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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

Yuxiang Li 1,

1. 

Department of Mathematics, Southeast University, Nanjing 210096, China

Received  November 2008 Revised  April 2009 Published  August 2009

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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