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September/October 2012 Isolated initial singularities for the viscous Hamilton-Jacobi equation
Marie Françoise Bidaut-Veron, Nguyen Anh Dao
Adv. Differential Equations 17(9/10): 903-934 (September/October 2012).

Abstract

Here we study the nonnegative solutions of the viscous Hamilton--Jacobi equation \begin{equation*} u_{t}-\Delta u+|\nabla u|^{q}=0 \end{equation*} in $Q_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ where $q>1,$ $T\in\left( 0,\infty\right] ,$ and $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N} $ containing $0,$ or $\Omega=\mathbb{R}^{N}$. We consider weak solutions with a possible singularity at the point $(x,t)=(0,0)$. We show that if $q\geq q_{\ast}=(N+2)/(N+1)$ the singularity is removable. For $1<q<q_{\ast}$, we prove the uniqueness of a very singular solution without condition as $% |x|\rightarrow\infty$; we also show the existence and uniqueness of a very singular solution of the Dirichlet problem in $Q_{\Omega,\infty},$ when $% \Omega$ is bounded. We give a complete description of the weak solutions in each case.

Citation

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Marie Françoise Bidaut-Veron. Nguyen Anh Dao. "Isolated initial singularities for the viscous Hamilton-Jacobi equation." Adv. Differential Equations 17 (9/10) 903 - 934, September/October 2012.

Information

Published: September/October 2012
First available in Project Euclid: 17 December 2012

zbMATH: 06108230
MathSciNet: MR2985679

Subjects:
Primary: 35B33, 35B65, 35D30, 35K15, 35K55

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.17 • No. 9/10 • September/October 2012
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