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Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations. (English) Zbl 1022.35006
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.

MSC:
35B42 Inertial manifolds
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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