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

We consider weak entropy measure-valued solutions of the Neumann initial-boundary value problem for the equation uₜ=[φ(u)]ₓₓ, where φ is nonmonotone. These solutions are obtained from the corresponding problem for the regularized equation uₜ=[φ(u)]ₓₓ+ε uₓₓₜ (ε>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.