Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension

Abstract

We explore the inverse problem of determining the reaction coefficient $q(x',t)$ in the time-fractional diffusion equation ${}^CD_t u-Lu+q(x',t)u=f$ in $\mathbb{R}^n\times(0,T]$, $x=(x',x_n)$, $0<\alpha<1$, from the trace of the solution on the hyperplane $\{x_n=0\}$. Here ${}^CD_t$ is the Gerasimov--Caputo derivative and $L$ is a uniformly elliptic second-order operator with variable, $x_n$-independent coefficients and without zero-order term. For $L=\Delta$ this problem was studied in dimensions $n=1,2,3$ by methods based on pointwise bounds of the kernels, which are not available for $n\ge4$. Our approach relies on the Eidelman--Kochubei fundamental solution: although its singularity on the diagonal depends on the dimension, we show that it satisfies integral bounds of the same form for all $n\ge1$. Using these bounds, we prove existence, uniqueness and stability of regular solutions of the direct problem with a H\"{o}lder continuous reaction coefficient. Without assuming additional smoothness of the solution, we show that the inverse problem is equivalent to a closed system of integral equations for $q$ and the derivatives $\partial_{x_n}u$, $\partial^2_{x_n}u$. For the inverse problem we establish a conditional Lipschitz stability estimate on the whole time interval and global uniqueness in every dimension, and prove local existence of a solution. In the case $a_{in}\equiv0$, $b_n\equiv0$ we obtain a stability estimate with explicit constants, which reduces to the known estimates for the Laplacian. An explicit two-dimensional example with a mixed coefficient and a normal drift shows that the hypotheses of the main theorems are compatible and that the corresponding terms in the trace condition cannot be omitted.