Backward problems in time for fractional diffusion-wave equation

TL;DR

This paper investigates the backward problem in time for a fractional diffusion-wave equation, establishing conditions under which the problem is well-posed depending on the terminal time T.

Contribution

It proves the existence of specific times T where the backward problem is well-posed for a fractional diffusion-wave equation with order between 1 and 2.

Findings

Backward problem is well-posed for T not in a specific countable set.

Existence of a unique accumulation point at zero for the set of T values.

Provides conditions for well-posedness depending on T.

Abstract

In this article, for a time-fractional diffusion-wave equation $\pppa u(x,t) = -Au(x,t)$, $0<t<T$ with fractional order $\alpha \in (1,2)$, we consider the backward problem in time: determine $u(\cdot,t)$, $0<t<T$ by $u(\cdot,T)$ and $\ppp_tu(\cdot,T)$. We proved that there exists a countably infinite set $\Lambda \in (0,\infty)$ with a unique accumulation point $0$ such that the backward problem is well-posed for $T \not\in \Lambda$.