The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations

Canadian mathematical bulletin

关键词: convexity;    generalized numerical range;    matrices;   

DOI  :  10.4153/CMB-2000-056-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Many classes of singularly perturbed reaction-diffusion equationspossess localized solutions where the gradient of the solution islarge only in the vicinity of certain points or interfaces in thedomain. The problems of this type that are considered are aninterface propagation model from materials science and anactivator-inhibitor model of morphogenesis. These two models areformulated as nonlocal partial differential equations. Resultsconcerning the existence of equilibria, their stability, and thedynamical behavior of localized structures in the interior and onthe boundary of the domain are surveyed for these two models. Byexamining the spectrum associated with the linearization of theseproblems around certain canonical solutions, it is shown that thenonlocal term can lead to the existence of an exponentially smallprincipal eigenvalue for the linearized problem. This eigenvalueis then responsible for an exponentially slow, or metastable,motion of the localized structure.

【 授权许可】

Unknown