## On the focusing critical semi-linear wave equation.(English)Zbl 1219.35144

The paper deals with small perturbations of stationary solutions to the equation $$\psi _{tt}-\Delta \psi -\psi ^5=0$$ in $$\mathbb R^+\times \mathbb R^3$$. The wave equation is written in the form of a Hamiltonian equation, and the spectrum of the linearized Hamiltonian is analyzed. Then the existence of a family of radial perturbations for the stationary solution $$\varphi =(3a)^{1/4}(1+|x|^2)^{-1/2}$$ (as a curve in the energy space $$H^1\times L_2$$) is proved. This leads to global solutions as the sum of a bulk term plus a scattering term, possessing a well-defined long time asymptotic behavior. The above family forms a co-dimension one manifold $${\mathcal M}$$ with the curve $$\varphi (\cdot,a)$$ as an attractor in $${\mathcal M}$$.

### MSC:

 35L71 Second-order semilinear hyperbolic equations 35P25 Scattering theory for PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 35B20 Perturbations in context of PDEs
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