Global uniqueness in an inverse problem for time fractional diffusion equations

Abstract

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d\ge 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T)\times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1)\cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of ${\mathbb{R}}^d$, two out the three coefficients $\rho$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $\rho {\partial }_t^{\alpha }u-\text{div}(a\nabla u)+qu=0$ on $(0,T)\times \Omega$ are recovered simultaneously.

References in corpus (3)

• Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
• Determination of time dependent factors of coefficients in fractional diffusion equations
• On existence and uniqueness of solutions for semilinear fractional wave equations

Cited by in corpus (1)

• Initial-boundary value problem for distributed order time-fractional diffusion equations