Isolated initial singularities for the viscous Hamilton-Jacobi equation

Abstract

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-Δu+|\nabla u|^{q}=0] in $Q_{\Omega ,T}=\Omega \times (0,T),$ where $q>1,T\in (0,\infty ],$ and $\Omega$ is a smooth bounded domain of $\mathbb{R}$ containing $0,$ or $\Omega ={\mathbb{R}}^N.$ We consider solutions with a possible singularity at point $(x,t)=(0,0).$ We show that if $q\ge q_{\ast }=(N+2)/(N+1)$ the singularity is removable.

32 pages