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

Krieger, J.; Schlag, W.

Am. J. Math. 129, No. 3, 843-913 (2007).

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 \({\mathcal M}\).

Reviewer: Marie Kopáčková (Praha)

MSC:

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

Keywords:

semilinear wave equation; critical exponent; perturbation; static solution; spectrum of the linearized Hamiltonian; bulk term; scattering term