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Semidiscretization and long-time asymptotics of nonlinear diffusion equations. (English) Zbl 1145.35028

The authors review some results conserning the long-time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of this nonlinear diffusion method is proposed and their numerical properties analyzed. Long-time asymptotic results by numerical simulation are demonstrated and several open problems are proposed. They show that for general nonlinear diffusion equation the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for euclidean Wasserstein distance. It can be proved that the family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities near zero.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35A35 Theoretical approximation in context of PDEs
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