Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition

Abstract

We consider the heat equation with a nonlinear boundary condition: (P) ∂_tu = Δu in R₊^N × (0,∞), ∂_νu = κ|u|^p−1u on ∂ R₊^N × (0,∞), u(x,0) = φ (x) in R₊^N, where R₊^N = {x= (x′,x_N) ∈ R^N: x_N > 0}, N ≥ 2, ∂_t = ∂/∂_t, ∂_ν = −∂/∂x_N, κ ∈ R, and p > 1 + 1/N. Let u be a solution of (P) satisfying sup_t>0(1 + t)^{(N/2)(1−1/q)}[||u(t)||_L^q(R+^N) + t^1/(2q)||u(t)||_L^q(∂R+^N)] < ∞, q ∈ [1,∞]. In this paper, under suitable assumptions of the initial function φ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.