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Metastability in the shadow system for Gierer-Meinhardt’s equations. (English) Zbl 1002.35014

The authors deal with the shadow system (\(D\to\infty, \tau\to 0\)) for the Gierer-Meinhardt system of activator \(U\) and inhibitor \(H\): \[ U_t = \epsilon^2 \Delta U - U + U^p H^{-q}, \]
\[ \tau H_t = D \Delta H - H + U^r \] on a bounded interval with zero Neumann boundary conditions, and coefficients \(p>1,q>0\), and \(r>1\) satisfying \((q r)/(p-1)>1\). They study the (meta)stability of the single internal spike of the shadow system via the spectrum of the linearized differential operator, extending the results obtained by J. Wei [Eur. J. Appl. Math. 10, No. 4, 353-378 (1999; Zbl 1014.35005)].

MSC:

35B25 Singular perturbations in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs

Citations:

Zbl 1014.35005
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