Theoretical and numerical aspects for the longtime behavior of nonlinear delay time Caputo fractional reaction–diffusion equations

Abstract

In this paper, we investigate the longtime behavior of time fractional reaction–diffusion equations with delay. The governing partial differential equation generalizes the Hutchinson, the Mackey–Glass and the Nicholson’s blowflies equations. Energy estimates, asymptotic stability and asymptotic contractivity of the problem are proved. The finite difference technique is used to discretize the time-fractional Caputo derivative, and the spectral Galerkin approach is used for the spatial approximation. Additionally, the ability to preserve asymptotic contractivity and stability rates can be proved for the numerical solution similarly as for the true solution. Finally, some numerical experiments are performed to confirm our findings. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.