Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations
type: Item
instance of: scholarly article
title: Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations
zbMATH Open document ID: 1022.35006
DOI: 10.1016/S0022-0396(02)00018-9
author: Anthony W. Leung
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: The following coupled parabolic-hyperbolic system is considered \[ \begin{aligned} u_t= \varepsilon^{-1} + \varepsilon^{-1}A(x)u +f(u,v), &\quad (x,t)\in\Omega \times[0,\infty),\\ v_{tt}+\mu v_t=\Delta v + B(x)v+g(u,v), &\quad (x,t)\in \Omega\times [0,\infty),\\ u=v=0, &\quad (x,t)\in\partial\Omega \times [0,\infty). \end{aligned} \] Assuming \(\varepsilon>0 \) sufficiently small and imposing appropriate conditions for \( A(x)\), \(B(x)\), \(f(u,v)\), \(g(u,v),\) the author proves the existence of an invariant manifold for the system above. He also proves that the asymptotic stabilty of an equilibrium restricted to the flow on the asymptotically stable invariant manifold implies the asymptotic stability of the equilibrium for the full system. Some examples of application of the results obtained are given.
Mathematics Subject Classification ID: 35B42
Mathematics Subject Classification ID: 35R20
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 1884218
zbMATH Keywords: invariant manifolds
zbMATH Keywords: parabolic-hyperbolic equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1884218