Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.