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arXiv · 1703.05168

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

Thomas Duyckaerts·Jianwei Yang
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Abstract

This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These spaces are local, thus adapted to finite speed of propagation, and related to a conservation law of the linear wave equation. We also construct the adapted profile decomposition.

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Thomas Duyckaerts, Jianwei Yang. 2017-03-15. Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations. https://doi.org/10.2140/apde.2018.11.983

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