Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for $u_t=Deltaphi(u)$

We investigate the long time asymptotics in L-+(1)( R) for solutions of general nonlinear diffusion equations u(t) = Delta phi(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities phi 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 phi(u) similar to u(m) at first order when u similar to 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.