Revista Matemática Iberoamericana

The Pressure Equation in the Fast Diffusion Range

Emmanuel Chasseigne and Juan Luis Vázquez

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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 $t\to \infty$.

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Rev. Mat. Iberoamericana, Volume 19, Number 3 (2003), 873-917.

First available in Project Euclid: 20 February 2004

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Primary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations

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. Iberoamericana 19 (2003), no. 3, 873--917.

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