Advances in Differential Equations

Isolated initial singularities for the viscous Hamilton-Jacobi equation

Marie Françoise Bidaut-Veron and Nguyen Anh Dao

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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.

Article information

Source
Adv. Differential Equations Volume 17, Number 9/10 (2012), 903-934.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355702927

Mathematical Reviews number (MathSciNet)
MR2985679

Zentralblatt MATH identifier
06108230

Subjects
Primary: 35K15: Initial value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 35B33: Critical exponents 35B65: Smoothness and regularity of solutions 35D30: Weak solutions

Citation

Bidaut-Veron, Marie Françoise; Dao, Nguyen Anh. Isolated initial singularities for the viscous Hamilton-Jacobi equation. Adv. Differential Equations 17 (2012), no. 9/10, 903--934. https://projecteuclid.org/euclid.ade/1355702927.


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