A reduction principle for coupled nonlinear parabolic-hyperbolic PDE
This paper concerns coupled systems of abstract parabolic and hyperbolic equations of the type \[ \left. \begin{aligned} w_{tt}+\gamma w_t &+ Lw= F(w,w_t,u)\\ u_t+ \nu Au &= G(w,w_t,u)\end{aligned}\right\}\;w\in W,\;u\in U. \] Here, \(W\) and \(U\) are Hilbert spaces, \(L\) and \(A\) are linear positive selfadjoint operators on \(W\) and \(U\), respectively, and \(\gamma\geq 0\) and \(\nu> 0\) are parameters. Initial-boundary value problems for PDEs of thermoelasticity can be modeled in this way. Under appropriate continuity and growth conditions on the nonlinearities \(F\) and \(G\), it is proved that the parabolic equation can be excluded by means of an invariant exponentially attracting Lipschitz manifold.
scientific article; zbMATH DE number 2163365