Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, p. 973-995

In this paper, we are concerned with peak solutions to the following one-dimensional Gierer–Meinhardt system with saturation: $\left\{\begin{array}{cc}0={ϵ}^{2}{A}^{″}-A+\frac{{A}^{2}}{H\left(1+\kappa {A}^{2}\right)}+\sigma ,\hfill & A>0,\phantom{\rule{0.166667em}{0ex}}x\in \left(-1,1\right),\hfill \\ 0=D{H}^{″}-H+{A}^{2},\hfill & H>0,\phantom{\rule{0.166667em}{0ex}}x\in \left(-1,1\right),\hfill \\ {A}^{\text{'}}\left(±1\right)={H}^{\text{'}}\left(±1\right)=0,\hfill \end{array}$ where $ϵ,D>0$, $\kappa ⩾0$, $\sigma ⩾0$. The saturation effect of the activator is given by the parameter κ. We will give a sufficient condition of κ for which point-condensation phenomena emerge. More precisely, for fixed $D>0$, we will show that the Gierer–Meinhardt system admits a peak solution when ε is sufficiently small under the assumption: κ depends on ε, namely, $\kappa =\kappa \left(ϵ\right)$, and there exists a limit ${\mathrm{lim}}_{ϵ\to 0}\kappa {ϵ}^{-2}={\kappa }_{0}$ for certain ${\kappa }_{0}\in \left[0,\infty \right)$.

DOI : https://doi.org/10.1016/j.anihpc.2010.01.003
Classification:  35K57,  35Q80,  92C15
Keywords: Gierer–Meinhardt system, Saturation effect, Pattern formation, Nonlinear elliptic system
@article{AIHPC_2010__27_4_973_0,
author = {Morimoto, Kotaro},
title = {Point-condensation phenomena and saturation effect for the one-dimensional Gierer--Meinhardt system},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {27},
number = {4},
year = {2010},
pages = {973-995},
doi = {10.1016/j.anihpc.2010.01.003},
zbl = {1202.34051},
mrnumber = {2659154},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_4_973_0}
}

Morimoto, Kotaro. Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, pp. 973-995. doi : 10.1016/j.anihpc.2010.01.003. http://www.numdam.org/item/AIHPC_2010__27_4_973_0/

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