Well-posedness of an initial value problem for fractional diffusion equation with Caputo–Fabrizio derivative

Abstract

In this paper, we consider the following Cauchy problem of space–time fractional diffusion equations CFD0,tαu+(−L)su=G(t,x;u),in(0,T]×Ω,u(t,x)=0,on(0,T]×∂Ω,u(0,x)=u0(x),inΩ,The time fractional derivative is taken in the sense of Caputo-Fabrizio type. We derive representation of solutions by using Laplace transform and we further establish the existence and uniqueness of the mild solution. Both the linear case and the nonlinear case with globally Lipschitz assumption, we investigate the regularity of the mild solution. For the case of locally Lipschitz assumption, we study the existence of local mild solutions to the problem and then a blow-up alternative is established. We also consider the problem of continuous dependence with respect to initial data.