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

2021/03/18 by Fabio Punzo, Punzo, Fabio
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2103.10352

openalex publication_date 2021/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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