Rates of decay to equilibria for pp-Laplacian type equations

The long-time asymptotics for pp-Laplacian type equations ρt=Δpρm=div(|∇ρm|p−2∇ρm) in RnRn is studied for p>1p>1 and m≥n−p+1n(p−1). The non-negative solutions of the equations are shown to behave asymptotically, as t→∞t→∞, 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 L1L1-norm. The rates are proved to be optimal for p=2p=2. The method used is based on mass transportation inequalities.