Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold

This paper determines when the Cauchy problem ∂tu=Δu-Vu+WupinM×(0,∞)u(·,0)=u0(·)inM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} {{\partial _t u} = \Delta u -Vu+ Wu^p} &{}\quad \text{ in } M \times (0, \infty ) \\ {u(\cdot ,0)= {u_0(\cdot )}} &{}\quad \text{ in } M \end{array} \right. \end{aligned}$$\end{document}has no global positive solution on a connected non-compact geodesically complete Riemannian manifold for a given triple (V, W, p). As the principal result of this paper, Theorem 1.1 optimally extends in a unified way most of the previous results in this subject (cf. Ishige in J Math Anal Appl 344:231–237, 2008; Pinsky in J Differ Equ 246(6):2561–2576, 2009; Zhang in Duke Math J 97:515–539, 1999; Zhang in J Differ Equ 170:188–214, 2001).