Large-time behavior of small solutions of a two-dimensional semilinear elliptic equation with a dynamical boundary condition
Марек Фила, Kazuhiro Ishige, Tatsuki Kawakami · Asymptotic Analysis · 2013
We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: −Δu=u p , x∈R 2 + , t>0, ∂ t u+∂ ν u=0, x∈∂R 2 + , t>0, u(x,0)=φ(x 1 )≥0, x=(x 1 ,0)∈∂R 2 + , where u=u(x,t), ∂ t :=∂/∂t, ∂ ν :=−∂/∂x 2 , R 2 + :={(x 1 ,x 2 ): x 1 ∈R,x 2 >0} and p>1. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel. This is an extension of previous results of the authors of this paper to the two-dimensional case.