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R. Donninger, B. Schörkhuber, P. Aichelburg:
"On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem";
Annales Henri Poincare, 1 (2011).

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 f0 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 f0. This is an indispensable prerequisite for the study of nonlinear
stability of the self-similar blow up which is conducted in the companion
paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In
particular, we prove that f0 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.

http://dx.doi.org/10.1007/s00023-011-0125-0