Rates of Decay to Equilibria for p-Laplacian Type Equations
Martial Agueh
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
agueh@andrew.cmu.edu

Abstract: The long-time asymptotics for $ p$-Laplacian type equations $ \rho_t=\Delta_p\rho^m=$div$ ({\vert\nabla\rho^m\vert^{p-2}\nabla\rho^m})$ in $ \mathbb{R}^n$, is studied for $ p>1$ and $ m\geq \frac{n-p+1}{n(p-1)}$. The non-negative solutions of the equations are shown to behave asymptotically, as $ t\rightarrow\infty$, like Barenblatt-type solutions, and the explicit rates of decay are established for the convergence of the relative energy, the convergence with respect to the Wasserstein distances and the convergence with respect to the $ L^1$-norm. The rates are proved to be optimal for $ p=2$. The method used is based on mass transportation inequalities.

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