Abstract
We investigate the long time asymptotics in L 1+ (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 d 2. In the particular case of φ(u) ∼ um at first order when u ~ 0, we also obtain an optimal rate of convergence in L 1 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.
Original language | English |
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Pages (from-to) | 127-149 |
Journal | Archive for Rational Mechanics and Analysis |
Volume | 180 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Apr 2006 |