MR3221839
Tatsuki KAWAKAMI
Tatsuki KAWAKAMI
Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition
Funkcialaj Ekvacioj. Serio Internacia
57
2014
57--89
http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-1/57_57.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3221839
We consider the heat equation with a nonlinear boundary condition, (P) $\partial_t u=\Delta u$ in ${{\mathbb R}^N_+}\times(0,\infty)$, $\partial_\nu u=\kappa|u|^{p-1}u$ on $\partial{{\mathbb R}^N_+}\times(0,\infty)$, $u(x,0)=\varphi(x)$ in ${{\mathbb R}^N_+}$, where ${{\mathbb R}^N_+}=\{x=(x',x_N)\in{\mathbb R}^N:x_N>0\}$, $N\ge 2$, $\partial_t=\partial/\partial t$, $\partial_\nu=-\partial/\partial x_N$, $\kappa\in{\mathbb R}$, and $p>1+1/N$. Let $u$ be a solution of (P) satisfying $\sup_{t>0}\,(1+t)^{(N/2)(1-1/q)}[\|u(t)\|_{L^q({{\mathbb R}^N_+})}+t^{1/(2q)}\|u(t)\|_{L^q(\partial{{\mathbb R}^N_+})}]<\infty$, $q\in[1,\infty]$. In this paper, under suitable assumptions of the initial function $\varphi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$.
Nonlinear boundary condition, Asymptotic expansion, Large time behavior.
35B40, 35K61.
57-57
2014
Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition
Tatsuki KAWAKAMI
Tatsuki KAWAKAMI
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