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.
Carrillo, J. A., Francesco, M. D., & Toscani, G. (2006). Intermediate asymptotics homogeneity and self-similarity: Long time behavior for u <inf>t</inf> = Δφ(u). Archive for Rational Mechanics and Analysis, 180(1), 127-149. https://doi.org/10.1007/s00205-005-0403-4