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Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation. (English) Zbl 0656.92004
The authors investigate the existence of positive solutions for reaction- diffusion systems of the following type: $$ (*)\quad \partial u\sb i/\partial t=\nu\sb i\Delta u\sb i-\mu\sb iu\sb i+g\sb i(x,u)\quad in\quad (0,\infty)\times \Omega \quad (i=1,2). $$ Here $\Omega \subseteq {\bbfR}\sp n$ is a bounded domain, $\nu\sb i,\mu\sb i\in {\bbfR}\sb+$ and $g\sb i(x,u)$ $(u\equiv (u\sb 1,u\sb 2)$; $i=1,2)$ are given functions of the form $$ g\sb 1(x,u)=\rho\sb 1(x,u)u\sp p\sb 1/u\sp q\sb 2+\sigma\sb 1(x),\quad g\sb 2(x,u)=\rho\sb 2(x,u)u\sp r\sb 1/u\sp s\sb 2+\sigma\sb 2(x). $$ It is assumed that the nonnegative exponents p, q, r, s satisfy the conditions $$ 0<(p-1)/r<q/(s+1); $$ moreover, $\sigma\sb i$ and $\rho\sb i$ satisfy suitable assumptions concerning nonnegativity, smoothness and boundedness. The system (*) is complemented with no-flux boundary conditions and initial conditions. The motivation for this study comes from mathematical models describing pattern formation [see e.g. {\it H. Meinhardt}, Models of biological pattern formation. (1982)]. Under additional assumptions on p and r, the existence of a unique positive solution to the above problem is established. Results concerning equilibrium solutions of (*) (with no-flux boundary conditions) are also derived; in particular, nonexistence of patterns is shown under suitable assumptions. The relationship with earlier results on the same subjects [{\it W.-M. Ni} and {\it I. Takagi}: Trans. Am. Math. Soc. 297, 351-368 (1986; Zbl 0635.35031)] is discussed.
Reviewer: A.Tesei

92B05General biology and biomathematics
35K57Reaction-diffusion equations
35J65Nonlinear boundary value problems for linear elliptic equations
35K60Nonlinear initial value problems for linear parabolic equations
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