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 CDt u−Lu+q(x',t)u=f in ℝn×(0,T], x=(x',xn), 0<α<1, from the trace of the solution on the hyperplane {xn=0}. Here CDt is the Gerasimov--Caputo derivative and L is a uniformly elliptic second-order operator with variable, xn-independent coefficients and without zero-order term. For L=Δ 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≥4. 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≥1. Using these bounds, we prove existence, uniqueness and stability of regular solutions of the direct problem with a Hölder 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 ∂xnu, ∂2xnu. 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 ain≡0, bn≡0 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.
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BibTeX
@misc{durdiev2026recovering,
title = {{Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension}},
author = {D. K. Durdiev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29763},
eprint = {2610.29763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29763},
}
RIS
TY - UNPB
TI - Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension
AU - D. K. Durdiev
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29763
UR - https://arxiv.org/abs/2610.29763
AB - 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 ℝ^n×(0,T], x=(x',x_n), 0<α<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=Δ 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≥4. 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≥1. Using these bounds, we prove existence, uniqueness and stability of regular solutions of the direct problem with a Hölder 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 ∂_x_nu, ∂^2_x_nu. 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≡0, b_n≡0 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.
ER -