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Blow-up of a critical
Sobolev norm for energy-subcritical and energy-supercritical
wave equations
Thomas Duyckaerts and Jianwei Yang |
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Vol. 11 (2018), No. 4, 983–1028
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Abstract |
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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. |
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Keywords
supercritical wave equation, Strichartz estimates,
scattering, blow-up, profile decomposition
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Mathematical Subject Classification 2010
Primary: 35L71
Secondary: 35B40, 35B44
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Milestones
Received: 6 April 2017
Revised: 2 August 2017
Accepted: 20 September 2017
Published: 12 January 2018
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Authors |
| Thomas Duyckaerts | |
| LAGA (UMR CNRS 7539) Université Paris 13 Sorbonne Paris Cité Villetaneuse France |
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| Jianwei Yang | |
| LAGA (UMR CNRS 7539) Université Paris 13 Sorbonne Paris Cité Villetaneuse France |
Beijing International Center for
Mathematical Research Peking University Beijing China |
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