Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u
t
= Δ?(u) |
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Authors: | José A Carrillo Marco Di Francesco Giuseppe Toscani |
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Institution: | 1. ICREA Institució Catalana de Recerca i Estudis Avan?ats and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193, Bellaterra, Spain 2. Dipartimento di Matematica, Università di L'Aquila, I-67100, L'Aquila, Italy 3. Dipartimento di Matematica, Università di Pavia, I-27100, Pavia, Italy
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Abstract: | We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations ut = 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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Keywords: | |
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