WELL-POSEDNESS AND REGULARIZATION FOR NONLOCAL DIFFUSION EQUATION WITH RIEMANN–LIOUVILLE DERIVATIVE

Abstract

In this paper, we are interested in studying the fractional diffusion equation with Riemann–Liouville as follows: [Formula: see text] with nonlocal in time condition. We are going to study the well-posedness of the above problem with some assumptions of the input data. On the other hand, in Hilbert scale and [Formula: see text] spaces, we provide several estimates of regularity results of the mild solution. We also establish the evaluation for gradient term of the mild solution. We also show that the nonlocal problem is ill-posed in the sense of Hadamard. We also derive the regularity result by applying Fourier truncation method. The main tool of the paper is to use some estimates of Wright functions and Sobolev embeddings. In addition, we also obtain a lower bound of the solution according to the input data.