Uniqueness of solution to boundary value problems for time-fractional wave equations

Abstract

We consider an initial boundary value problem in a bounded domain $\Omega$ over a time interval $(0,T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic operator has time-independent coefficients and is not necessarily symmetric. We prove that if for arbitrarily chosen subdomain $\omega \subset \Omega$ and $T>0$, a solution to the problem vanishes in $\omega \times (0,T)$, then $u=0$ in $\Omega \times (0,T)$. The uniqueness does not require any geometric condition on $\omega$.