# The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

@article{Killip2010TheRD,
title={The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions},
author={Rowan Killip and Monica Visan},
journal={arXiv: Analysis of PDEs},
year={2010}
}
• Published 9 February 2010
• Mathematics
• arXiv: Analysis of PDEs
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2} \frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

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## References

SHOWING 1-10 OF 37 REFERENCES

### The defocusing energy-supercritical nonlinear wave equation in three space dimensions

• Mathematics
• 2010
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter)

### High frequency approximation of solutions to critical nonlinear wave equations

• Mathematics
• 1999
This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □u + |u|4 = 0 in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /],

### Energy-Critical NLS with Quadratic Potentials

• Mathematics
• 2006
We consider the defocusing -critical nonlinear Schrödinger equation in all dimensions (n ≥ 3) with a quadratic potential . We show global well-posedness for radial initial data obeying ∇u 0(x), xu

### Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

• Mathematics
• 2009
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak