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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 $$u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)+ g(x,u(x,t))$$ are studied and stability and bifurcation results are obtained by studying the perturbation of the spectrum of the linearization $$u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)$$ caused by $g$. It is shown that a centre manifold of any dimension exists for some $a$, $b$, and $\tau$. Moreover, it is shown that if $\pm i\omega_0,\pm i\omega_1,\dots,\pm i\omega_{m-1}$ are the $2m$ pure imaginary eigenvalues of the linearized equation, then the numbers $\omega_0,\omega_1,\dots,\omega_{m-1}$ are rationally independent and as $\tau$ increases through the critical value, the nonlinear equation undergoes a bifurcation from $2\ell$ eigenvalues, where $\ell$ is the sum of the multiplicities of the $m$ eigenvalues.

35R10Partial functional-differential equations
35K57Reaction-diffusion equations
Full Text: DOI
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