Large time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition
In this paper, the semilinear elliptic equation with a dynamical boundary condition \[ \begin{cases} -\Delta u = u^p, & x \in \mathbb{R}_+^N, \;t>0, \\ \partial_t u + \partial_\nu u = 0, & x \in \partial\mathbb{R}_+^N, \;t>0, \\ u(x,0)=\varphi(x') \geq 0, & x = (x',0) \in \partial\mathbb{R}_+^N \end{cases} \] is considered, where \(N \geq 3\), \(\mathbb{R}_+^N:=\{ x=(x',x_N) : x' \in \mathbb{R}^{N-1}, x_N>0 \}\), \(u=u(x,t)\), \(\Delta\) is the \(N\)-dimensional Laplacian, \(\partial_t:=\partial/\partial t\), \(\partial_\nu:=-\partial/\partial x_N\), and \(p>1\). It is shown that if \(p \leq (N+1)/(N-1)\), then there are no nontrivial global solutions, and that if \(p>(N+1)/(N-1)\), then there is a global minimal solution for a small \(\varphi\). Furthermore, the asymptotic behavior of the global minimal solution is also studied.
scientific article; zbMATH DE number 6141571