Minimal solutions of a semilinear elliptic equation with a dynamical boundary condition (Q280827)

The authors study properties of positive solutions of a semilinear elliptic equation with a linear dynamical boundary condition NEWLINENEWLINE\[NEWLINE \begin{cases} -\Delta u =f(u), & x\in \mathbb R^N_+, t>0, \\ NEWLINE\partial_t u+\partial_{\nu}u =0, & x\in \partial \mathbb R^N_+, t>0, \\NEWLINEu(x,0)=\varphi(x'), & x=(x',0)\in \partial \mathbb R^N_+, \end{cases} NEWLINE\]NEWLINE NEWLINEwhere \(N\geq 2, \) \(\partial_\nu = -\partial/\partial_{x_N},\) \(\varphi \) is a nonnegative measurable function in \(\mathbb R^{N-1}\) and \(f\) is a nondecreasing continuous function in \(\mathbb R\) such that \(f(0)\geq 0.\) It is established semigroup property for minimal solutions. It is shown that every local-in-time solution can be extended globally, and it is revealed a relationship between minimal solutions of the time-dependent problem and minimal solutions of a corresponding stationary problem.