An Inverse Source Problem in a Variable-Order Time-Fractional Diffusion PDE

Abstract

We study an inverse source problem for a semilinear diffusion equation involving a Caputo-type time-fractional derivative whose order is a function of time. The equation is considered in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $d\ge 1$, and is supplemented with homogeneous Dirichlet boundary conditions. The source term is taken to be separable, $h\left(t\right)f\left(x\right)$, where the temporal component $h\left(t\right)$ is unknown. This quantity is to be identified from spatially localized measurements $m\left(t\right)$ of the solution. In this setting, we establish existence and uniqueness results in suitable function spaces, thereby demonstrating the well-posedness of the corresponding inverse source problem.

1. Problem Formulation

The Caputo–Volterra fractional derivative offers a natural and flexible framework for describing diffusion and evolution processes with memory. In contrast to classical time derivatives, it accounts for the influence of the entire past history of the system on its current state, a feature that is essential in the modeling of phenomena such as anomalous diffusion, viscoelastic behavior, and heat transfer in heterogeneous media.

Its Volterra structure highlights the causal character of the model, ensuring that only past states affect the present and thereby preserving physical consistency. At the same time, the Caputo-type formulation is advantageous from both theoretical and practical viewpoints, since it accommodates standard initial conditions expressed through integer-order derivatives, which are often accessible in experimental settings.

From a mathematical perspective, this approach strikes a balance between modeling fidelity and analytical feasibility. The weakly singular memory kernel captures long-range temporal dependence while remaining integrable, which allows the use of functional-analytic techniques to establish existence, uniqueness, and stability of solutions. In addition, the Caputo–Volterra derivative is particularly well suited for inverse problems, as it provides a transparent description of how past sources shape present observations.

A survey of the literature reveals that several definitions of fractional derivatives and integrals have been proposed within the framework of variable-order (VO) fractional calculus, extending their classical constant-order (CO) counterparts (see [1,2,3,4,5,6,7,8,9,10]). In this context, a natural way to introduce a time-dependent Riemann–Liouville kernel is given by

$$\left(g_{1-\beta \left(t\right)}\right)\left(t\right)={\displaystyle \frac{t^{-\beta \left(t\right)}}{\mathrm{\Gamma }(1-\beta (t\left)\right)}}t>0,0<\beta \left(t\right)<1.$$

(1)

We consider a function $\beta :[0,T]\to [\underline{\beta },\overline{\beta }]$ such that

$$0<\underline{\beta }<\overline{\beta }<1,\beta \in W^{1,\infty }\left([0,T]\right).$$

(2)

There is no warranty that $\left(g_{1-\beta \left(t\right)}\right)\left(t\right)$ is monotone. One can just consider, e.g., $\beta \left(t\right)=0.6+0.2sin\left(2\pi t\right)$ (cf. [11]) to observe the non-monotonic behavior of the kernel $\left(g_{1-\beta \left(t\right)}\right)\left(t\right)$.

In addition to the CO and VO fractional derivatives, the literature also considers distributed-order (DO) fractional derivatives; see, for example, [12,13,14,15].

While variable-order derivatives allow the fractional order to vary as a function of time or space—thereby capturing dynamic memory effects—distributed-order derivatives treat the order itself as a continuous variable integrated over a prescribed range. This leads to a superposition of fractional behaviors across multiple scales. As a result, the DO framework provides a powerful generalization for modeling complex systems with multiscale memory, whereas the VO approach is more suitable for processes in which memory evolves locally in time or space (e.g., lava flows). In this sense, VO derivatives offer a functional description of non-integer order dynamics, while DO derivatives provide a distributional one.

The VO Riemann–Liouville kernel (1) has the singularity of the type $t^{-\beta \left(0\right)}$ near $t=0_{+}$. The paper [11] introduced a function $B\left(t\right)$ defined as follows:

$$\begin{array}{cc}\hfill {\displaystyle \left(g_{1-\beta \left(t\right)}\right)\left(t\right)}& {\displaystyle =\left(g_{1-\beta \left(0\right)}\right)\left(t\right)+B\left(t\right).}\hfill \end{array}$$

(3)

In this way, the VO Riemann–Liouville kernel $\left(g_{1-\beta \left(t\right)}\right)\left(t\right)$ can be decomposed into the sum of a CO kernel—retaining the same principal singularity—and a continuous function. This representation makes it possible to express the VO Caputo fractional derivative in terms of its CO counterpart together with a singular Volterra operator; see (7). The resulting singularity is of weak type. Moreover, it was shown in [11] that for any $\epsilon >0$ one has

$$\left|B\left(t\right)\right|=\mathcal{O}\left(t^{1-\beta \left(0\right)-\epsilon }\right)+\mathcal{O}\left(t^{1-\beta \left(t\right)}\right)=\mathcal{O}\left(t^{1-\overline{\beta }-\epsilon }\right)$$

(4)

and

$$|B^{\prime }\left(t\right)|=\mathcal{O}\left(t^{-\beta \left(0\right)-\epsilon }\right).$$

(5)

We denote by ∗ the convolution on the positive half-line, defined as

$$(k\ast v)\left(t\right)={\int }_0^{t}k(t-s)v\left(s\right)\mathrm{d}s.$$

The Caputo VO fractional derivative is given by (Please note that variable order derivative $\beta \left(t\right)$ depends on time, and it differs from a variably distributed-order time-fractional derivative (cf. [16]). By variably distributed-order the fractional derivative depends on an additional parameter, which is time-independent.)

$$\left({\partial }_t^{\beta \left(t\right)}u\right)\left(t\right)=\left(g_{1-\beta \left(t\right)}\ast {\partial }_{t}u\right)\left(t\right)={\int }_0^t{\displaystyle \frac{{(t-s)}^{-\beta (t-s)}}{\mathrm{\Gamma }(1-\beta (t-s\left)\right)}}{\partial }_{t}u\left(s\right)\mathrm{d}s$$

(6)

for $t>0,0<\beta \left(t\right)<1$.

So, we are able to rewrite the VO–Caputo fractional derivative (6) in an equivalent form using (3)–(5) and integration by parts as follows:

$$\begin{array}{cc}\hfill {\displaystyle \left({\partial }_t^{\beta \left(t\right)}u\right)\left(t\right)}& {\displaystyle =\left(g_{1-\beta \left(t\right)}\ast {\partial }_{t}u\right)\left(t\right)}\hfill \\ & {\displaystyle \stackrel{\left(3\right)}{=}\left(\left(g_{1-\beta \left(0\right)}+B\right)\ast {\partial }_{t}u\right)\left(t\right)}\hfill \\ & {\displaystyle =\left({\partial }_t^{\beta \left(0\right)}u\right)\left(t\right)+{\int }_0^{t}B(t-s){\partial }_{t}u\left(s\right)\mathrm{d}s}\hfill \\ & {\displaystyle =\left({\partial }_t^{\beta \left(0\right)}u\right)\left(t\right)+{\left.B(t-s)u\left(s\right)\right|}_{s=0}^{s=t}+{\int }_0^t{B}^{\prime }(t-s)u\left(s\right)\mathrm{d}s}\hfill \\ & {\displaystyle =\left({\partial }_t^{\beta \left(0\right)}u\right)\left(t\right)-B\left(t\right)u\left(0\right)+{\int }_0^t{B}^{\prime }(t-s)u\left(s\right)\mathrm{d}s.}\hfill \end{array}$$

(7)

We would like to point out that the function $B\left(t\right)$ is bounded on $[0,T]$ by (4) and $B\left(0\right)=0$. The derivative $B^{\prime }\left(t\right)$ admits a weak singularity at $t=0$; see (5). A weak singularity of a kernel means that the kernel becomes unbounded at some point, but not strongly enough to destroy integrability.

We first recall the notion of an abstract Volterra operator.

Definition 1

(see [17]). Let $S_t=[0,t],t\in [0,T]$. The operator

$$E:L^{\infty }\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)\to L^{\infty }\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$$

is said to be a Volterra operator in $L^2\left(\mathrm{\Omega }\right)$ iff

$$\begin{array}{cc}{\displaystyle }\hfill & \left\{u\left(s\right)=v\left(s\right)a.e.in S_t\right\}⟹\left\{E\left(u\right)\left(s\right)=E\left(v\right)\left(s\right)a.e.in S_t\right\}\hfill \end{array}$$

for any $t\in [0,T]$.

We consider a (nonlinear) Volterra operator $E\left(V\right)\left(t\right)$ obeying for any $t\in [0,T]$

$$∥E\left(U\right)\left(t\right)-E\left(V\right)\left(t\right)∥\le C{\int }_0^t{(t-s)}^{-\overline{\alpha }}∥U\left(s\right)-V\left(s\right)∥\mathrm{d}s$$

(8)

for some $0\le \overline{\alpha }<1$ and for any admissible arguments U and V. This estimate expresses a stability property of the operator E: it shows that the difference between the outputs corresponding to two inputs is controlled by the history of the difference between those inputs.

The presence of a weakly singular kernel reflects the nonlocal-in-time nature of the problem. In systems with memory, the present state depends on past values in a non-uniform way: recent history has a stronger influence than distant history. This type of weighting is characteristic of fractional and hereditary models and provides a realistic description of diffusion processes with memory effects.

From an analytical point of view, such an inequality is essential for establishing well-posedness. It allows one to apply Grönwall-type arguments adapted to weakly singular kernels, which leads to results on uniqueness, continuous dependence on data, and stability of solutions.

In summary, the inequality quantifies how perturbations propagate in time in a system with memory and provides the fundamental tool for proving stability and robustness of the model. Some examples of such operators can be found in [18]. The simplest example is the linear operator $E\left(U\right)\left(t\right)={\int }_0^t{(t-s)}^{-\overline{\alpha }}U\left(s\right)\mathrm{d}s$.

We consider a general second-order linear differential operator $\mathcal{L}$ defined by

$$\mathcal{L}\left(u\right)=\nabla ·\left(-\mathbf{A}\left(x\right)\nabla u\right)+c\left(x\right)u,$$

(9)

where

$$\begin{array}{cc}\hfill & A\left(x\right)={\left(a_{i,j}\left(x\right)\right)}_{i,j=1,\dots ,d},a_{ij}\in L^{\infty }\left(\mathrm{\Omega }\right),A^T=A,\hfill \\ \hfill & c\in L^{\infty }\left(\mathrm{\Omega }\right),c\left(x\right)\ge 0\mathrm{for}\mathrm{all}x\in \mathrm{\Omega },\hfill \end{array}$$

(10)

and there exists a constant $C_0>0$ such that

$$\sum _{i,j=1}^d{a}_{ij}\left(x\right){\xi }_i{\xi }_j⩾C_0{\left|\xi \right|}^2,\mathrm{for}\mathrm{all}x\in \overline{\mathrm{\Omega }}\mathrm{and}\xi \in {\mathbb{R}}^d.$$

(11)

Since $\mathcal{L}$ is a symmetric uniformly elliptic operator, the spectrum of $\mathcal{L}$ is entirely composed of discrete eigenvalues and counting according to the (finite) multiplicities, we can set: $0<{\lambda }_1\le {\lambda }_2\le \cdots$. By $e_n\in H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)$ we denote the eigenfunction corresponding to ${\lambda }_n$ and $\mathcal{L}{e}_n={\lambda }_n{e}_n$. Then the sequence ${\left\{e_n\right\}}_{n\in \mathbb{N}}$ creates an orthonormal basis in $L^2\left(\mathrm{\Omega }\right)$. (cf. [19,20], §6.5.1).

We study the following inverse source problem:

Find a couple $\left(u\right(x,t),h(t\left)\right)$ obeying

$$\begin{array}{ccc}\hfill {\displaystyle \left({\partial }_t^{\beta \left(t\right)}u(x,·)\right)\left(t\right)+\mathcal{L}u(x,t)}& {\displaystyle =f\left(x\right)h\left(t\right)+F\left(x,t,u(x,t),E\left(u\right(x,·\left)\right)\left(t\right)\right)}\hfill & {\displaystyle \mathrm{i}\mathrm{n} \mathrm{\Omega }\times (0,T]}\hfill \\ \hfill u(x,t)& =0\hfill & \mathrm{o}\mathrm{n} \mathrm{\Gamma }\times (0,T]\hfill \\ \hfill u(x,0)& {\displaystyle =u_0\left(x\right)}\hfill & \mathrm{i}\mathrm{n} \mathrm{\Omega }\hfill \end{array}$$

(12)

along with the local in space measurement

$$m\left(t\right)=\left(\omega ,u\left(t\right)\right)={\int }_{\mathrm{\Omega }}\omega \left(x\right)u(x,t)\mathrm{d}x,\omega \in H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right),\left(f,\omega \right)\ne 0.$$

(13)

The conditions on $u_0,f$ and F are formulated later in (14). Solvability of the direct problem (12) has been addressed in [11]. However a few questions still remain open:

• The order function $\beta$ is required to satisfy (2). Ideally, one would like to allow the full range $0<\beta \left(t\right)<1$.
• Assuming ${\mathcal{L}}^{\gamma }{u}_0\in L^2\left(\mathrm{\Omega }\right)$ for some $0\le \gamma <1$, it was shown in [11] that ${\mathcal{L}}^{\gamma }u\in C\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$. It would be desirable to extend this result to the case $\gamma =1$.

The goal of this paper is to extend the proof technique developed in [11] to establish the well-posedness of the inverse source problem (ISP) (12), (13), as formulated in Theorem 1. We note that the overposed data (13) correspond to invasive measurements occurring inside the domain $\mathrm{\Omega }$. Ideally, one would like to replace these with non-invasive (boundary) measurements; this remains a topic for future research.

Highlights and Added Value

Inverse source problems are a central topic in applied mathematics and engineering, where the goal is to reconstruct hidden sources that drive a system from limited observational data. These problems are notoriously challenging due to their ill-posed nature and the scarcity or incompleteness of available measurements. Nevertheless, they are of great practical importance, with applications ranging from imaging and nondestructive testing to environmental monitoring and medical diagnostics (see, e.g., [21,22,23,24] and the references therein).

A significant body of work has also addressed ISPs for fractional PDEs, reflecting the increasing use of fractional models; see, for instance, [19,25,26]. In contrast, much less attention has been paid to VO time-fractional problems. While some papers analyze VO time-fractional direct problems [27,28,29], to the best of our knowledge, publications on ISPs in this setting are very hard to find. The only related work we are aware of is [18], where VO derivatives appear only as lower-order terms on the right-hand side of parabolic or wave-type equations.

To the best of our knowledge, inverse source problems in the framework of (12) have not been investigated in the existing literature. This paper therefore fills a fundamental gap by establishing, for the first time, the unique solvability of the inverse source problem (12), (13) for a class of models involving nonlinear Volterra operators.

The results presented in Theorem 1 represent a substantial advance in the theory of inverse problems for fractional partial differential equations. In particular, they provide the first rigorous treatment of inverse source problems for variable-order time-fractional PDEs within this analytical framework, thereby opening a new direction for both theoretical investigation and practical applications.

2. Well Posedness

We assume that the initial datum and the right-hand-side obey (for some $C>0$)

$$\begin{array}{cc}\hfill {\displaystyle u_0,f}& {\displaystyle \in L^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill ∥F(t,z(t\left)\right)∥& {\displaystyle \le C\left(1+∥z\left(t\right)∥\right)}\hfill \\ \hfill ∥F(t,z(t\left)\right)-F(t,w(t\left)\right)∥& {\displaystyle \le C∥z\left(t\right)-w\left(t\right)∥.}\hfill \end{array}$$

(14)

for each time t and all $L^2\left(\mathrm{\Omega }\right)$-measurable admissible argument functions $z\left(t\right),w\left(t\right)$. The measurement satisfies

$$m\in C\left([0,T]\right),|m^{\prime }\left(t\right)|\le C{t}^{-\delta }t\in [0,T],0\le \delta <1.$$

(15)

Applying the measurement operator to the governing PDE (12) we get

$$\left({\partial }_t^{\beta \left(t\right)}m\right)\left(t\right)+\left(\mathcal{L}u\left(t\right),\omega \right)=h\left(t\right)\left(f,\omega \right)+\left(F\left(t,u\left(t\right),E\left(u\right)\left(t\right)\right),\omega \right),$$

which implies that

$$h\left(t\right)={\displaystyle \frac{1}{\left(f,\omega \right)}}\left(\left({\partial }_t^{\beta \left(t\right)}m\right)\left(t\right)+\left(u\left(t\right),\mathcal{L}\omega \right)-\left(F\left(t,u\left(t\right),E\left(u\right)\left(t\right)\right),\omega \right)\right).$$

(16)

Therefore the ISP to find $\left(u\right(x,t),h(t\left)\right)$ obeying (12), (13) can be reformulated as a direct setting: Determine u satisfying (12), along with (16). So, we can write

$$\begin{array}{ccc}\hfill {\displaystyle \left({\partial }_t^{\beta \left(t\right)}u(x,·)\right)\left(t\right)+\mathcal{L}u(x,t)}& {\displaystyle =\tilde{F}\left(x,t,u(x,t),E\left(u\right(x,·\left)\right)\left(t\right)\right)}\hfill & {\displaystyle \mathrm{i}\mathrm{n} \mathrm{\Omega }\times (0,T]}\hfill \\ \hfill u(x,t)& =0\hfill & \mathrm{o}\mathrm{n} \mathrm{\Gamma }\times (0,T]\hfill \\ \hfill u(x,0)& {\displaystyle =u_0\left(x\right)}\hfill & \mathrm{i}\mathrm{n} \mathrm{\Omega },\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill {\displaystyle \tilde{F}\left(x,t,u(x,t),E\left(u\right(x,·\left)\right(t)\right)}& {\displaystyle :=F\left(x,t,u(x,t),E\left(u\right(x,·\left)\right(t)\right)}\hfill \\ & {\displaystyle +\frac{f\left(x\right)}{\left(f,\omega \right)}\left(\left({\partial }_t^{\beta \left(t\right)}m\right)\left(t\right)+\left(u\left(t\right),\mathcal{L}\omega \right)-\left(F\left(t,u\left(t\right),E\left(u\right)\left(t\right)\right),\omega \right)\right).}\hfill \end{array}$$

Now, we provide some estimates for $\tilde{F}$. Using (15) we have

$$|\left({\partial }_t^{\beta \left(t\right)}m\right)\left(t\right)|\le C{\int }_0^t{(t-s)}^{-\beta (t-s)}\left|m^{\prime }\left(s\right)\right|\mathrm{d}s\le C{\int }_0^t{(t-s)}^{-\overline{\beta }}{s}^{-\delta }\mathrm{d}s\le C.$$

So, we can easily see that if (14) is valid then also $\tilde{F}$ obeys (14).

So, the following theorem is valid.

Theorem 1.

Assume (2), (8), (9), (10), (11), (13), (14), (15). Then

(i) 

The problem (17) admits a unique solution $u\in C\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$. This solution can be expressed by an implicit form

$$\begin{array}{cc}\hfill {\displaystyle u(x,t)}& {\displaystyle =\sum _{n=1}^{\infty }{e}_n\left(x\right)\left[\left(u_0,e_n\right)E_{\beta \left(0\right),1}\left(-{\lambda }_n{t}^{\beta \left(0\right)}\right)\right.}\hfill \\ & {\displaystyle \left.+{\int }_0^t\left(R\left(\tau \right),e_n\right){(t-\tau )}^{\beta \left(0\right)-1}{E}_{\beta \left(0\right),\beta \left(0\right)}\left(-{\lambda }_n{(t-\tau )}^{\beta \left(0\right)}\right)d\tau \right].}\hfill \end{array}$$

(18)

where

$$R\left(\tau \right)=\tilde{F}(x,\tau ,u(x,\tau ),E\left(u(x,·)\right)\left(\tau \right)).$$

If we additionally assume that ${\mathcal{L}}^{\gamma }{u}_0\in L^2\left(\mathrm{\Omega }\right)$ for some $0<\gamma <1$ then ${\mathcal{L}}^{\gamma }u\in C\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$.

(ii) 

Let m be any function obeying (15) and $m\left(0\right)=\left(u\left(0\right),\omega \right)$. Then the ISP (12), (13) admits a unique solution $u\in C\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$ and $h\in L^{\infty }\left([0,T]\right)$.

Proof. 

(i) This part follows directly from ([11], Thm. 5). Please note that the conditions (14) imposed on the initial data $u_0$, f and on the right-hand side F are sufficient to ensure the solvability of the auxiliary problem (17). At present, the author is not aware whether these conditions can be further relaxed.

(ii) Let m be any function obeying (15), and $m\left(0\right)=\left(u\left(0\right),\omega \right)$. We denote $h\left(t\right)$ by (16). From the just proved part $\left(i\right)$ we see that the setting (17) admits a unique solution $u\in C\left([0,T],L^2\left(\mathrm{\Omega }\right)\right)$. The fact that $h\in L^{\infty }\left([0,T]\right)$ follows from (16). Please note that for continuity of $h\left(t\right)$ we would need the continuity in time of $F\left(x,t,u(x,t),E\left(u\right(x,·\left)\right)\left(t\right)\right)$.

Testing the governing PDE from (17) by $\omega$ we easily obtain that

$$\left({\partial }_t^{\beta \left(t\right)}m\right)\left(t\right)=\left({\partial }_t^{\beta \left(t\right)}\left(u,\omega \right)\right)\left(t\right).$$

Using (7) we deduce that

$$\begin{array}{cc}\hfill {\displaystyle \left({\partial }_t^{\beta \left(0\right)}[m-\left(u,\omega \right)]\right)\left(t\right)}& {\displaystyle =B\left(t\right)[m\left(0\right)-\left(u\left(0\right),\omega \right)]-{\int }_0^t{B}^{\prime }(t-s)[m\left(s\right)-\left(u\left(s\right),\omega \right)]\mathrm{d}s}\hfill \\ & {\displaystyle =-{\int }_0^t{B}^{\prime }(t-s)[m\left(s\right)-\left(u\left(s\right),\omega \right)]\mathrm{d}s}\hfill \end{array}$$

and applying the convolution operator $g_{\beta \left(0\right)}\ast$ to both sides of this equation we obtain

$$m\left(t\right)-\left(u\left(t\right),\omega \right)=-\left(g_{\beta \left(0\right)}\ast \left(B^{\prime }\ast [m-\left(u,\omega \right)]\right)\right)\left(t\right)=-\left(\left(g_{\beta \left(0\right)}\ast B^{\prime }\right)\ast [m-\left(u,\omega \right)]\right)\left(t\right).$$

Using (1) and (5) we can write for small $\epsilon >0$ that

$$\begin{array}{cc}\hfill {\displaystyle |\left(g_{\beta \left(0\right)}\ast B^{\prime }\right)\left(t\right)|}& {\displaystyle \le C{\int }_0^t{(t-s)}^{\beta \left(0\right)-1}{s}^{-\beta \left(0\right)-\epsilon }\mathrm{d}s}\hfill \\ & {\displaystyle =C{t}^{-\epsilon }\mathcal{B}(\beta \left(0\right),1-\beta \left(0\right)-\epsilon )}\hfill \\ & {\displaystyle \le C{t}^{-\epsilon },}\hfill \end{array}$$

where $\mathcal{B}$ denotes the usual Beta function.

Thus we see that

$$|m\left(t\right)-\left(u\left(t\right),\omega \right)|\le C{\int }_0^t{(t-s)}^{-\epsilon }\left|m\left(s\right)-\left(u\left(s\right),\omega \right)\right|\mathrm{d}s.$$

Applying a generalized Grönwall lemma, (cf. [30], Lemma 7.1.1, [31], Lemma 1) we conclude that

$$m\left(t\right)=\left(u\left(t\right),\omega \right)t\in [0,T].$$

Therefore the couple $u,h$ solves the ISP (12), (13).

If $u_k,h_k$ ($k=1,2$) solve (12), (13), then $u_k$ solves (17). So, $u_k$ is uniquely determined by the part $\left(i\right)$. Therefore if $u_1=u_2$ we get by (16)

$$\begin{array}{cc}\hfill {\displaystyle |h_1\left(t\right)-h_2\left(t\right)|}& {\displaystyle =\left|\frac{\left(F\left(t,u_1\left(t\right),E\left(u_1\right)\left(t\right)\right)-F\left(t,u_2\left(t\right),E\left(u_2\right)\left(t\right)\right),\omega \right)}{\left(f,\omega \right)}\right|}\hfill \\ & {\displaystyle \stackrel{\left(14\right)}{\le }C\left(∥u_1\left(t\right)-u_2\left(t\right)∥+∥E\left(u_1\right)\left(t\right)-E\left(u_2\right)\left(t\right)∥\right)}\hfill \\ & {\displaystyle \stackrel{\left(8\right)}{\le }C\left(∥u_1\left(t\right)-u_2\left(t\right)∥+{\int }_0^t{(t-s)}^{-\overline{\alpha }}∥u_1\left(s\right)-u_2\left(s\right)∥\mathrm{d}s\right)}\hfill \\ & {\displaystyle =0.}\hfill \end{array}$$

This concludes the proof of uniqueness. □

3. Conclusions

This article investigates the well-posedness of an abstract inverse source problem associated with a variable-order time-fractional diffusion equation. In particular, it addresses the existence and uniqueness of weak solutions. The governing time-fractional model (12) is supplemented by a measurement condition of the form (13). As a first step, the unknown source term $h\left(t\right)$ is eliminated through a reformulation based on the observed data $m\left(t\right)$. This leads to an auxiliary direct problem, for which well-posedness is established.

The main result, stated in Theorem 1, shows that the solution of the auxiliary problem coincides with the unique solution of the original inverse source problem (12), (13). Consequently, the theorem provides a rigorous foundation for the solvability of the inverse problem under consideration.

Moreover, Theorem 1 identifies a set of sufficient conditions ensuring the well-posedness of the inverse source problem. Whether these conditions can be further relaxed remains an interesting topic for future research.