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The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. (English) Zbl 0983.35020

Summary: The stability properties of an \(N\)-spike equilibrium solution to a simplified form of the Gierer-Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically in the limit of small activator diffusivity \(\varepsilon\). The equilibrium solution consists of a sequence of spikes of equal height. The two classes of eigenvalues that must be considered are the \(O(1)\) eigenvalues and the \(O(\varepsilon^2)\) eigenvalues, which are referred to as the large and small eigenvalues, respectively. The spike pattern is stable when the parameters in the Gierer-Meinhardt model are such that both sets of eigenvalues lie in the left half-plane. For a certain range of these parameters and for \(N\geq 2\) and \(\varepsilon\to 0\), it is shown that the \(O(1)\) eigenvalues are in the left half-plane only when \(D< D_N\), where \(D_N\) is some explicit critical value of the inhibitor diffusivity \(D\). For \(N\geq 2\) and \(\varepsilon\to 0\), it is also shown that the small eigenvalues are real and that they are negative only when \(D< D^*_N\), where \(D^*_N\) is another critical value of \(D\), which satisfies \(D^*_N< D_N\). Thus, when \(N\geq 2\) and \(\varepsilon\ll 1\), the spike pattern is stable only when \(D< D^*_N\). An explicit formula for \(D^*_N\) is given. For the special case \(N= 1\), it is shown that a one-spike equilibrium solution is stable when \(D< D_1(\varepsilon)\), where \(D_1(\varepsilon)\) is exponentially large as \(\varepsilon\to 0\), and is unstable when \(D> D_1(\varepsilon)\). An asymptotic formula for \(D_1(\varepsilon)\) is given. Finally, the dynamics of a one-spike solution is studied by deriving a differential equation for the trajectory of the center of the spike.

MSC:

35B35 Stability in context of PDEs
35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations

Software:

NAG; nag; d03pcf
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References:

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