Long-Time Behavior of Solutions to a Class of Forward-Backward Parabolic Equations

Abstract

We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem for the equation $u_t=[\phi(u)]_{xx}$, where $\phi$ is nonmonotone. These solutions are obtained from the corresponding problem for the regularized equation $u_t=[\phi(u)]_{xx}+\varepsilon u_{xxt}$ ($\varepsilon>0$) by a vanishing viscosity method and satisfy a family of suitable entropy inequalities. Relying on a strong version of these inequalities, we prove exhaustive results concerning the long-time behavior of solutions.