On the equivalence between p-Poincaré inequalities and Lr–Lq regularization and decay estimates of certain nonlinear evolutions

Consider a p-homogeneous functional E(p) (p>2) and suppose that a weighted Poincaré inequality involving it holds. Then all solutions u(t) to the evolution equation driven by the associated weighted p-Laplacian satisfy, for any 10, the bound . Such bound is in fact equivalent to the Poincaré inequality. There are examples in which the Poincaré inequality holds but the evolution does not map Lq0 into L∞ for any t and any q0≠∞. Moreover, if a p-logarithmic Sobolev inequality holds then the Poincaré inequality is shown to hold too, therefore the previous regularization result is valid. Finally, the weighted Sobolev-type inequality ‖u‖q⩽CE(p)(u) (q0, ε>0, .