Uniqueness for fractional nonsymmetric diffusion equations and an application to an inverse source problem

TL;DR

This paper establishes the uniqueness of solutions to time-fractional diffusion equations with non-symmetric elliptic operators, demonstrating that solutions are uniquely determined by boundary data on small spatial regions, with implications for inverse source problems.

Contribution

It proves a uniqueness result for fractional diffusion equations with non-symmetric operators, extending previous symmetric cases and applying spectral and Laplace transform techniques.

Findings

Solution is uniquely determined by boundary data on small regions.

The method applies to general time-fractional PDEs with non-symmetric operators.

The proof uses spectral decomposition and Laplace transform techniques.

Abstract

In this paper, we discuss the uniqueness for solution to time-fractional diffusion equation $\partial_t^\alpha (u-u_0) + Au=0$ with the homogeneous Dirichlet boundary condition, where an elliptic operator $-A$ is not necessarily symmetric. We prove that the solution is identically zero if its normal derivative with respect to the operator $A$ vanishes on an arbitrary small part of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition, and is valid for more general time-fractional partial differential equations, including those involving non symmetric operators.