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) no. 4, pp. 973-995.

In this paper, we are concerned with peak solutions to the following one-dimensional Gierer–Meinhardt system with saturation:

$$\{\begin{array}{cc}0={ϵ}^2{A}^{″}-A+\frac{A^2}{H(1+\kappa A^2)}+\sigma ,\hfill & A>0,x\in (-1,1),\hfill \\ 0=D{H}^{″}-H+A^2,\hfill & H>0,x\in (-1,1),\hfill \\ A^{\text{'}}(\pm 1)=H^{\text{'}}(\pm 1)=0,\hfill \end{array}$$

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 ${\lim }_{ϵ\to 0}\kappa {ϵ}^{-2}={\kappa }_0$ for certain ${\kappa }_0\in [0,\infty )$.

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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