Backward problems in time for fractional diffusion-wave equation

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 $T \not\in \Lambda$.