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Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. (English) Zbl 1183.35202

The Cauchy problem to the equation \(u_{tt}-\Delta u=u|u|^{4/(N-2)}\) with initial data in the energy space in \(\mathbb R^{N+1},\) where \(N=3,4,5,\) is studied. Let us suppose that the energy integral of the initial data is smaller than the energy integral of the stationary solution \(W\) to the equation. Then the following alternative is proved. If the \(H^1\)-norm of the initial data is smaller than the \(H^1\)-norm of \(W\) then global well-posedness and scattering hold. If the \(H^1\)-norm of the initial data is larger than the \(H^1\)-norm of \(W\) then the solution \(u\) breaks down in a finite time.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B44 Blow-up in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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