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A metastable spike solution for a nonlocal reaction-diffusion model. (English) Zbl 0956.35011

An asymptotic reduction of the Gierer-Meinhardt activator-inhibitor system in the limit of large inhibitor diffusitivity leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit of small activator diffusitivity, a one-spike solution to this nonlocal model is constructed. The spectrum of the eigenvalue problem associated with the linearization of the nonlocal model around such an isolated spike solution is studied. Explicit metastable spike dynamics are derived by a projection method, which enforces a limiting solvability condition on the solution to the linearized problem.

MSC:

35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
35C20 Asymptotic expansions of solutions to PDEs
35P15 Estimates of eigenvalues in context of PDEs

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