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

Kenig, Carlos E.; Merle, Frank

Acta Math. 201, No. 2, 147-212 (2008).

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.

Reviewer: Marie Kopáčková (Praha)

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

Keywords:

semi-linear equation; Cauchy problem; well-posedness; global solution; blow-up; energy space; energy integral; stationary solution

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