On the two-dimensional Gierer-Meinhardt system with strong coupling. (English) Zbl 0955.35006

Wei, Juncheng; Winter, Matthias

SIAM J. Math. Anal. 30, No. 6, 1241-1263 (1999).

The authors study the Gierer-Meinhardt system \[ A_t= \varepsilon^2\Delta A- A+ A^p/H^q,\quad A> 0\quad\text{in }\Omega, \]
\[ \tau H_t= D\Delta H- H+ A^r/H^s,\quad H>0\quad\text{in }\Omega, \]
\[ {\partial A\over\partial\nu}= {\partial H\over\partial\nu}= 0\quad\text{on}\quad \partial\Omega. \] The unknowns \(A= A(x,t)\), \(H= H(x,t)\) represent the concentrations at a point \(x\in\Omega\) at time \(t\) of some biochemicals called activator and inhibitor, \(\varepsilon\), \(\tau\), \(D\) are positive constants and the exponents \(p\), \(q\), \(r\), \(s\) are assumed to satisfy the conditions \[ 1< p<\Biggl({N+ 2\over N-2}\Biggr)_+,\quad q> 0,\quad r>0,\quad s\geq 0,\quad 0<{p- 1\over q}<{r\over s+1}, \] where \(((N+ 2)/(N- 2))_+\) is \((N+ 2)/(N- 2)\) if \(N\geq 3\) and is \(+\infty\) if \(N= 1\) or 2. Using the Lyapunov-Schmidt reduction method, the authors study the associated stationary system for \(\varepsilon\to 0\) in the two-dimensional case, constructing solutions with isolated condensation points.

Reviewer: Otto Liess (Darmstadt)

MSC:

35B25	Singular perturbations in context of PDEs
92C40	Biochemistry, molecular biology
35B40	Asymptotic behavior of solutions to PDEs
35J55	Systems of elliptic equations, boundary value problems (MSC2000)
92C15	Developmental biology, pattern formation
35K50	Systems of parabolic equations, boundary value problems (MSC2000)

Keywords:

pattern formation; mathematical biology; strong coupling; Lyapunov-Schmidt reduction method; solutions with isolated condensation points