On the existence and uniqueness of solutions to a nonlinear variable order time-fractional reaction–diffusion equation with delay

Abstract

In this article, our purpose is to study the existence and uniqueness of a solution to a damped variable order fractional subdiffusion equation with time delay. Under weak assumptions on the data, we prove the uniqueness of a weak solution to the problem under consideration. The method of semi-discretization is extended to this kind of time fractional parabolic problem with delay in the case that the time delay parameter s>0 satisfies s⩽T, where T denotes the final time. As a consequence, two a priori estimates are predicted based on a discrete variational formulation of the problem. The existence of the problem’s weak solution on the time frame 0,⌊Ts⌋s is established by the aid of these derived a priori estimates. The paper is closed by introducing a fully discrete scheme based on Galerkin Legendre spectral approximation for the spatial operator and the backward Euler difference approximation for the temporal variable order operator. Accordingly, the accuracy and efficiency of the proposed scheme are justified by giving some numerical experiments for the sake of clearness.