On the System of Reaction-Diffusion Equations in a Limited Region

Nikita Borisov, Vladimir Nefedov

Abstract


This paper addresses problems of dynamics and long-term behavior of replicator (nonlinear) systems of partial differential equations. The primary focus is on the influence of the spatial factor on the behavior of distributed systems described by partial differential equations. A general problem formulation with Neumann, Dirichlet, and Robin boundary conditions is considered, and both spatially homogeneous and inhomogeneous stationary equilibrium states are analyzed. The stability of these states is investigated using spectral analysis and the energy method, including generalizations for various types of boundary conditions. The paper demonstrates that for sufficiently large diffusion coefficients, solutions tend to a stationary regime, with Dirichlet and Robin conditions enhancing stability compared to Neumann conditions. Examples, such as the Fisher-Kolmogorov equation and a two-component system, are provided to illustrate the application of the proposed methods. The results emphasize the importance of accounting for boundary conditions and diffusion in predicting the long-term behavior of reaction-diffusion systems.

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References


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