Isolated initial singularities for the viscous Hamilton-Jacobi equation

TL;DR

This paper investigates the behavior of solutions to the viscous Hamilton-Jacobi equation near isolated singularities, establishing conditions under which such singularities are removable based on the exponent q.

Contribution

It provides a criterion for the removability of isolated singularities in solutions to the viscous Hamilton-Jacobi equation depending on the value of q.

Findings

Singularities are removable if q ≥ (N+2)/(N+1).

Solutions with singularities at (0,0) are characterized by the value of q.

The study extends understanding of singularity behavior in viscous Hamilton-Jacobi equations.

Abstract

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-\Delta u+|\nabla u|^{q}=0] in $Q_{\Omega,T}=\Omega\times(0,T),$ where $q>1,T\in(0,\infty] ,$ 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.