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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:

$\left(*\right)\phantom{\rule{1.em}{0ex}}\partial {u}_{i}/\partial t={\nu }_{i}{\Delta }{u}_{i}-{\mu }_{i}{u}_{i}+{g}_{i}\left(x,u\right)\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}\left(0,\infty \right)×{\Omega }\phantom{\rule{1.em}{0ex}}\left(i=1,2\right)·$

Here ${\Omega }\subseteq {ℝ}^{n}$ is a bounded domain, ${\nu }_{i},{\mu }_{i}\in {ℝ}_{+}$ and ${g}_{i}\left(x,u\right)$ $\left(u\equiv \left({u}_{1},{u}_{2}\right)$; $i=1,2\right)$ are given functions of the form

${g}_{1}\left(x,u\right)={\rho }_{1}\left(x,u\right){u}_{1}^{p}/{u}_{2}^{q}+{\sigma }_{1}\left(x\right),\phantom{\rule{1.em}{0ex}}{g}_{2}\left(x,u\right)={\rho }_{2}\left(x,u\right){u}_{1}^{r}/{u}_{2}^{s}+{\sigma }_{2}\left(x\right)·$

It is assumed that the nonnegative exponents p, q, r, s satisfy the conditions

$0<\left(p-1\right)/r

moreover, ${\sigma }_{i}$ and ${\rho }_{i}$ 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. 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 [W.-M. Ni and I. Takagi: Trans. Am. Math. Soc. 297, 351-368 (1986; Zbl 0635.35031)] is discussed.

Reviewer: A.Tesei
##### MSC:
 92B05 General biology and biomathematics 35K57 Reaction-diffusion equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial value problems for linear parabolic equations