# Revista Matemática Iberoamericana

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**Volume 19, Issue 3, 2003, pp. 873–917**

**DOI: 10.4171/RMI/373**

Published online: 2003-12-31

The Pressure Equation in the Fast Diffusion Range

Emmanuel Chasseigne^{[1]}and Juan Luis Vázquez

^{[2]}(1) Université François Rabelais, Tours, France

(2) Universidad Autónoma de Madrid, Spain

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 $t\to \infty$.

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

Chasseigne Emmanuel, Vázquez Juan Luis: The Pressure Equation in the Fast Diffusion Range. *Rev. Mat. Iberoam.* 19 (2003), 873-917. doi: 10.4171/RMI/373