## Advances in Differential Equations

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

### Isolated initial singularities for the viscous Hamilton-Jacobi equation

Marie Françoise Bidaut-Veron and Nguyen Anh Dao

#### 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