## Analysis & PDE

• Anal. PDE
• Volume 11, Number 4 (2018), 983-1028.

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

#### Abstract

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.

#### Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 983-1028.

Dates
Revised: 2 August 2017
Accepted: 20 September 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.apde/1517454161

Digital Object Identifier
doi:10.2140/apde.2018.11.983

Mathematical Reviews number (MathSciNet)
MR3749374

Zentralblatt MATH identifier
06830004

Subjects
Primary: 35L71: Semilinear second-order hyperbolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35B44: Blow-up

#### Citation

Duyckaerts, Thomas; Yang, Jianwei. Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations. Anal. PDE 11 (2018), no. 4, 983--1028. doi:10.2140/apde.2018.11.983. https://projecteuclid.org/euclid.apde/1517454161

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