Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations
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.
scientific article; zbMATH DE number 1884218