Relative Newtonian Potentials of Radial Functions and Asymptotics in Nonlinear Diffusion

Abstract

The Newtonian potential is introduced in a relative sense for radial functions. In this way one may treat the potential theory for a larger class of functions in a unified manner for all dimensions $d\ge1$. As a result, Newton's theorem is revised in terms of relative potentials, which is a simpler argument that works for all dimensions $d\ge1$. This relative potential is then used to obtain the $L^1$-convergence order $O(t^{-1})$ as $t\to\infty$ for radially symmetric solutions to the porous medium and fast diffusion equations. The technique is also applied to radial solutions of the p-Laplacian equations to obtain the same convergence order.