Large-time behavior of small solutions of a two-dimensional semilinear elliptic equation with a dynamical boundary condition

Abstract

We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition:

−Δu=u^p, x∈R²₊, t>0,

∂_tu+∂_νu=0, x∈∂R²₊, t>0,

u(x,0)=φ(x₁)≥0, x=(x₁,0)∈∂R²₊,

where u=u(x,t), ∂_t:=∂/∂t, ∂_ν:=−∂/∂x₂, R²₊:={(x₁,x₂): x₁∈R,x₂>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.