Time-fractional diffusion equations have attracted considerable attention in recent years and have become a very active area of research in both pure and applied mathematics. Despite the extensive literature, several aspects of the theory remain only partially understood. In this talk, I will discuss recent results concerning initial value problems for time-fractional evolution equations in Banach spaces. The first part will focus on forward problems in L^p-spaces and on a framework based on the vector-valued Laplace transform, which is particularly useful in view of applications to inverse problems. The second part will be devoted to time-fractional transport-type equations, where the underlying operator is not necessarily of elliptic type. In this setting, I will present a fractional extension of classical semigroup generation results, together with applications to transport, Boltzmann, and Schrödinger equations. If time permits, I will also discuss a stability result for an inverse source problem in a time-fractional diffusion equation. This talk is based on joint works with F. Gölegeleyen, G. Floridia, S.-E. Chorfi, and R. Ashurov.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006