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The Gierer \(\&\) Meinhardt system: The breaking of homoclinics and multi-bump ground states. (English) Zbl 1003.34025

The authors consider the following second-oder system of ordinary differential equations \[ \begin{aligned} u''- u+{u^2\over v}= 0,\quad &\text{in }\mathbb{R},\\ \sigma^{-2} v''- v+ u^2= 0,\quad &\text{in }\mathbb{R},\end{aligned}\tag{1} \] and study homoclinic solutions satisfying \[ u,v> 0\quad\text{in }\mathbb{R},\quad \lim_{|x|\to\infty} u(x)= \lim_{|x|\to \infty} v(x)= 0. \] The following theorem is proved: Given \(N\geq 1\), there exists a number \(\sigma_N> 0\) such that, for any \(0< \sigma< \sigma_N\), there exist solutions \((u_\sigma, v_\sigma)\) to (1) and points \(\xi^*_1< \xi^*_2<\cdots< \xi^*_N\) such that \[ \lim_{\sigma\to 0} \Biggl|\sigma u_\sigma(x)- {e^{-\sigma|x|}\over N \int^\infty_0 U^2} \sum^N_{i=1} U(x- \xi^*_i)\Biggr|= 0,\quad\lim_{\sigma\to 0} \Biggl|\sigma v_\sigma(x)- {e^{-\sigma|x|}\over N\int^\infty_0 U^2} \Biggr|=0, \] uniformly in \(x\), and \[ \xi^*_i= \xi^*_1+(i- 1)|\ln\sigma|+ 0(1) \] as \(\sigma\to 0\), for \(i= 1,\dots, N\), besides, \(u_\sigma(x)= u_\sigma(-x)\), \(v_\sigma(x)= v_\sigma(-x)\) and \[ u_\sigma(x)\leq Ce^{-|x- \xi^*_N|}\quad\text{as }x\to \pm \infty, \] where \(U(x)= 6\cdot e^x(1+ e^x)^{-2}\).

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35C20 Asymptotic expansions of solutions to PDEs
35J60 Nonlinear elliptic equations
35K99 Parabolic equations and parabolic systems
92C15 Developmental biology, pattern formation
92C40 Biochemistry, molecular biology
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References:

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