On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz-Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as $t\to+\infty$. Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.