The 3-D Nonlinear Hyperbolic–Parabolic Problems: Invariant Manifolds

We investigate the existence of invariant manifolds for a coupled problem of nonlinear hyperbolic–parabolic PDEs on a 3-D torus. The problem arises usually in the study of wave propagation phenomena with viscous damping which are heat generating. The spectral gap condition could fail for it. We prove that there exists a Lipschitz manifold which is locally invariant under the semiflow. The local asymptotic stability and regularity of the manifold are also considered. Moreover, under more assumptions on the nonlinearity, it is proved that the manifold is provided with the feature as that global manifold usually holds, i.e., it contains the global attractor. Through it all, no large damping and heat diffusivity are needed.