Classical unique continuation property for multi-term time-fractional evolution equations

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 evolution equation, we have already shown it by assuming that the solutions are zero for $t\le 0$ (see [12]). Here the strongly elliptic operator for this evolution 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 $t\le 0$. We have achieved this aim by first using the usual Holmgren transformation together with the argument in [12] 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 result and argument in [12] which use 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 evolution equation.