Isolated initial singularities for the viscous Hamilton-Jacobi equation

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.