Rate of approach to the steady state for a diffusion-convection equation on annular domains

Abstract

In this paper, we study the asymptotic behavior of global solutions of the equation $u_t=\Delta u+e^{|\nabla u|}$ in the annulus $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M\geq 0$ 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 blowup rate in infinite time is obtained.

Keywords

• convergence
• steady state
• gradient blowup