On a class of equations with variable parabolicity direction

Abstract

We consider the following pseudoparabolic regularization of a forward-backward quasilinear diffusion equation: $u_t=\Delta \phi(u)+\varepsilon\Delta u_t$ ($\varepsilon>0$). As suggested by several models of the applied sciences, the function $\phi$ is nonmonotone and vanishing at infinity. We investigate the limit points of the set of solutions to the associated Neumann problem as $\varepsilon\to 0$, proving existence of suitable weak solutions of the original ill-posed equation. Qualitative properties of such solutions are also addressed.