Stability Results for Backward Nonlinear Diffusion Equations with Temporal Coupling Operator of Local and Nonlocal Type
Pages 1665 - 1700
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.
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DOI:10.1137/sjnaam.60.4
Issue’s Table of Contents© 2022, Society for Industrial and Applied Mathematics.
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Published: 01 January 2022
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