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Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions. (English) Zbl 1155.34031

The aim of this paper is to study boundary layer type solutions to the singularly perturbed scalar PDE (*) \(\varepsilon^2 (u_ t - u_{xx}) + F(u,x,t) =0,\;x\in(0,1),t>0,\) subject to Dirichlet boundary conditions at \(x=0,1\) and \(2\pi\) periodicity condition in time \(t\). Here \(\varepsilon>0\) is a small parameter. The forcing term \(F\) is assumed to be a \(2\pi\) periodic function in \(t\) variable. In the paper the authors presented results on the qualitative asymptotic analysis of have developed asymptotic expansions and has alternating boundary layer type solutions of the bistable scalar parabolic equation (*). It should be noted that the dynamical system approach based on the notions like e.g. phase-plane analysis; transversal intersection of stable and unstable manifolds of slow manifolds; heteroclinic and/or homoclinic orbits; is often used.

MSC:

34E05 Asymptotic expansions of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
35K55 Nonlinear parabolic equations
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References:

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