Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds

TL;DR

This paper investigates the conditions under which semilinear parabolic equations have global solutions on negatively curved Riemannian manifolds, emphasizing the role of curvature bounds in initial data.

Contribution

It establishes a connection between curvature bounds and the initial conditions that guarantee global existence of solutions.

Findings

Global existence depends critically on curvature bounds.

Certain initial conditions ensure solutions exist for all time.

Curvature influences the behavior of solutions significantly.

Abstract

We are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds.