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December, 2003 The Pressure Equation in the Fast Diffusion Range
Emmanuel Chasseigne, Juan Luis Vázquez
Rev. Mat. Iberoamericana 19(3): 873-917 (December, 2003).

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 $\gamma$. While the range $\gamma < 0$ is well understood, qualitative and analytical novelties appear for $\gamma>0$. Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for $\gamma>\max\{N/2,1\}$ the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class $\mathcal{W}_s$ of local weak solutions with constant support; initial data can be any nonnegative measurable function $v_{0}$ (infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the $p$-trace with $p=-\gamma < 0$, since the standard concepts of initial trace do not produce uniqueness. Here are some additional properties: the solutions turn out to be classical for $t>0$, 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 $t\to \infty$.

Citation

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Emmanuel Chasseigne. Juan Luis Vázquez. "The Pressure Equation in the Fast Diffusion Range." Rev. Mat. Iberoamericana 19 (3) 873 - 917, December, 2003.

Information

Published: December, 2003
First available in Project Euclid: 20 February 2004

zbMATH: 1073.35128
MathSciNet: MR2053567

Subjects:
Primary: 35K55 , 35K65

Keywords: fast diffusion , measure as initial trace , non-uniqueness , optimal initial data , pressure equation , well-posed problem

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.19 • No. 3 • December, 2003
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