## 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.

### MSC:

 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

### Keywords:

blow-up profile; perturbative analysis
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### References:

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