Vasil’eva, Adelaida B.; Kalachev, Leonid V. Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions. (English) Zbl 1155.34031 Abstr. Appl. Anal. 2006, Article ID 52856, 21 p. (2006). 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. Reviewer: Daniel Ševčovič (Bratislava) Cited in 2 Documents MSC: 34E05 Asymptotic expansions of solutions to ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 35K55 Nonlinear parabolic equations Keywords:singular perturbations; scalar parabolic equation; alternating boundary layer type solution; phase-plane analysis PDFBibTeX XMLCite \textit{A. B. Vasil'eva} and \textit{L. V. Kalachev}, Abstr. Appl. Anal. 2006, Article ID 52856, 21 p. (2006; Zbl 1155.34031) Full Text: DOI EuDML References: [1] N. D. Alikakos, P. W. Bates, and X. Chen, “Periodic traveling waves and locating oscillating patterns in multidimensional domains,” Transactions of the American Mathematical Society, vol. 351, no. 7, pp. 2777-2805, 1999. · Zbl 0929.35067 · doi:10.1090/S0002-9947-99-02134-0 [2] J. Keener and J. Sneyd, Mathematical Physiology, vol. 8 of Interdisciplinary Applied Mathematics, Springer, New York, 1998. · Zbl 0913.92009 · doi:10.1007/b98841 [3] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, New York, 2nd edition, 1993. · Zbl 0779.92001 [4] N. N. Nefedov, “An asymptotic method of differential inequalities for the investigation of periodic contrast structures: existence, asymptotics, and stability,” Differential Equations, vol. 36, no. 2, pp. 298-305, 2000. · Zbl 0979.35150 · doi:10.1007/BF02754216 [5] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14, Springer, New York, 2nd edition, 2001. · Zbl 1027.92022 [6] A. B. Vasil’eva, “Periodic solutions to a parabolic problem with a small parameter multiplying the derivatives,” Computational Mathematics and Mathematical Physics, vol. 43, no. 7, pp. 932-943, 2003. · Zbl 1136.35306 [7] A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, vol. 14 of SIAM Studies in Applied Mathematics, SIAM, Pennsylvania, 1995. · Zbl 0823.34059 [8] A. B. Vasil’eva, A. P. Petrov, and A. A. Plotnikov, “On the theory of alternating contrast structures,” Computational Mathematics and Mathematical Physics, vol. 38, no. 9, pp. 1471-1480, 1998. · Zbl 0965.35011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.