On temporal regularity and polynomial decay of solutions for a class of nonlinear time-delayed fractional reaction–diffusion equations

Abstract

This paper is devoted to analyzing the regularity in time and polynomial decay of solutions for a class of fractional reaction–diffusion equations (FrRDEs) involving delays and nonlinear perturbations in a bounded domain of $R^d$. By establishing some regularity estimates in both time and space variables of the resolvent operator, we present results on the Hölder and $C^1$-regularity in time of solutions for both time-delayed linear and semilinear FrRDEs. Based on the aforementioned results, we study the existence, uniqueness, and regularity of solutions to an identification problem subjected to the delay FrRDE and the additional observations given at final time. Furthermore, under quite reasonable assumptions on nonlinear perturbations and the technique of measure of noncompactness, the existence of decay solutions with polynomial rates for the problem under consideration is shown.