Classical unique continuation property for multi-terms time fractional diffusion equations

TL;DR

This paper extends the unique continuation property for multi-term time fractional diffusion equations to cases without the assumption that solutions are zero for negative times, using Holmgren transformations and coordinate changes.

Contribution

It removes the previous restriction on solutions being zero for t ≤ 0, establishing the classical UCP for multi-term time fractional diffusion equations.

Findings

Established UCP without zero initial condition assumption

Used Holmgren transformations to derive local UCP

Extended UCP to broader class of solutions

Abstract

As for the unique continuation property (UCP) of solutions in $(0,T)\times\Omega$ with a domain $\Omega\subset{\mathbb R}^n,\,n\in{\mathbb N}$ for a multi-terms time fractional diffusion equation, we have already shown it by assuming that the solutions are zero for $t\le0$ (see \cite{LN2019}). Here the strongly elliptic operator for this diffusion equation can depend on time and the orders of its time fractional derivatives are in $(0,2)$. This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for $t\le0$. We have achieved this aim by first using the usual Holmgren transformation together with the argument in \cite{LN2019} to derive the UCP in $(T_1,T_2)\times B_1$ for some $0<T_1<T_2<T$ and a ball $B_1\subset\Omega$. Then if $u$ is the solution of the equation with $u=0$ in $(T_1,T_2)\times B_1$, we show $u=0$ also in $((0,T_1]\cup[T_2,T))\times B_r$ for some $r<1$ by using the argument in \cite{LN2019} which uses two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time fractional diffusion equation.