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Effects of a degeneracy in the competition model. II: Perturbation and dynamical behaviour. (English) Zbl 1042.35017

This is the second part of the author in the study of the competition model

u t-Δu=λu-b(x)u 2 -cuvv t-Δv=μv-v 2 -duv,(1)

where xΩ and t0, Ω denotes a smooth bounded in N (N2), b(x)0, λ,μ,c and d are positive constants. Moreover, the author supposes that u and v satisfy homogeneous Dirichlet boundary conditions on Ω. The aim of the author is to understand the effects of the degeneracy of b(x) on (1). Here, based on results obtained in Part I [ibid. 181, 92–132 (2002; Zbl 1042.35016)] the author shows that both the classical and generalized steady-states of (1) occur naturally as the limits, when ε0, of the positive classical solutions of the perturbed system

-Δu=λu-b ( x ) + εu 2 -cuv-Δv=μv-v 2 -duvu| Ω =0,v| Ω =0,(2)

where ε>0 is a constant.


MSC:
35K50Systems of parabolic equations, boundary value problems (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
92D25Population dynamics (general)