## 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

### Keywords:

multi-bump; Gierer and Meinhardt system; shadow system
Full Text:

### References:

 [1] DOI: 10.1080/03605309408821059 · Zbl 0814.35042 [2] DOI: 10.1007/s002050050072 · Zbl 0906.35049 [3] DOI: 10.1016/0022-1236(86)90096-0 · Zbl 0613.35076 [4] DOI: 10.1007/BF00289234 [5] DOI: 10.1080/03605308408820335 · Zbl 0546.35053 [6] DOI: 10.1016/0022-0396(88)90147-7 · Zbl 0676.35030 [7] Ni W.-M., Notices of Amer. Math. Soc. 45 (1) pp 9– (1998) [8] DOI: 10.1090/S0002-9947-1986-0849484-2 [9] DOI: 10.1002/cpa.3160440705 · Zbl 0754.35042 [10] DOI: 10.1016/0022-0396(86)90119-1 · Zbl 0627.35049 [11] DOI: 10.1137/S0036141098347237 · Zbl 0955.35006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.