On the Fractional Diffusion Equation Associated With Exponential Source and Operator With Exponential Kernel

Abstract

In this paper, we investigate the well-posedness of mild solutions of the time-fractional diffusion equation with an exponential source function and the Caputo-Fabrizio derivative of a fractional order a is an element of ( 0 , 1 ). Some linear estimates of the solution kernels on Hilbert scale spaces are constructed using a spectrum of the Dirichlet Laplacian. Based on the Banach fixed point theorem, the global existence and uniqueness of the small-data mild solution are approved. This work is considered the first study on the time-fractional diffusion equation with a nonlinear function for all common dimensions of 1, 2, and 3.