The large-time decay toward intermediate self--similar profiles for dissipative PDE's is classically related with scaling invariance properties and homogeneity of the involved nonlinearities (see for instance the porous medium equation or the nonlinear scalar conservation law). A moment renormalization procedure involving contraction results with respect to Wasserstein distances has been recently employed to generalize the concept of self-similarity for non homogeneous dissipative PDE's (where no "classical" self similar profiles can be detected) and to prove their asymptotic stability after a suitable scaling. This talk will provide an overview of the existing results, by presenting them in a unified framework. Open problems and works in progress related with further properties of the intermediate profiles and with possible generalization to nonlocal transport PDE's will be also presented.
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