### Nonlinear diffusions, hypercontractivity and the optimal $L\sp p$-Euclidean logarithmic Sobolev inequality

Manuel Del Pino, Jean Dolbeault & Ivan Gentil
The equation ut = ∆p(u1/(p−1)) for p > 1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focuse on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the Lp-Euclidean logarithmic Sobolev inequality. A...
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