Publication

On stable self-similar blow up for equivariant wave maps: The linearized problem

Roland Donninger
2012
Journal paper
Abstract

We consider co-rotational wave maps from (3+1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f0f_0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f0f_0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in a companion paper. In particular, we prove that f0f_0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1/2. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

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