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

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November 2009, 8(6): 1917-1924. doi: 10.3934/cpaa.2009.8.1917

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

1.	Department of Mathematics, Southeast University, Nanjing 210096, China

Received November 2008 Revised April 2009 Published August 2009

Abstract
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In this short paper, we obtain the asymptotic behavior of the global solutions of a viscous Hamilton-Jacobi equation $u_t=\Delta u+|\nabla u|^p$ 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 a constant $M_c>0$ such that the problem admits a unique steady state if and only if $M\leq M_c$. When $M < M_c$, the global solution converges in $C^1(\overline{B_{r,R}})$ to the unique regular steady state. When $M=M_c$, the global solution converges in $C(\overline{B_{r,R}})$ to the unique singular steady state, and the grow-up rate of $||u_\nu(t)||_{L^\infty(\partial B_r)}$ in infinite time is obtained.

Keywords: gradient blow-up., Viscous Hamilton-Jacobi equations, asymptotic behavior.

Mathematics Subject Classification: Primary: 35K60, 35K65, 35B4.

Citation: Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917

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