On a terminal value problem for parabolic reaction–diffusion systems with nonlocal coupled diffusivity terms

Abstract

In this article, we are interested in investigating the nonlocal nonlinear reaction– diffusion system with final conditions. This problem is called backward in time problem, or terminal value problem which is understood as redefining the previous distributions when the distribution data at the terminal observation are known. There are three main goals presented in this paper. First, we prove that the problem is ill-posed (often called as unstable property) in the sense of Hadamard. Our next propose is to provide a modified quasi-reversibility model to stabilize the ill-posed problem. Using some techniques and tools of Faedo–Galerkin method, we prove the existence of the unique weak solution of the regularized problem. Further, we investigate error estimates between the sought solution and the regularized solution in L2(Ω)− and H1(Ω)− norms. The final aim of this paper is to give some numerical results to demonstrate that our method is useful and effective.