In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating diffusions (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→+∞. Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.
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