Fundamental Solutions and Asymptotic Behaviour for the $p$-Laplacian Equation

Abstract

We establish the uniqueness of fundamental solutions to the p -Laplacian equation \mathrm {(PLE)} \; u_t = \mathrm {div} (|Du|^{p-2}Du), \; p > 2, defined for x \in \mathbb R^N , 0 < t < T . We derive from this result the asymptotic behaviour of nonnegative solutions with finite mass, i.e. such that u(\cdotp, t) \in L^1(\mathbb R^N) . Our methods also apply to the porous medium equation \mathrm {(PME)} \; u_t = \Delta (u^m), \; m > 1, giving new and simpler proofs of known results. We finally introduce yet another method of proving asymptotic results based on the idea of asymptotic radial symmetry. This method can be useful in dealing with more general equations.