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Boundedness and blow up for the general activator-inhibitor model. (English) Zbl 0834.35069

Summary: This paper deals with the general activator-inhibitor model

${u}_{t}=d{\Delta }u-\mu u+{u}^{p}{v}^{-q}+\sigma ,\phantom{\rule{2.em}{0ex}}{v}_{t}=D{\Delta }v-\nu v+{u}^{r}{v}^{-s}$

with Neumann boundary conditions. We show that the solutions of the model are bounded all the time for each pair of initial value if $r>p-1$ and $rq>\left(p-1\right)\left(s-1\right)$, and that they will blow up in a finite time for some initial values if either $r>p-1$ with $rq<\left(p-1\right)\left(s+1\right)$ or $r.

##### MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 92D25 Population dynamics (general) 35B40 Asymptotic behavior of solutions of PDE 35B35 Stability of solutions of PDE
##### Keywords:
holomorphic semigroup; activator-inhibitor model; blow up
##### References:
 [1] Gierer, A. and H. Meinhardt. A Theory of Biological Pattern Formation.Kybernetik, 1972, 12: 30–39. · doi:10.1007/BF00289234 [2] Rothe, F. Global Solutions of Reaction-Diffusion Equations. Lecture Notes in Mathematics 1072, Springer-Verlag, Berlin, 1984. [3] Wu Jianhua and Li Yanling. Global Classical Solution for the Activator-Inhibitor Model.Acta Mathematicae Applicatae Sionica (in Chinese), 1990, 13: 501–505. [4] Henry, D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.