On stable self-similar blowup for equivariant wave maps

Abstract

We consider corotational wave maps from (3 + 1) Minkowski space into the 3-sphere. This is an energy supercritical model that is known to exhibit finite-time blowup via self-similar solutions. The ground state self-similar solution f0 is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0. © 2011 Wiley Periodicals, Inc.