ON MOTION OF AN ELASTIC WIRE IN A RIEMANNIAN MANIFOLD AND SINGULAR PERTURBATION

R.E. Caflish and J.H. Maddocks analyzed the dynamics of a plana r slender elastic rod. We consider a thin elastic rod in an N-dimensional riemannian manifold. The former model represents an elastic rod with positive thi ckness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear pl ate equation. We prove the short time existence of solutions. We also discuss the be aviour of the solution when the resistance goes to infinity, and find that the solutio n converges to a solution of a gradient flow equation.