Isolated initial singularities for the viscous Hamilton-Jacobi equation

32 pagesHere we study the nonnegative solutions of the viscous Hamilton-Jacobi equation \[ u_{t}-\Delta u+|\nabla u|^{q}=0 \] 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 solutions with a possible singularity at point $(x,t)=(0,0).$ We show that if $q\geq q_{\ast }=(N+2)/(N+1)$ the singularity is removable. For $