Dynamics of Partial Differential Equations

Volume 2 (2005)

Multiple-spike ground state solutions of the Gierer-Meinhardt equations for biological activator-inhibitor systems

Pages: 187 – 260

DOI: https://dx.doi.org/10.4310/DPDE.2005.v2.n3.a1

Author

Yuncheng You (Department of Mathematics, University of South Florida, Tampa, Florida, U.S.A.)

Abstract

In many biological pattern formation processes and in some chemical or biochemical reactions, an activator-inhibitorsystem of two reaction-diffusion equations serves as a mathematical model, typically, the Gierer-Meinhardt equations.This type of model equations features two largely different diffusion coefficients and an essentially nonlocal nonlinearity.In this paper, the one-dimensional Gierer-Meinhardt equations are considered,$\begin{gather*}A_t=d\Delta A-A+\frac{A^2}H=0\quad\text{in }\mathbb{R}, \\H_t=D\Delta H-H+A^2=0\quad\text{in }\mathbb{R}, \\A, H>0\text{ and }A, H\to 0\text{ as }|x|\to\infty,\end{gather*}$where $\sigma^2=d/D\ll 1$. By the Lyapunov-Schmidt method, a sharp order-estimate of the number $k$ of multiplespikes of the ground state solutions is made. The $k$-spike solutions are constructed by adding small perturbation to thefunction which has $k$ appropriately distributed spikes resembling the solution of the problem$\begin{gather*}\Delta u-u+u^2=0\quad\text{in }\mathbb{R}, \\0<u\to 0\text{ as }|x|\to\infty.\end{gather*}$The main result is that, for sufficiently small $\sigma>0$, there exists such a ground state solution with$k=\const\sigma^{-\beta}$, where $0<\beta<1/2$ and $\beta$ can be arbitrarily close to $1/2$. In the proof of thisconjecture, a priori estimates of linear and nonlinear parts are conducted by means of cut-off decomposition,sharp calculations of multiple spike interactions at all levels, and finally a fine-tuned adjustment of spike centers.

Keywords

Gierer-Meinhardt equation, multiple-spike solution, activator-inhibitor system, pattern formation, Lyapunov-Schmidt method

2010 Mathematics Subject Classification

Primary 35B25, 35B40, 35B45. Secondary 35Jxx, 92C15, 92C40.

Published 1 January 2005