Abstract.
We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations u t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ um at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.
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Carrillo, J., Francesco, M. & Toscani, G. Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u). Arch. Rational Mech. Anal. 180, 127–149 (2006). https://doi.org/10.1007/s00205-005-0403-4
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DOI: https://doi.org/10.1007/s00205-005-0403-4