This issuePrevious ArticleRigorous derivation of the Landau equation in the weak coupling limitNext ArticleSymmetry and monotonicity for a system of integral equations
Stabilization towards the steady state for a viscous Hamilton-Jacobi equation
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.