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Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations. (English) Zbl 0881.35120

Nonlinear delay-diffusion equations of the form

u t =u xx (x,t)+a(x)u(x,t)+bu(x,t-τ)+g(x,u(x,t))

are studied and stability and bifurcation results are obtained by studying the perturbation of the spectrum of the linearization

u t =u xx (x,t)+a(x)u(x,t)+bu(x,t-τ)

caused by g. It is shown that a centre manifold of any dimension exists for some a, b, and τ. Moreover, it is shown that if ±iω 0 ,±iω 1 ,,±iω m-1 are the 2m pure imaginary eigenvalues of the linearized equation, then the numbers ω 0 ,ω 1 ,,ω m-1 are rationally independent and as τ increases through the critical value, the nonlinear equation undergoes a bifurcation from 2 eigenvalues, where is the sum of the multiplicities of the m eigenvalues.

MSC:
35R10Partial functional-differential equations
35K57Reaction-diffusion equations