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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 =dΔu-μu+u p v -q +σ,v t =DΔv-νv+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>(p-1)(s-1), and that they will blow up in a finite time for some initial values if either r>p-1 with rq<(p-1)(s+1) or r<p-1.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
92D25Population dynamics (general)
35B40Asymptotic behavior of solutions of PDE
35B35Stability of solutions of PDE
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.