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Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. (English) Zbl 1197.35023

The authors consider existence and multiplicity of stationary states for the Gierer-Meinhard system with saturation:
\[ \left\{ \begin{alignedat}{2} A_t&=\varepsilon^2\Delta A-A+\frac{A^2}{H(1+kA^2)}, &&\qquad\text{in }\Omega\times(0,\infty),\\ \tau H_t&=D\Delta H-H+A^2, &&\qquad\text{in }\Omega\times(0,\infty),\\ \frac{\partial A}{\partial\nu}&=\frac{\partial H}{\partial\nu}=0, &&\qquad\text{on }\partial\Omega\times(0,\infty),\\ A&>0,\;H>0, &&\qquad\text{in }\Omega\times(0,\infty).\\ \end{alignedat}\right.\tag{1} \]
Here \(\Omega\subseteq\mathbb{R}^N\) is a bounded smooth domain, \(2\leq N\leq 5\), \(\tau\geq0\), and \(\varepsilon, k>0\). Moreover, they assume that \(\Omega\) is rotationally symmetric with respect to the \(x_N\)-axis, and that \(k=k(\varepsilon)\) and \(\varepsilon\) have the dependence \(\lim_{\varepsilon\to0} 4k\varepsilon^{-2N}|\Omega|^2=k_0\), for some \(k_0\in[0,\infty)\) that is sufficiently small.
Fixing a subset \(\{P_1,P_2,\dots,P_m\}\) of the finite set of intersections of the \(x_N\)-axis with \(\partial\Omega\) the following result is obtained: If \(\varepsilon\) is sufficiently small and \(D\) sufficiently large then there exists a stationary solution to (1) that has its mass concentrated near the points \(P_i\). These spikes are individually approximated, asymptotically as \(\varepsilon\to 0\) and \(D\to\infty\), by the rescaled unique solution of a suitable limit equation posed in \(\mathbb{R}^N\).
A key point in the proof is that the symmetry condition on \(\Omega\) allows to construct a unique symmetric multi-peak solution of a related nonlinear elliptic equation that depends on a suitably defined new parameter \(\delta\). Uniqueness in turn leads to continuous dependence of this solution on \(\delta\) and allows to obtain a multi-peak stationary state for the shadow system (where \(D=\infty\)). Finally, to obtain a stationary state for the original equation (1) the implicit function theorem is used.

MSC:

35B25 Singular perturbations in context of PDEs
35J57 Boundary value problems for second-order elliptic systems
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C15 Developmental biology, pattern formation
35J60 Nonlinear elliptic equations
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