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Renormalization and blow up for charge one equivariant critical wave maps. (English) Zbl 1139.35021
Summary: We prove the existence of equivariant finite time blow-up solutions for the wave map problem from \(\mathbb R^{2+1}\rightarrow S^{2}\) of the form \(u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)\) where \(u\) is the polar angle on the sphere, \(Q(r)=2\arctan r\) is the ground state harmonic map, \(\lambda (t)= t^{-1-\nu}\), and \(\mathcal{R}(t,r)\) is a radiative error with local energy going to zero as \(t \rightarrow 0\). The number \(\nu>\frac{1}{2}\) can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.

35B40 Asymptotic behavior of solutions to PDEs
35Q75 PDEs in connection with relativity and gravitational theory
35P25 Scattering theory for PDEs
35L70 Second-order nonlinear hyperbolic equations
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