recent works have highlighted inverse and identification problems for memory kernels within heat equations [1,15,17,21,22], as well as the stability and regularity analysis of fractional parabolic models in both theoretical and applied settings [2,8,13,14,20]. For the analysis and application of convolution-type memory terms in evolution equations, we refer to [16,18,19]. Advances in modeling heat conduction with fractional Cattaneo and generalized fractional operators are presented in [6,23,24,25,26,27], complementing the rigorous inverse problem frameworks for fractional PDEs established in [28,29,30]. The practical need for these fractional and memory-dependent formulations is further supported by experimental studies indicating nonlocal heat transport phenomena at micro- and nano-scales [8,9,31], emphasizing the relevance of these models in geophysics, viscoelastic materials, and biological tissue modeling [12,32,33,34,35,36].

Recent developments in fractional modeling further demonstrate the breadth of applications of nonlocal operators. Examples include the fractional-order Ambartsumian equation with exponential decay kernels, fractional delayed models of plankton dynamics, and fractional kinetic models describing heterogeneous catalytic processes such as the hydrogenolysis of glycerol. These works illustrate the wide relevance of fractional calculus across physics, chemistry, and biology [37,38,39,40].

2. Problem Statement: Cauchy Problem for a Loaded Heat Equation

We consider the Cauchy problem for a heat equation that involves a time-dependent diffusion coefficient, a fractional integral contribution, and a memory term of convolution type on the domain $(x,t)\in {\mathbb{R}}^n\times (0,T)$:

• Cauchy Problem. Find a function $u(x,t)$ satisfying

$${\partial }_{t}u(x,t)-\delta \left(t\right)\Delta u(x,t)=\lambda D_{0t}^{-\alpha }u(x_1,\dots ,x_{n-1},t)+{\int }_0^t\kappa (x,s)u(x,t-s)ds,$$

(1)

together with the condition

$$u(x,0)=\phi \left(x\right),x\in {\mathbb{R}}^n,$$

(2)

where $\Delta$ denotes the Laplace operator in the spatial coordinates $x_1,\dots ,x_n$, and $D_{0t}^{-\alpha }$ is the fractional integration operator in the Riemann–Liouville with index $\alpha \in (0,1)$. The diffusion coefficient $\delta \left(t\right)$ and function $\phi \left(x\right)$ are assumed to belong to the class

$$\delta \left(t\right)\in E:=\left\{\delta \left(t\right)\in C^1[0,T]:0<{\delta }_0<\delta \left(t\right)\le {\delta }_1<\infty \right\},$$

(3)

$$\phi \left(x\right)\in H^{l+2}\left({\mathbb{R}}^n\right),\phi \left(x\right)\le {\phi }_0=\mathrm{const}>0,$$

(4)

were ${\delta }_0,{\delta }_1>0$ are constants ensuring the uniform parabolic nature of the operator. We denote the boundary trace of u at $x_n=0$ by $u(x_1,\dots ,x_{n-1},t)$. Problems of this class naturally appear in the modeling of anomalous diffusion processes with hereditary effects as well as in viscoelastic materials.

We assume that the kernel of the memory operator admits a separable representation of the form

$$\kappa (x,t)=\psi \left(x_n\right)\chi (x^{\prime },t),x^{\prime }=(x_1,\dots ,x_{n-1}),$$

and satisfies the regularity conditions

$$\psi \left(x_n\right)\in H^{l+2}\left(\mathbb{R}\right),\chi (x^{\prime },t)\in H^{l,(l+2)/2}\left({\overline{\mathbb{R}}}_T^{n-1}\right).$$

(5)

These assumptions ensure well-posedness in appropriate Hölder spaces while accommodating the nonlocal and memory-dependent nature of the problem. In order to formulate and analyze the existence and uniqueness results for the fractional parabolic equation, it is necessary to work within suitable functional spaces that capture the regularity of solutions both in space and in time. Since the problem involves spatial derivatives and temporal fractional operators of parabolic type, the natural setting is provided by Hölder and anisotropic Hölder spaces, which ensure the continuity and smoothness properties required by classical and fractional parabolic theory (see, e.g., [10,11,13,14]).

Definition 1.

We call a function u an element of the Holder space $H^l(\Omega )$ if it has continuous derivatives up to order l in the domain $\Omega \subset {\mathbb{R}}^n$, and these derivatives satisfy the standard Holder continuity conditions. Similarly, the anisotropic Holder space $H^{l,l/2}\left(R_T^n\right)$ consists of functions that are l-Holder continuous in space and $l/2$-Holder continuous in time in $R_T^n={\mathbb{R}}^n\times (0,T)$.

For detailed definitions and properties, we refer to [13].

3. Well-Posedness of the Classical Problem

Using the representation of the fundamental solution to the heat operator with time-dependent diffusivity [13], the original Cauchy problem can be reformulated as a Volterra-type integral relation of the form

$$\begin{array}{c}u(x,t)={\int }_{{\mathbb{R}}^n}\phi \left(s\right)\Gamma (x-s,\zeta \left(t\right))ds+\hfill \\ +{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )u(y,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)d\alpha dy\hfill \\ +{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}u(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy,\hfill \end{array}$$

(6)

where $\zeta \left(t\right)={\int }_0^t\delta \left(s\right)ds,{\zeta }^{-1}\left(t\right)\mathrm{denotes}\mathrm{its}\mathrm{inverse},$ and $\Gamma (x-y,\zeta (t)-s)$ is the fundamental solution associated with the operator ${\partial }_t-\delta \left(t\right)\Delta .$

Theorem 1.

Let $\alpha \in (0,1)$ and $t>1$. Assume there exist constants $k_0,h_0$ with $k_0{h}_0<1$, and suppose that

$$0<\left|\lambda \right|<(1-k_0{h}_0){\displaystyle \frac{\Gamma (\alpha +j)}{j!}},j=1,2,\dots ,$$

and that the functions $\delta \left(t\right)\in E$, $\phi \left(x\right)$, $\psi \left(x_n\right)$, and $\chi (x^{\prime },t)$ satisfy the standard regularity conditions (4)–(5). Then the associated integral equation admits a unique classical solution in $H^{l+2,(l+2)/2}\left({\mathbb{R}}_T^n\right)$.

Proof of Theorem 1.

Existence of the Solution. The solution is constructed using the method of successive approximations. Consider the sequence ${\left\{u_j(x,t)\right\}}_{j=0}^{\infty }$ defined as follows:

$$u_0(x,t)={\int }_{{\mathbb{R}}^n}\phi \left(s\right)\Gamma (x-s,\zeta \left(t\right))ds,$$

and

$$u_1(x,t)={\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{u}_0(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )u_0(y,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)d\alpha dy.$$

$$\dots \dots \dots$$

For $j\ge 1$, define the iterates

$$u_j(x,t)={\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{u}_{j-1}(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{{\mathbb{R}}^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )u_{j-1}(\xi ,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)d\alpha dy,$$

(7)

for $(x,t)\in {\mathbb{R}}_T^n$ and $j=1,2,\dots$.

Under the assumptions (3)–(5), we set

$${\phi }_0:={\left|\phi \left(x\right)\right|}^l,h_0:={\left|\psi \left(x_n\right)\right|}^l,k_0:={\parallel \chi (x^{\prime },t)\parallel }_T^{l,l/2}.$$

By estimating the norm of the successive approximations $u_j\left(x,t\right)$ from the integral Equation (6), we obtain convergence under the stated conditions. Specifically, we obtain

$${\left|u_0(x,t)\right|}_T^{l+2,(l+2)/2}\le {\int }_{R^n}{\left|\phi \left(y\right)\right|}_T^l\Gamma (x-y,\zeta \left(t\right))dy\le {\phi }_0,$$

$${\left|u_1(x,t)\right|}_T^{l+2,(l+2)/2}\le {\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\left|\delta \left({\zeta }^{-1}\left(s\right)\right)\right|}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}$$

$$\times {\left|u_0(y^{\prime },\beta )\right|}_T\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\left|\delta \left({\zeta }^{-1}\left(s\right)\right)\right|}}$$

$$\times {\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left|\psi \left(y_n\right)\right|}_T^{l,l/2}{\left|\chi (y^{\prime },\alpha )\right|}_T{\left|u_0(y,{\zeta }^{-1}\left(s\right)-\alpha )\right|}_T^{\prime }\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha$$

$$\le {\phi }_0{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}{\displaystyle \frac{t^{\alpha }}{\alpha }}+{\phi }_0{\displaystyle \frac{{\delta }_1{h}_0{k}_{0}T}{{\delta }_0}}{\displaystyle \frac{t}{1!}}={\phi }_0{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\left({\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\displaystyle \frac{t^{\alpha }}{\alpha }}+k_0{h}_0{\displaystyle \frac{t}{1!}}\right).$$

(8)

In view of the assumptions of Theorem 1 and using the relation ${\int }_{{\mathbb{R}}^n}\Gamma (x-\xi ,\zeta \left(t\right)-\tau )d\xi =1,$ we obtain the following estimate:

$${\left|u_1(x,t)\right|}_T^{l+2,(l+2)/2}\le {\phi }_0{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\left({\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\displaystyle \frac{t^{\alpha }}{\alpha }}+k_0{h}_0{\displaystyle \frac{t}{1!}}\right)\le {\phi }_0{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\left[k_0{h}_0+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha +1\right)}}\right]{\displaystyle \frac{t}{1!}},$$

By continuing this process for $j=0,1,2,\dots$, we obtain:

$${\left|u_j(x,t)\right|}_T^{l+2,(l+2)/2}\le {\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\left|\delta \left({\zeta }^{-1}\left(s\right)\right)\right|}}{\int }_{R^n}$$

$$\times {\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{\left|u_{j-1}(y^{\prime },\beta )\right|}_T\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left|\psi \left(y_n\right)\right|}_T^{l,l/2}$$

$$\times {\left|\chi (y^{\prime },\alpha )\right|}_T{\left|u_{j-1}(y,{\zeta }^{-1}\left(s\right)-\alpha )\right|}_T\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha \le$$

$$\le {\phi }_0{\left({\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\right)}^j\left(k_0{h}_0+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha +1\right)}}\right)\left(k_0{h}_0+{\displaystyle \frac{2\left|\lambda \right|}{\Gamma \left(\alpha +2\right)}}\right)...\left(k_0{h}_0+{\displaystyle \frac{j!\left|\lambda \right|}{\Gamma \left(\alpha +j\right)}}\right){\displaystyle \frac{t^j}{j!}},$$

As a consequence, one can represent the solution in the form of the functional series

$$\sum _{j=0}^{\infty }{u}_j(x,t).$$

(9)

Based on the assumptions stated in Theorem 1, the following estimate holds:

$$\left[k_0{h}_0+{\displaystyle \frac{1!\left|\lambda \right|}{\Gamma (\alpha +1)}}\right]\le A,\dots ,\left[k_0{h}_0+{\displaystyle \frac{j!\left|\lambda \right|}{\Gamma (\alpha +j)}}\right]\le A,$$

that is, each multiplicative factor can be bounded by a constant $A\in \left(0,1\right)$ for $j=0,1,2,\dots$. Based on this, we estimate the functional series (9) by a numerical series over the domain $\left(x,t\right)\in R_T^n$ as follows:

$$\sum _{j=0}^{\infty }\left|u_j(x,t)\right|\le \sum _{j=0}^{\infty }{\phi }_0{\left({\displaystyle \frac{{\delta }_{1}TA}{{\delta }_0}}\right)}^j{\displaystyle \frac{t^j}{j!}}\le {\phi }_{0}exp\left({\displaystyle \frac{{\delta }_1{T}^{2}A}{{\delta }_0}}\right).$$

By applying the Weierstrass M-test for uniform convergence of functional series, we conclude that the above series converges uniformly. Hence, the sequence $u_j(x,t)$ constructed from the integral Equation (6) converges uniformly to a limit function $u(x,t)$. This establishes the existence of a solution to the Cauchy problem (1)–(2) within the class $H^{l+2,(l+2)/2}\left(R_T^n\right)$ and shows that the integral Equation (6) indeed admits a solution.

• Uniqueness of the Solution. To prove uniqueness, we proceed by contradiction. Assume that the integral Equation (6) admits two distinct solutions:

$$u^{\left(1\right)}(x,t)={\int }_{R^n}\phi \left(s\right)\Gamma (x-s,\zeta \left(t\right))ds+$$

$$+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{u}^{\left(1\right)}(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )u^{\left(1\right)}(y,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha ,$$

$$u^{\left(2\right)}(x,t)={\int }_{R^n}\phi \left(s\right)\Gamma (x-s,\zeta \left(t\right))ds+$$

$$+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{u}^{\left(2\right)}(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )u^{\left(2\right)}(y,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha .$$

Now, suppose there are two classical solutions $u^{\left(1\right)}$ and $u^{\left(2\right)}$ to the integral Equation (6). Considering their difference, denoted by $Z(x,t)$, we have

$$\begin{array}{c}Z(x,t)={\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}Z(y^{\prime },\beta )\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+\hfill \\ +{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\delta \left({\zeta }^{-1}\left(s\right)\right)}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}\psi \left(y_n\right)\chi (y^{\prime },\alpha )Z(y,{\zeta }^{-1}\left(s\right)-\alpha )\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha .\hfill \end{array}$$

(10)

This leads to a homogeneous Volterra integral equation. For every fixed $t\in [0,T]$, let us denote the supremum of $\left|Z\right(x,t\left)\right|$ over all $x\in {\mathbb{R}}^n$ by

$$\overline{Z}\left(t\right)=\underset{x\in {\mathbb{R}}^n}{sup}\left|Z(x,t)\right|.$$

From the integral relation (10), it follows that

$$\overline{Z}\left(t\right)\le {\displaystyle \frac{{\delta }_1}{{\delta }_0}}\left({\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}·{\displaystyle \frac{T^{\alpha }}{\alpha }}+k_0{h}_{0}T\right){\int }_0^{{\delta }_{1}t}\overline{Z}\left(\tau \right)d\tau .$$

Applying the Gronwall–Bellman inequality, we get

$$\overline{Z}\left(t\right)\equiv 0,t\in [0,T].$$

Hence, $Z(x,t)\equiv 0$ in the domain ${\overline{R}}_T^n$; it follows that the two possible solutions agree identically. This establishes the uniqueness of the solution to the integral Equation (6). □

Theorem 2.

If

$$0<\left|\lambda \right|<1-k_0{h}_0·{\displaystyle \frac{\Gamma (j\alpha +1)}{\Gamma \left(\alpha \right(j-1)+2)}},j=1,2,\dots ,$$

(11)

where $\alpha \in (0,1)$ and $0<t<1$, and if the conditions (3)–(5) are satisfied, then the integral Equation (6) admits a unique solution in $H^{l+2,(l+2)/2}\left(R_T^n\right).$

Proof of Theorem  2.

The proof proceeds in the same spirit as that of Theorem 1, employing the method of successive approximations. Under the assumptions of Theorem 2, together with conditions (3)–(5), and by introducing analogous notations, we derive the following bounds for the terms of the sequence $\left\{u_j(x,t)\right\}$:

$${\left|u_0(x,t)\right|}_T^{l+2,(l+2)/2}\le {\int }_{R^n}{\left|\phi \left(s\right)\right|}_T^l\Gamma (x-s,\zeta \left(t\right))ds\le {\phi }_0,$$

$${\left|u_1(x,t)\right|}_T^{l+2,(l+2)/2}\le {\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\left|\delta \left({\zeta }^{-1}\left(s\right)\right)\right|}}\times$$

$$\times {\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left({\zeta }^{-1}\left(s\right)-\beta \right)}^{\alpha -1}{\left|u_0(y^{\prime },\beta )\right|}_T\Gamma (x-y,\zeta \left(t\right)-s)d\beta dy+$$

$$+{\int }_0^{\zeta \left(t\right)}{\displaystyle \frac{ds}{\left|\delta \left({\zeta }^{-1}\left(s\right)\right)\right|}}{\int }_{R^n}{\int }_0^{{\zeta }^{-1}\left(s\right)}{\left|\psi \left(y_n\right)\right|}_T^{l,l/2}\times$$

$$\times {\left|\chi (y^{\prime },\alpha )\right|}_T^{l+2,(l+2)/2}{\left|u_0(y,{\zeta }^{-1}\left(s\right)-\alpha )\right|}_T\Gamma (x-y,\zeta \left(t\right)-s)dyd\alpha \le$$

$$\le {\phi }_0{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left(\alpha \right)}}{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}{\displaystyle \frac{t^{\alpha }}{\alpha }}+{\phi }_0{\displaystyle \frac{{\delta }_1{h}_0{k}_{0}T}{{\delta }_0}}{\displaystyle \frac{t}{1!}}\le {\phi }_0{\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\left[{\displaystyle \frac{k_0{h}_0\Gamma \left(\alpha +1\right)}{\Gamma \left(2\right)}}+\left|\lambda \right|\right]{\displaystyle \frac{t^{\alpha }}{\Gamma \left(\alpha +1\right)}},$$

$$\dots \dots \dots \dots$$

$${\left|u_j(x,t)\right|}_T^{l+2,(l+2)/2}\le$$

$$\le {\phi }_0{\left({\displaystyle \frac{{\delta }_{1}T}{{\delta }_0}}\right)}^j\left[k_0{h}_0{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{\Gamma \left(2\right)}}+\left|\lambda \right|\right]\left[k_0{h}_0{\displaystyle \frac{\Gamma \left(2\alpha +1\right)}{\Gamma \left(\alpha +2\right)}}+\left|\lambda \right|\right]...$$

$$...\left[k_0{h}_0{\displaystyle \frac{\Gamma \left(j\alpha +1\right)}{\Gamma \left(\alpha \left(j-1\right)+2\right)}}+\left|\lambda \right|\right]{\displaystyle \frac{t^{\alpha j}}{\Gamma \left(j\alpha +1\right)}},$$

As a result, by constructing the functional series as in (9) and denoting $A\in (0,1)$, we obtain the following estimate of the functional series by a finite series in the domain $(x,t)\in R_T^n$:

$$\sum _{j=0}^{\infty }\left|u_j(x,t)\right|\le \sum _{j=0}^{\infty }{\phi }_0{\left({\displaystyle \frac{{\delta }_{1}TA}{{\delta }_0}}\right)}^j{\displaystyle \frac{t^{\alpha j}}{\left(\alpha j\right)!}}\le \sum _{j=0}^{\infty }{\phi }_0{\left({\left[{\displaystyle \frac{{\delta }_{1}TA}{{\delta }_0}}\right]}^{\frac{1}{\alpha }}\right)}^{\alpha j}{\displaystyle \frac{t^{\alpha j}}{\left(\alpha j\right)!}}$$

$$\le {\phi }_0\sum _{j=0}^{\infty }{\left({\left[{\displaystyle \frac{{\delta }_{1}TA}{{\delta }_0}}\right]}^{\frac{1}{\alpha }}T\right)}^{\alpha j}{\displaystyle \frac{1}{\left(\alpha j\right)!}}={\phi }_0\sum _{j=0}^{\infty }{\left({\left[{\displaystyle \frac{{\delta }_{1}A}{{\delta }_0}}\right]}^{\frac{1}{\alpha }}{T}^{1+\frac{1}{\alpha }}\right)}^{\alpha j}{\displaystyle \frac{1}{\left(\alpha j\right)!}}$$

$$={\phi }_{0}exp\left({\left[{\displaystyle \frac{{\delta }_{1}A}{{\delta }_0}}\right]}^{\frac{1}{\alpha }}{T}^{1+\frac{1}{\alpha }}\right).$$

By the Weierstrass theorem on uniform convergence, the series converges uniformly, and therefore $u_j(x,t)$ tends to a function $u(x,t)\in H^{l+2,(l+2)/2}\left(R_T^n\right)$, which solves the Cauchy problem (1)–(2) and the integral Equation (6).

The uniqueness in this theorem is proved in a manner similar to that of Theorem 1. □

4. Inverse Problem for Recovering the Memory Term in a Fractionally Loaded Equation

In this section, we address the inverse problem of simultaneously determining the unknown function $u(x,t)$ and the kernel $\chi (x_1,\dots ,x_{n-1},t)$ in the convolution integral, for $(x,t)\in R_T^n$. The precise formulation of the problem is given below.

Inverse problem. Find functions $u(x,t)$ and $\chi (x^{\prime },t)$, defined for $(x,t)\in {\mathbb{R}}_T^n$, that satisfy the following system:

$$u_t(x,t)-\delta \left(t\right)\Delta u(x,t)=\lambda D_{0t}^{-\alpha }u(\overline{x},t)+{\int }_0^t\kappa (x,s)u(x,t-s)ds,$$

(12)

$$u(x,0)=\phi \left(x\right),x\in {\mathbb{R}}^n.$$

(13)

$${\left.u(x,t)\right|}_{x_n=0}=g(x^{\prime },t),g(x^{\prime },0)=\phi (x^{\prime },0),(x^{\prime },t)\in {\overline{R}}_T^{n-1},$$

(14)

where $D_{0t}^{-\alpha }$ denotes the Riemann-Liouville fractional operator [41] of order $\alpha >0$. Here $\phi \left(x\right)$ and $g(x^{\prime },t)$ are given functions representing the initial and boundary data, respectively. The convolution kernel $\kappa (x,t)$ is assumed to be separable in the spatial variable x:

$$\kappa (x,t)=\psi \left(x_n\right)\chi (x^{\prime },t),x^{\prime }=(x_1,\dots ,x_{n-1}),\overline{x}=(0,0,x_3,\dots ,x_n),$$

with $\psi \left(x_n\right)\in H^{l+2}\left(\mathbb{R}\right)$ and $\chi (x^{\prime },t)\in H^{l,(l+2)/2}\left({\overline{R}}_T^{n-1}\right)$. The coefficient $\delta \left(t\right)$ is assumed to belong to the class E.

$${\overline{R}}_T^{n-2}=\left\{(\overline{x},t)|\overline{x}\in {\mathbb{R}}^{n-2},t\in \left[0,T\right]\right\},{\overline{R}}_T^{n-1}=\left\{(x^{\prime },t)|x^{\prime }\in {\mathbb{R}}^{n-1},t\in \left[0,T\right]\right\},$$

with $0<l<1$ and $\lambda \in \mathbb{R}$. The inverse problem consists in finding $u(x,t)$ and $\chi \left(x^{\prime },t\right)$ from the system (12)–(14).

Theorem 3.

If $g(x^{\prime },t)\in H^{l+4,(l+4)/2}\left({\overline{R}}_T^{n-1}\right),$ with $g(x^{\prime },0)=\phi \left(x^{\prime }\right)$, and

$$\phi \left(x\right)\in H^{l+2}\left({\mathbb{R}}^n\right),\phi \left(x\right)\le {\phi }_0=\mathrm{const}>0,$$

then the inverse problem admits a unique solution in the domains $R_T^n$ and ${\overline{R}}_T^{n-1}$, respectively.

Proof of Theorem 3.

The proof will be carried out using the theory of integral equations. We first show that the inverse problem can be reformulated as a system of Volterra-type integral equations.

To this end, we differentiate the governing equation with respect to t, which yields:

$$u_{tt}-\delta \left(t\right)\Delta u_t={\delta }^{\prime }\left(t\right)\Delta u+{\int }_0^t\psi \left(x_n\right)\chi (x^{\prime },s)u_t(x,t-s)ds+\psi \left(x_n\right)\chi (x^{\prime },t)u(x,0)$$

$$-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{t}^{\alpha -1}\phi \left(\overline{x}\right)-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\vartheta (\overline{x},s)ds,$$

where

$$\Delta u={\displaystyle \frac{u_t}{\delta \left(t\right)}}-{\displaystyle \frac{\lambda }{\delta \left(t\right)}}{D}_{0t}^{-\alpha }u(\overline{x},t)-{\displaystyle \frac{1}{\delta \left(t\right)}}{\int }_0^t\psi \left(x_n\right)\chi (x^{\prime },s)u(x,t-s)ds.$$

We introduce the notation

$$\vartheta (x,t)=u_t(x,t).$$

The function $\vartheta (x,t)$ is governed by the following equivalent formulation:

$$\begin{array}{c}{\vartheta }_t-\delta \left(t\right)\Delta \vartheta ={(ln\delta \left(t\right))}^{\prime }\vartheta (x,t)-{(ln\delta \left(t\right))}^{\prime }{\int }_0^t\psi \left(x_n\right)\chi (x^{\prime },s)u(x,t-s)ds\hfill \\ +{\int }_0^t\psi \left(x_n\right)\chi (x^{\prime },s)\vartheta (x,t-s)ds+\psi \left(x_n\right)\chi (x^{\prime },t)\phi \left(x\right)\hfill \\ -{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{t}^{\alpha -1}\phi \left(\overline{x}\right)-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\vartheta (\overline{x},s)ds\hfill \\ -{(ln\delta \left(t\right))}^{\prime }{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}u(\overline{x},s)ds.\hfill \end{array}$$

(15)

with the initial and boundary conditions:

$${\vartheta (x,t)|}_{t=0}=\delta \left(0\right)\Delta \phi \left(x\right),$$

(16)

$${\vartheta |}_{x_n=0}=g_t(x^{\prime },t),g_t(x^{\prime },t){|}_{t=0}=\delta \left(0\right)\Delta \phi (x^{\prime },0).$$

(17)

By differentiating Equations (15) and (16) with respect to $x_n$, we derive an equivalent problem for ${\vartheta }_{x_n}(x,t)$:

$${\left({\vartheta }_{x_n}\right)}_t-\delta \left(t\right)\Delta {\vartheta }_{x_n}={(ln\delta \left(t\right))}^{\prime }{\vartheta }_{x_n}(x,t)-{(ln\delta \left(t\right))}^{\prime }\psi \left(x_n\right){\int }_0^t\chi (x^{\prime },\tau )u_{x_n}(x,t-s)ds$$

$$-{(ln\delta \left(t\right))}^{\prime }{\psi }^{\prime }\left(x_n\right){\int }_0^t\chi (x^{\prime },s)u(x,t-s)ds+{\psi }^{\prime }\left(x_n\right){\int }_0^t\chi (x^{\prime },s)\vartheta (x,t-s)ds$$

$$+\psi \left(x_n\right){\int }_0^t\chi (x^{\prime },s){\vartheta }_{x_n}(x,t-s)ds+{\psi }^{\prime }\left(x_n\right)\chi (x^{\prime },t)\phi \left(x\right)+\psi \left(x_n\right)\chi (x^{\prime },t){\phi }_{x_n}\left(x\right)$$

$$-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{t}^{\alpha -1}{\phi }_{x_n}\left(\overline{x}\right)-{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\vartheta }_{x_n}(\overline{x},s)ds$$

$$-{(ln\delta \left(t\right))}^{\prime }{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{u}_{x_n}(\overline{x},s)ds,$$

(18)

$${\vartheta }_{x_n}(x,o)=\delta \left(0\right)\Delta {\phi }_{x_n}\left(x\right).$$

(19)

Thus, we arrive at the system of relations (15)–(18), from which the functions $\vartheta (x,t)$, $\chi (x^{\prime },t)$, and ${\vartheta }_{x_n}(x,t)$ are determined.

Next, differentiating Equations (15) and (16) twice with respect to $x_n$ and introducing the notation

$$\omega (x,t)={\vartheta }_{x_n{x}_n}(x,t),$$

we obtain additional relations. Consequently, the inverse problem of determining the functions $u(x,t)$ and $\chi (x^{\prime },t)$ in (12)–(14) is equivalent to the problem of finding $\vartheta (x,t)$, $\chi (x^{\prime },t)$, and $\omega (x,t)$ from the system (15)–(18), together with the derived relations for $\omega (x,t)$.

Furthermore, following arguments similar to those in [13], and taking into account the properties of integral equations together with the imposed conditions, we establish the unique solvability of the resulting system. This, in turn, directly leads to the proof of Theorem 3. □

5. Conclusions

This study analyzed a time-fractional loaded heat equation with memory and nonlocal effects relevant to anomalous heat conduction. We formulated the corresponding Cauchy problem, derived its integral representation, and proved existence and uniqueness results within anisotropic Hölder spaces.

The proposed framework extends classical parabolic theory to models with hereditary and memory-dependent terms and establishes stability and uniqueness for the associated inverse problem. The obtained results provide a rigorous basis for analyzing diffusion processes in complex media and for identifying memory kernels in fractional systems.

Future research will focus on numerical implementation, extension to nonlinear and multidimensional settings, and comparison with experimental data on anomalous heat transfer in advanced materials.