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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.

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