Global existence for nonlinear diffusion with conformable operator using Banach fixed point theorem

Abstract

In this work, we are interested to focus on a fractional diffusion equation with conformable derivative which contain the time dependent coefficients which occurs in many application models. By using some given assumptions, we consider the global solution of the problem. Moreover, the convergence of the mild solution when fractional order tends to $1^-$ is presented. This research can be considered as one of the first results on the topic related to conformable problem with time-dependent coefficients. In the simple case of coefficient, we show the global regularity for the mild solution in $L^p$ space. The main techniques of this work is to use Banach fixed point theorem, $L^p-L^q$ heat semigroup and some complex evaluations and techniques.