Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system

Morimoto, Kotaro

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 973-995 / Harvested from Numdam

Résumé

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 $ \epsilon ,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 (\epsilon )$, and there exists a limit $ {\mathrm{lim}}_{\epsilon \to 0}\kappa {\epsilon }^{-2}={\kappa }_{0}$ for certain $ {\kappa }_{0}\in [0,\infty )$.

Détail

Publié le : 2010-01-01

MR 2659154 | Zbl 1202.34051

DOI : https://doi.org/10.1016/j.anihpc.2010.01.003
Classification:  35K57,  35Q80,  92C15

Bibliographie

[1] M. Mimura, M. Tabata, Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal. 11 (1980), 613-631 | MR 579554 | Zbl 0438.34014

[2] M.A. Del Pino, A priori estimates and applications to existence–nonexistence for a semilinear elliptic system, Indiana Univ. Math. J. 43 (1994), 77-129 | MR 1275454 | Zbl 0803.35055

[3] M.A. Del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc. 347 (1995), 4807-4837 | MR 1303116 | Zbl 0853.35009

[4] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), 30-39

[5] D. Iron, M. Ward, J. Wei, The stability of spike solutions to the one-dimensional Gierer–Meinhardt model, Phys. D 150 (2001), 25-62 | MR 1818735 | Zbl 0983.35020

[6] H. Jiang, W.-M. Ni, A priori estimates of stationary solutions of an activator–inhibitor system, Indiana Univ. Math. J. 56 (2007), 681-730 | MR 2317543 | Zbl 1125.35018

[7] T. Kolokolnikov, W. Sun, M.J. Ward, J. Wei, The stability of a stripe for the Gierer–Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst. 5 (2006), 313-363 | MR 2237150 | Zbl 1210.35016

[8] K. Kurata, K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer–Meinhardt system with saturation, Commun. Pure Appl. Anal. 7 (2008), 1443-1482 | MR 2425018 | Zbl 1197.35023

[9] Y. Miyamoto, An instability criterion for activator–inhibitor systems in a two-dimensional ball, J. Differential Equations 229 (2006), 494-508 | MR 2263564 | Zbl 1154.35371

[10] K. Morimoto, Construction of multi-peak solutions to the Gierer–Meinhardt system with saturation and source term, Nonlinear Anal. 71 (2009), 2532-2557 | MR 2532780 | Zbl 1178.35040

[11] W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, Handb. Differ. Equ. vol. I, North-Holland, Amsterdam (2004), 157-233 | MR 2103689 | Zbl 1129.35401

[12] W.-M. Ni, P. Poláčik, E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc. 353 (2001), 5057-5069 | MR 1852094 | Zbl 0981.35018

[13] W.-M. Ni, I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator–inhibitor type, Trans. Amer. Math. Soc. 297 (1986), 351-368 | MR 849484 | Zbl 0635.35031

[14] W.-M. Ni, I. Takagi, Point condensation generated by a reaction–diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math. 12 (1995), 327-365 | MR 1337211 | Zbl 0843.35006

[15] Y. Nishiura, Global structure of bifurcating solutions of some reaction–diffusion systems, SIAM J. Math. Anal. 13 (1982), 555-593 | MR 661590 | Zbl 0501.35010

[16] K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 359-401 | MR 1621343 | Zbl 0902.35040

[17] I. Takagi, Point-condensation for a reaction–diffusion system, J. Differential Equations 61 (1986), 208-249 | MR 823402 | Zbl 0627.35049

[18] J. Wei, Existence and stability of spikes for the Gierer–Meinhardt system, Handb. Differ. Equ. vol. V, North-Holland (2008), 487-585 | MR 2497911 | Zbl 1223.35007

[19] J. Wei, M. Winter, On the two-dimensional Gierer–Meinhardt system with strong coupling, SIAM J. Math. Anal. 30 (1999), 1241-1263 | MR 1718301 | Zbl 0955.35006

[20] J. Wei, M. Winter, Spikes for the two-dimensional Gierer–Meinhardt system: the weak coupling case, J. Nonlinear Sci. 11 (2001), 415-458 | MR 1871278 | Zbl 1141.35345

[21] J. Wei, M. Winter, Spikes for the Gierer–Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations 178 (2002), 478-518 | MR 1879835 | Zbl 1042.35005

[22] J. Wei, M. Winter, On the Gierer–Meinhardt system with saturation, Commun. Contemp. Math. 6 (2004), 259-277 | MR 2057842 | Zbl 1066.35023

[23] J. Wei, M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer–Meinhardt system in $ {R}^{1}$, Methods Appl. Anal. 14 (2007), 119-163 | MR 2437100 | Zbl 1195.35033

[24] J. Wei, M. Winter, Stationary multiple spots for reaction–diffusion systems, J. Math. Biol. 57 (2008), 53-89 | MR 2393208 | Zbl 1141.92007