zbMATH — the first resource for mathematics

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

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
Full Text: DOI