Global existence of positive solutions for semilinear parabolic equations in a half-space

Abstract

We prove the global existence of continuous solutions of the semilinear parabolic problem $\Delta u- {\partial \over {\partial t}}u+ V u^p =0$ in $\mathbb{R}^n_+\times (0,\infty)$, where ${\mathbb{R}^n_+}$ is a half-space in ${\mathbb{R}^n},\, n\geq 3$ . The potential $V$ is in some functional class ${\mathcal K}^{\infty}$. Our approach uses the Shauder fixed-point theorem.