# Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications

@article{Kenig2008NondispersiveRS,
title={Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications},
author={Carlos E. Kenig and Frank Merle},
journal={American Journal of Mathematics},
year={2008},
volume={133},
pages={1029 - 1065}
}
• Published 27 October 2008
• Mathematics
• American Journal of Mathematics
In this paper we establish optimal pointwise decay estimates for non-dispersive (compact) radial solutions to non-linear wave equations in 3 dimensions, in the energy supercritical range. As an application, we show for the full energy supercritical range, in the defocusing case, that if the scale invariant Sobolev norm of a radial solution remains bounded in its maximal interval of existence, then the solution must exist for all times and scatter.
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## References

SHOWING 1-10 OF 40 REFERENCES

### Global and Unique Weak Solutions of Nonlinear Wave Equations

While the uniqueness of the energy class solutions of semi-linear wave equations with supercritical nonlinearity remains an open problem, there has been a considerable progress recently in

### 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" /],

### STABILITY OF ENERGY-CRITICAL NONLINEAR SCHR¨ ODINGER EQUATIONS IN HIGH DIMENSIONS

• Mathematics
• 2005
We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schrodinger equations in dimen- sions n 3, for solutions which have large, but finite,

### Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions

• Mathematics
• 2007
We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrodinger equation iut+ u = |u|4/nu for large, spherically symmetric,

### Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation

• Mathematics
• 2002
In this paper, we are interested in the phenomenon of blow up in finite time (or formation of singularity in finite time) of solutions of the critical generalized KdV equation. Few results are known

### Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class

Using the harmonic map heat flow, we construct an energy class for wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a

### The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

• Mathematics
• 2007
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2)

### Semilinear Schrodinger Equations

Preliminaries The linear Schrodinger equation The Cauchy problem in a general domain The local Cauchy problem Regularity and the smoothing effect Global existence and finite-time blowup Asymptotic