Unique continuation property for multi-terms time fractional diffusion equations

TL;DR

This paper establishes a unique continuation property for multi-term time fractional diffusion equations using Carleman estimates, pseudo-differential operators, and a novel argument for global continuation, advancing understanding of solution behavior.

Contribution

It introduces new Carleman estimates and a method for proving global unique continuation for multi-term time fractional diffusion equations with general elliptic operators.

Findings

Proved local unique continuation using a Holmgren type transformation.

Developed a new argument for global unique continuation.

Established Carleman estimates for multi-term fractional diffusion equations.

Abstract

A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $\alpha\,\,(0<\alpha<2)$ and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuation of solutions. Here the global unique continuation means as follows. If u is a solution of the multi-terms time fractional diffusion equation in a domain over the time interval $(0,T)$, then a zero set of solution over a subdomain of $\Omega$ can be continued to $(0,T)\times\Omega$.