Existence and Blow-up of Smooth Solutions for Cahn-Hilliard Equation with Inertial Term

The motivation of this paper is to systematically study how the inertial term affects the behavior of the solution of the Cahn-Hilliard equation. When there is an inertial term, the equation becomes a parabolic-hyperbolic one. To overcome the difficulties from large perturbation and the hyperbolicity, a few elaborate energy estimates should be given, which are based on choosing suitable space of the smooth solution, even some inequalites in negative Sobolev space. More precisely, for 1-dimensional(1D) non-viscous case and 1D, 2D, and 3D viscous case, the global existence of the smooth solutions are given. Moreover, several blow-up results are established by using the convex method. The results exhibit the interplay between the viscosity and the inertial term for the behavior of the smooth solution.