## 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

NAG; nag; d03pcf
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### References:

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