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: {0=ϵ 2 A -A+A 2 H(1+κA 2 )+σ,A>0,x(-1,1),0=DH -H+A 2 ,H>0,x(-1,1),A ' (±1)=H ' (±1)=0, where ϵ,D>0, κ0, σ0. 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, κ=κ(ϵ), and there exists a limit lim ϵ0 κϵ -2 =κ 0 for certain κ 0 [0,).
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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