The main goal of this paper is twofold. The first one is to investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation on a noncompact Riemannian manifold M,

where \(1<p<2\), \(V\in L_ {\text {loc}}^1(\varOmega ) \), \(q>0 \), \(\lambda \in {\mathbb {R}}\), \(\varOmega \) is bounded and has a smooth boundary in M and \(\varDelta _{p,g}\) is the p-Laplacian on M. The second one is to obtain Hardy- and Leray-type inequalities with remainder terms on a Riemannian manifold M that provide us concrete potentials to use in the partial differential equation we are interested in. In particular, we obtain explicit (mostly sharp) constants for these inequalities on the hyperbolic space \({\mathbb {H}}^n\).

本文的主要目标是双重的。第一个是研究非紧黎曼流形M上以下非线性抛物型偏微分方程正解的不存在性，

其中\（1 <p <2 \），\（V \ in L_ {\ text {loc}} ^ 1（\ varOmega）\），\（q> 0 \），\（\ lambda \ in {\ mathbb {R}} \），\（\ varOmega \）是有界的，并且在M中具有平滑边界，而\（\ varDelta _ {p，g} \）是M上的p -Laplacian 。第二个方法是在黎曼流形M上获得带有余项的Hardy型和Leray型不等式，这为我们提供了可用于我们感兴趣的偏微分方程中的具体潜力。双曲空间\（{\ mathbb {H}} ^ n \）上的这些不等式。