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 alpha in (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 xed 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.