On a wave equation corresponding to geodesics (Q1917025)

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 \(t \to \infty\).