preprint

Inserted: 2 mar 2025
Last Updated: 2 mar 2025

Year: 2025

Abstract:

We consider an initial value problem for time-fractional evolution equation in Banach space. The operator satisfies a decay condition of resolvent which is common as a generator of analytic semigroup, and in particular, we can treat the $L^p$ case over a bounded domain and a uniform elliptic operator. First we construct a solution operator by means of suitable Laplace transform, and we establish the well-posedness in classes such as weak solution and strong solutions. We discuss also mild solutions local in time for semilinear time-fractional evolution equations. Finally we apply the result on the well-posedness to an inverse problem of determining an initial value and we establish the uniqueness for the inverse problem.