Uniqueness of inverse source problems for
time-fractional diffusion equations with
singular functions in time

Let $\Omega\subset\mathbb{R}^{d}$ ($d\in\mathbb{N}:=\{1,2,\ldots\}$) be a bounded domain with a smooth boundary $\partial\Omega$, and let $\bm{\nu}=\bm{\nu}(\bm{x})$ be the unit outward normal vector of $\partial\Omega$. We set

where $a_{ij}=a_{ji}\in C^{1}(\overline{\Omega})$, $b_{j}\in C(\overline{\Omega})$ ($1\leq i,j\leq d$), $c\in C(\overline{\Omega})$ and we assume that there exists a constant $\kappa>0$ such that

We consider an initial-boundary value problem for a time-fractional diffusion equation whose source term is described by $\mu(t)f(\bm{x})$, where $f$ is a spatial distribution of the source and $\mu$ is a temporal change factor. We can describe a governing initial-boundary value problem as follows:

Here, for $0<\alpha<1$, we can formally define the pointwise Caputo derivative as

Owing to their capability of describing memory effects, time-fractional partial differential equations such as (2) have gathered consistent popularity among multidisciplinary researchers as models for anomalous diffusion and viscoelasticity (e.g. [2, 3, 9]). Though the history of fractional calculus can be traced back to Leibniz, the main focus on fractional equations was biased to the construction of explicit and approximate solutions via special functions and transforms until the last decades due to the needs from applied science. It has only been started recently that problems like (2) are formulated in appropriate function spaces using modern mathematical tools, followed by rapidly increasing literature on their fundamental theories, numerical analysis and inverse problems. Here we do not intend to give a complete bibliography, but only refer to several milestone works [4, 8, 14, 22] and the references therein.

Especially, the source term in (2) takes the form of separated variables, where $\mu(t)$ and $f(\bm{x})$ describe the time evolution and the spatial distribution of some contaminant source, respectively. Therefore, the determination of $\mu(t)$ or $f(\bm{x})$ turns out to be important in the context of environmental issues, which motivates us to propose the following problem.

Indeed, Problem 1.1 includes two inverse problems, namely, the determination of $\mu(t)$ by the single point observation and that of $f(\bm{x})$ by the partial interior observation of $u$. Both problems have been studied intensively in the last decade, especially among which the uniqueness was already known in literature. We can refer to many works, but here only to [18, 16, 19] for the inverse $t$-source problem and [10, 13, 12, 15] for the inverse $\bm{x}$-source problem. For a comprehensive survey on Problem 1.1 especially before 2019, we refer to Liu, Li and Yamamoto [17].

However, it reveals that the existing papers mainly discuss regular temporal components $\mu$, e.g. in $C^{1}[0,T]$ or $L^{2}(0,T)$. Such restrictions exclude a wide class of singular functions represented by the Dirac delta function, which corresponds with point sources in practice. This encourages us to reconsider Problem 1.1 in function spaces with lower regularity. More precisely, in this article we are mainly concerned with the case of $\mu=\mu(t)$ in a Sobolev space of negative order, in particular $\mu\not\in L^{2}(0,T)$.

For such less regular $\mu$, we cannot expect the differentiability of $u(\bm{x},\,\cdot\,)$ in time and we must redefine the Caputo derivative $\mathrm{d}_{t}^{\alpha}$ in (2). In this case, the unique existence of a solution to the initial-boundary value problem is more delicate, and an adequate formulation is needed.

This article is composed of 6 sections and 1 appendix. Redefining $\mathrm{d}_{t}^{\alpha}$ and thus problem (2) in negative Sobolev spaces, in Sect. 2 we state the main result in this paper. Then Sect. 3–4 are devoted to the two key ingredients for treating the singular system, i.e., the transfer to another regular system in $L^{2}$ and Duhamel’s principle in ${}_{\alpha}H(0,T)$-space. Next, the proof of Theorem 2.5 is completed in Sect. 5. Finally, Sect. 6 provides concluding remarks, and the proof of a technical detail is postponed to Appendix A.

To start with, we first define a fractional derivative for $v\in L^{2}(0,T)$ which extends the domain of $\mathrm{d}_{t}^{\alpha}$ and formulate the initial-boundary value problem. To this end, we introduce function spaces and operators. Set the forward and backward Riemann-Liouville integral operators as

We define an operator $\tau:L^{2}(0,T)\longrightarrow L^{2}(0,T)$ by $(\tau v)(t):=v(T-t)$ and obviously $\tau$ is an isomorphism. We set

By $H^{\alpha}(0,T)$ we denote the Sobolev-Slobodecki space with the norm $\|\cdot\|_{H^{\alpha}(0,T)}$ defined by

(e.g., Adams [1]). Then we further introduce the function spaces

Regarding the operators $J_{\alpha},J^{\alpha}$ and the spaces ${}_{\alpha}H(0,T),{}^{\alpha}H(0,T)$, we have the following lemma.

As for the proof of Lemma 2.1(i), see Gorenflo, Luchko and Yamamoto [8], Kubica, Ryszewska and Yamamoto [14]. Lemma 2.1(ii) can be readily derived from Lemma 2.1(i), the identity $J^{\alpha}v=\tau(J_{\alpha}(\tau v))$ and the fact that $\tau$ is an isomorphism.

Based on the spaces ${}_{\alpha}H(0,T)$ and ${}^{\alpha}H(0,T)$, let

be the Gel’fand triples, where $(\,\cdot\,)^{\prime}$ denotes the dual space. Let $X,Y$ be Hilbert spaces and $K:X\longrightarrow Y$ be a bounded linear operator with its domain $\mathcal{D}(K)=X$. Then we recall that $K^{\prime}$ is the maximal operator among $\widehat{K}:Y^{\prime}\longrightarrow X^{\prime}$ with $\mathcal{D}(\widehat{K})\subset Y^{\prime}$ such that

Then we can prove the following lemma.

Now we can redefine the Caputo derivative for functions in $L^{2}(0,T)$.

Now we are ready to propose the initial-boundary value problem:

where

Henceforth we assume the existence of such a solution $u$, although it can be verified during the succeeding arguments.

Now we are well prepared to state the main result on the uniqueness for Problem 1.1 with singularity in time.

In Theorem 2.5(ii), in terms of (8) we can verify

Therefore, the data $u(\bm{x}_{0},\,\cdot\,)\in{}^{-\beta}H(0,T)$ makes sense. The proof of (9) is given in Appendix A.

In Theorem 2.5(ii), we can further study and prove the uniqueness in the case where we replace assumption (7) by more general conditions, e.g., $b_{j}\neq 0$ and $c$ is not necessarily non-negative, but we omit details. Moreover, here we omit the treatments for the case $1<\alpha<2$.

Since ${}^{\beta}H(0,T)\subset C[0,T]$ by $\beta>1/2$, we see that the following $\mu$ are in ${}^{-\beta}H(0,T)$: $\mu\in L^{1}(0,T)$ and $\mu(t)=\sum_{k=1}^{N}r_{k}\delta_{a_{k}}(t)$, where $r_{k}\in\mathbb{R}$, $0<a_{k}<T$ and $\delta_{a_{k}}$ is the Dirac delta function: $\langle\delta_{a_{k}},\varphi\rangle=\varphi(a_{k})$ for $\varphi\in C[0,T]$. In particular, from Theorem 2.5(ii) we can directly derive

For the case of $\mu\in L^{2}(0,T)$ or $\mu\in L^{1}(0,T)$, there are rich references but we are here restricted to [10, 12, 13, 18]. In particular, [12] considers also the case $0<\alpha\leq 2$ and variable $\alpha(\bm{x})$ for $\mu\in L^{1}(0,T)$, and we can apply our method to their formulated inverse problems, as mentioned in Sect. 6.

The main purpose of this section is the proof of

In terms of Proposition 3.1, the unique existence of the solution $u$ to (4)–(5) can be clarified via $v:=(J^{\beta})^{\prime}u$ in (10).

In this section, we establish Duhamel’s principle which transforms a solution to an initial-boundary value problem without the inhomogeneous term to a solution to (11). Such a principle is known and see e.g. [18] and we can refer also to the survey [17] and Umarov [24]. Here we reformulate the formula in the space ${}_{\alpha}H(0,T)$ in order to apply within our framework.

We first show the following key lemma.