Note on a time fractional diffusion equation with time dependent variables coefficients

Abstract

In this short paper, we study time fractional diffusion equations with time-dependent coefficients. The derivative operator that appears in the main equation is Riemann-Liouville. The main purpose of the paper is to prove the existence of a global solution. Due to the nonlocality of the derivative operator, we cannot represent the solution directly when the coefficient depends on time. Using some new transformations and techniques, we investigate the global solution. This paper can be considered as one of the first results on the topic related to problems with time-dependent coefficients. Our main tool is to apply Fourier analysis method and combine with some estimates of Mittag-Lefler functions and some Sobolev embeddings.