del Pino, Manuel; Kowalczyk, Michał; Chen, Xinfu The Gierer \(\&\) Meinhardt system: The breaking of homoclinics and multi-bump ground states. (English) Zbl 1003.34025 Commun. Contemp. Math. 3, No. 3, 419-439 (2001). 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 \(U(x)= 6\cdot e^x(1+ e^x)^{-2}\). Reviewer: Anatolij Ivan Kolosov (Khar’kov) Cited in 25 Documents 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 PDFBibTeX XMLCite \textit{M. del Pino} et al., Commun. Contemp. Math. 3, No. 3, 419--439 (2001; Zbl 1003.34025) Full Text: DOI References: [1] DOI: 10.1080/03605309408821059 · Zbl 0814.35042 · doi:10.1080/03605309408821059 [2] DOI: 10.1007/s002050050072 · Zbl 0906.35049 · doi:10.1007/s002050050072 [3] DOI: 10.1016/0022-1236(86)90096-0 · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0 [4] DOI: 10.1007/BF00289234 · doi:10.1007/BF00289234 [5] DOI: 10.1080/03605308408820335 · Zbl 0546.35053 · doi:10.1080/03605308408820335 [6] DOI: 10.1016/0022-0396(88)90147-7 · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7 [7] Ni W.-M., Notices of Amer. Math. Soc. 45 (1) pp 9– (1998) [8] DOI: 10.1090/S0002-9947-1986-0849484-2 · doi:10.1090/S0002-9947-1986-0849484-2 [9] DOI: 10.1002/cpa.3160440705 · Zbl 0754.35042 · doi:10.1002/cpa.3160440705 [10] DOI: 10.1016/0022-0396(86)90119-1 · Zbl 0627.35049 · doi:10.1016/0022-0396(86)90119-1 [11] DOI: 10.1137/S0036141098347237 · Zbl 0955.35006 · doi:10.1137/S0036141098347237 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.