This paper determines when the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} {{\partial _t u} = \Delta u -Vu+ Wu^p} &{}\quad \text{ in } M \times (0, \infty ) \\ {u(\cdot ,0)= {u_0(\cdot )}} &{}\quad \text{ in } M \end{array} \right. \end{aligned}$$

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).

中文翻译：

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黎曼流形上带电势的半线性抛物方程的整体正解

本文确定何时出现柯西问题

$$ \ begin {aligned} \ left \ {\ begin {array} {ll} {{\ partial _t u} = \ Delta u -Vu + Wu ^ p}＆{} \ quad \ text {in} M \ times 0，\ infty）\\ {u（\ cdot，0）= {u_0（\ cdot}}}＆{} \ quad \ text {in} M \ end {array} \ right。\ end {aligned} $$

对于给定的三元（V，W，  p）， 在连通的非紧致测地完全黎曼流形上没有全局正解。作为本文的主要结果，定理1.1最佳地以统一的方式扩展了该学科先前的大多数结果（参见Ishige在J Math Anal Appl 344：231–237，2008; Pinsky在J Differ Equ 246（6）中：2561–2576，2009; Zhang在Duke Math J 97：515–539，1999; Zhang在J Differ Equ 170：188–214，2001）中。