JournalsprimsVol. 32 , No. 3DOI 10.2977/prims/1195162854

Asymptotic Behavior of Blowup Solutions of a Parabolic Equation with the <i>p</i>-Laplacian

  • Ataru Fujii

    University of Tokyo, Japan
  • Masahito Ohta

    University of Tokyo, Japan
Asymptotic Behavior of Blowup Solutions of a Parabolic Equation with the <i>p</i>-Laplacian cover

Abstract

We consider the blowup problem for ut = Δpu + ∣ up–2_u_ (x ∈ Ω, t > 0) under the Dirichlet boundary condition and p > 2. We derive sufficient conditions on blowing up of solutions. In particular, it is shown that every non-negative and non-zero solution blows up in a finite time if the domain Ω is large enough. Moreover, we show that every blowup solution behaves asymptotically like a self-similar solution near the blowup time. The Rayleigh type quotient introduced in Lemma A plays an important role throughout this paper.