Stability Results for Backward Nonlinear Diffusion Equations with Temporal Coupling Operator of Local and Nonlocal Type
Tran Thi Khieu, Hoang‐Hung Vo
Abstract
In this paper, we investigate the problem of reconstructing the historical distribution for a nonlinear diffusion equation, in which the diffusion is driven by not only a nonlocal operator but also a locally one. The problem naturally arises in many real-world applications including the biological population dynamic where a population competes for the resources and diffuses by a combination of classical and nonlocal dispersal processes. By using the Banach fixed point theorem and some appropriate estimates, we first construct an example to show the ill-posedness of the problem. Next, we propose a filter regularization method written in form of a nonlinear Volterra integral equation, in combination with the new technique recently developed calling the globally Lipschitz approximation. Finally, several numerical tests, with a combination of the finite difference schemes and the fast Fourier transform (FFT) algorithm, are also presented to illustrate the theoretical results.