On stable self‐similar blowup for equivariant wave maps

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.