JournalsrmiVol. 19, No. 3pp. 873–917

The Pressure Equation in the Fast Diffusion Range

  • Emmanuel Chasseigne

    Université François Rabelais, Tours, France
  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
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Abstract

We consider the following degenerate parabolic equation

v_{t}=v\Delta v-\gamma|\nabla v|^{2}\quad\mbox{in $\mathbb{R}^{N} \times(0,\infty)$,}

whose behaviour depends strongly on the parameter γ\gamma. While the range γ<0\gamma < 0 is well understood, qualitative and analytical novelties appear for γ>0\gamma>0. Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for γ>max{N/2,1}\gamma>\max\{N/2,1\} the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class Ws\mathcal{W}_s of local weak solutions with constant support; initial data can be any nonnegative measurable function v0v_{0} (infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the pp-trace with p=γ<0p=-\gamma < 0, since the standard concepts of initial trace do not produce uniqueness. Here are some additional properties: the solutions turn out to be classical for t>0t>0, the support is constant in time, and not all of them can be obtained by the vanishing viscosity method. We also show that singular measures are not admissible as initial data, and study the asymptotic behaviour as tt\to \infty.

Cite this article

Emmanuel Chasseigne, Juan Luis Vázquez, The Pressure Equation in the Fast Diffusion Range. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 873–917

DOI 10.4171/RMI/373