Affiliations: Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia. E-mail: fila@fmph.uniba.sk | Mathematical Institute, Tohoku University, Sendai, Japan. E-mail: ishige@math.tohoku.ac.jp | Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Japan. E-mail: kawakami@ms.osakafu-u.ac.jp
Note: [] Corresponding author: Marek Fila, Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovakia. E-mail: fila@fmph.uniba.sk
Abstract: We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: −Δu=up, 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.
Keywords: semilinear elliptic equation in a half-plane, dynamical boundary condition, large time behavior
