JournalsrmiVol. 4, No. 2pp. 339–354

Fundamental Solutions and Asymptotic Behaviour for the pp-Laplacian Equation

  • Shoshana Kamin

    Tel-Aviv University, Israel
  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
Fundamental Solutions and Asymptotic Behaviour for the $p$-Laplacian Equation cover
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Abstract

We establish the uniqueness of fundamental solutions to the pp-Laplacian equation

(PLE)  ut=div(Dup2Du),  p>2,\mathrm {(PLE)} \; u_t = \mathrm {div} (|Du|^{p-2}Du), \; p > 2,

defined for xRNx \in \mathbb R^N, 0<t<T0 < t < T. We derive from this result the asymptotic behaviour of nonnegative solutions with finite mass, i.e. such that u(,t)L1(RN)u(\cdotp, t) \in L^1(\mathbb R^N). Our methods also apply to the porous medium equation

(PME)  ut=Δ(um),  m>1,\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.

Cite this article

Shoshana Kamin, Juan Luis Vázquez, Fundamental Solutions and Asymptotic Behaviour for the pp-Laplacian Equation. Rev. Mat. Iberoam. 4 (1988), no. 2, pp. 339–354

DOI 10.4171/RMI/77