Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Abstract

In this paper, we study a class of backward problems for nonlinear fractional super-diffusion equations in Banach spaces. We consider the time fractional derivative in the sense of Caputo type. First, we establish some results for the existence of the mild solutions. Moreover, we obtain regularity results of the first order and fractional derivatives of mild solutions. These conclusions are mainly based on fixed point theorems and properties of \(\alpha \)-resolvent family as well as Mittag-Leffler functions. Finally, two applications are provided to illustrate the efficiency of our results.