Asymptotic $L^1$ -convergence order for the porous medium equation via potential and entropy comparison

Abstract

The $L^{1}$ -convergence order of the porous medium equation $u_{t} = Δ(u^{m})$ to the Barenblatt solution is studied. The optimal convergence order willbe obtained using the potential comparison method for the compactly supported radial symmetric initial data when the space dimension is not equal to two. Also the optimal convergence order for the p-Laplacian equation will be obtained under the same constraint on initial data. Therefore, we can conclude the potential comparison method is sufficiently strong under the constraint on the initial data.