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Stability of singularly perturbed solutions to systems of reaction- diffusion equations. (English) Zbl 0638.35010
This paper is concerned with systems of reaction-diffusion equations of the form: $(1)\quad u_ t=\epsilon^ 2u_{xx}+f(u,v),\quad v_ t=Dv_{xx}+g(u,v),$ (t,x)$$\in (0,\infty)\times I$$, $$I=(0,1)$$, $$u_ x=0=v_ x$$, (t,x)$$\in (0,\infty)\times \partial I$$, $$\partial I=\{0,1\}$$, where $$\epsilon$$ is a small parameter and $$D>0$$. The authors present a stability theorem for large amplitude solutions of (1), for general D, which provides a framework of global assumptions to be imposed on the nonlinearities to obtain stability. They show that a violation of these assumptions may cause instability. A key role in the theory is played by a spectral analysis of the linearized eigenvalue problem associated to (1).
Reviewer: D.Huet

##### MSC:
 35B25 Singular perturbations in context of PDEs 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K45 Initial value problems for second-order parabolic systems 35B35 Stability in context of PDEs 35K57 Reaction-diffusion equations
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