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In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully...
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