JournalsifbVol. 13, No. 2pp. 271–295

Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent

  • Razvan Gabriel Iagar

    Universidad Autónoma de Madrid, Spain
  • Philippe Laurençot

    Université de Toulouse, Toulouse, France
  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent cover
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Abstract

We consider the problem posed for x\in &#8477^N and t>0t>0 with nonnegative and compactly supported initial data. We take the exponent p>2p>2 which corresponds to slow pp-Laplacian diffusion. The main feature of the paper is that the exponent qq takes the critical value q=p1q=p-1 which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term uq|\nabla u|^q and the diffusive term Δpu\Delta_p u have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium happens, so that the large-time behaviour of the solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation Wp1=W|\nabla W|^{p-1}=W, with logarithmic time corrections. The asymptotic rescaled profile is a kind of sandpile with a cusp on top, and it is independent of the space dimension.

Cite this article

Razvan Gabriel Iagar, Philippe Laurençot, Juan Luis Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent. Interfaces Free Bound. 13 (2011), no. 2, pp. 271–295

DOI 10.4171/IFB/258