Spectral Properties and Linear Stability of Self-Similar Wave Maps

Abstract

We study co--rotational wave maps from $(3+1)$--Minkowski space to the three--sphere $S^3$. It is known that there exists a countable family $\{f_n\}$ of self--similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well--posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from $f_n$ by letting $n\to \infty$.

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References in corpus (5)

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Cited by in corpus (3)

• On stable self-similar blow up for equivariant wave maps: The linearized problem
• On stable self-similar blow up for equivariant wave maps
• A Proof for the Mode Stability of a Self-similar Wave Map