ON THE GIERER–MEINHARDT SYSTEM WITH SATURATION

We consider the following shadow Gierer–Meinhardt system with saturation: where ∊>0 is a small parameter, τ≥0, k>0 and Ω⊂Rn is smooth bounded domain. The case k=0 has been studied by many authors in recent years. Here we give some sufficient conditions on k for the existence and stability of stable spiky solutions. In the one-dimensional case we have a complete answer to the stability behavior. Central to our study are a parameterized ground-state equation and the associated nonlocal eigenvalue problem (NLEP) which is solved by functional analysis arguments and the continuation method.

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