Operator approach for time-fractional evolution equations in Banach spaces

Abstract

Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $α\in (0,1)$ in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.