Global Solutions of Semilinear Parabolic Equations with Drift Term on Riemannian Manifolds

TL;DR

This paper investigates conditions for the existence or non-existence of global solutions to semilinear heat equations with drift on Riemannian manifolds, highlighting the influence of curvature and initial data size.

Contribution

It provides new criteria linking curvature conditions and initial data size to the global solvability of semilinear heat equations with drift on Cartan-Hadamard manifolds.

Findings

Global solutions do not exist for large initial data under certain curvature conditions.

Global existence is achieved for small initial data with appropriate drift conditions.

Ricci curvature decay at infinity affects solution existence.

Abstract

We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.