On a wave equation corresponding to geodesics
type: Item
instance of: scholarly article
title: On a wave equation corresponding to geodesics
zbMATH Open document ID: 0854.58038
author: Norihito Koiso
published in: Osaka Journal of Mathematics
publication date: 1997-01-22T00:00:00Z
review text: Let \(M\) be a Riemannian manifold. A hyperbolic equation of \(\gamma = \gamma (t,x)\) \[ \nabla^2_t \gamma + \mu \partial_t \gamma = \nabla^2_x \gamma,\tag{H} \] where the coefficient \(\mu\) represents the resistance, is considered. The following result is proved. Theorem. Let \(M\) be a complete Riemannian manifold and \(\mu\) a constant. Let \(\gamma_0 (x)\) be a \(C^\infty\) closed curve on \(M\) and \(\gamma_1(x)\) a \(C^\infty\) vector field along \(\gamma_0\). Then the equation (H) with initial data \(\gamma(0,x) = \gamma_0(x)\) and \(\partial_t \gamma(0,x) = \gamma_1(x)\) has a unique solution \(\gamma(t,x)\) on \(\mathbb{R} \times S^1\). If \(M\) is compact and \(\mu > 0\), then the solution almost converges to geodesics: that is \(\partial_t \gamma \to 0\) and \(\nabla^2_x \gamma \to 0\) when \(t \to \infty\).
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 902858
zbMATH Keywords: wave equation
zbMATH Keywords: convergence
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: geodesics
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 902858