Stabilization towards the steady state for a viscous Hamilton-Jacobi equation

Abstract

In this short paper, we obtain the asymptotic behavior of the globalsolutions of a viscous Hamilton-Jacobi equation $u_t=\Deltau+|\nabla u|^p$ in $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and$u(x,t)=M$ on $\partial B_R$. It is proved that there exists aconstant $M_c>0$ such that the problem admits a unique steady stateif and only if $M\leq M_c$. When $M < M_c$, the global solutionconverges in $C^1(\overline{B_{r,R}})$ to the unique regular steadystate. When $M=M_c$, the global solution converges in$C(\overline{B_{r,R}})$ to the unique singular steady state, and thegrow-up rate of $||u_\nu(t)||_{L^\infty(\partial B_r)}$ in infinitetime is obtained.