Remarks on a 1-D nonlocal in time fractional diffusion equation with inhomogeneous source

摘要

In this paper, we deal with the fractional diffusion equation with Riemann-Liouville in the form D0α+ u ? uxx = F(x, t). Under some various assumptions of the input data ψ, F, we study the wellposedness of our problem. To achieve our purpose, we use the techniques of Fourier series expansion in Hilbert scales. In particular, we apply Fourier analysis method and combine with some estimates of Mittag-Lefler functions to establish the existence and uniqueness of solutions to our problem on the Sobolev spaces.