Vol. 11, No. 4, 2018

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Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations

Thomas Duyckaerts and Jianwei Yang

Vol. 11 (2018), No. 4, 983–1028

We consider a wave equation in three space dimensions, with a power-like nonlinearity which is either focusing or defocusing. The exponent is greater than 3 (conformally supercritical) and not equal to 5 (not energy-critical). We prove that for any radial solution which does not scatter to a linear solution, an adapted scale-invariant Sobolev norm goes to infinity at the maximal time of existence. The proof uses a conserved generalized energy for the radial linear wave equation, new Strichartz estimates adapted to this generalized energy, and a bound from below of the generalized energy of any nonzero solution outside wave cones. It relies heavily on the fact that the equation does not have any nontrivial stationary solution. Our work yields a qualitative improvement on previous results on energy-subcritical and energy-supercritical wave equations, with a unified proof.

supercritical wave equation, Strichartz estimates, scattering, blow-up, profile decomposition
Mathematical Subject Classification 2010
Primary: 35L71
Secondary: 35B40, 35B44
Received: 6 April 2017
Revised: 2 August 2017
Accepted: 20 September 2017
Published: 12 January 2018
Thomas Duyckaerts
Université Paris 13
Sorbonne Paris Cité
Jianwei Yang
Université Paris 13
Sorbonne Paris Cité
Beijing International Center for Mathematical Research
Peking University