# Scattering for the radial 3D cubic wave equation

@article{Dodson2015ScatteringFT, title={Scattering for the radial 3D cubic wave equation}, author={Benjamin Dodson and Andrew Lawrie}, journal={Analysis \& PDE}, year={2015}, volume={8}, pages={467-497} }

Consider the Cauchy problem for the radial cubic wave equation in 1+3 dimensions with either the focusing or defocusing sign. This problem is critical in $\dot{H}^{\frac{1}{2}} \times \dot{H}^{-\frac{1}{2}}$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time-interval of existence, then the solution must in fact be global-in-time and scatter to free waves as $t \to \pm \infty$.

## 36 Citations

### Scattering for radial energy-subcritical wave equations

- Mathematics
- 2016

In this paper, we study the focusing and defocusing energy--subcritical, nonlinear wave equation in $\mathbb{R}^{1+d}$ with radial initial data for $d = 4,5$. We prove that if a solution remains…

### Global well-posedness and scattering for the radial, defocusing, cubic nonlinear wave equation

- Mathematics
- 2018

In this paper we prove global well-posedness and scattering for the defocusing, cubic, nonlinear wave equation on $\mathbf{R}^{1 + 3}$ with radial initial data lying in the critical Sobolev space…

### Long time behaviour of finite-energy radial solutions to energy subcritical wave equation in higher dimensions

- Mathematics
- 2019

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the…

### Inward/outward Energy Theory of Wave Equation in Higher Dimensions.

- Mathematics
- 2019

We consider the semi-linear, defocusing wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in $\mathbb{R}^d$ with $1+4/(d-1)\leq p < 1+4/(d-2)$. We generalize the inward/outward energy theory…

### Exterior scattering of non-radial solutions to energy subcritical wave equations.

- Mathematics
- 2020

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq…

### Scattering for radial energy-subcritical wave equations in dimensions 4 and 5

- Mathematics
- 2017

ABSTRACT In this paper, we consider the focusing and defocusing energy-subcritical, nonlinear wave equation in ℝ1+d with radial initial data for d = 4,5. We prove that if a solution remains bounded…

### Scattering of solutions to the defocusing energy subcritical semi-linear wave equation in 3D

- Mathematics
- 2015

ABSTRACT In this paper, we consider a semi-linear, energy sub-critical, defocusing wave equation in the three-dimensional space with p∈[3,5). We prove that if initial data are radial so that , where…

### Inward/outward energy theory of non-radial solutions to 3D semi-linear wave equation

- MathematicsAdvances in Mathematics
- 2020

### Energy distribution of radial solutions to energy subcritical wave equation with an application on scattering theory

- Mathematics
- 2018

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are…

### Scattering for defocusing energy subcritical nonlinear wave equations

- MathematicsAnalysis & PDE
- 2020

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is…

## References

SHOWING 1-10 OF 54 REFERENCES

### Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

- Mathematics
- 2012

For the critical focusing wave equation $${\square u = u^5 \, {\rm on} \, \mathbb{R}^{3+1}}$$□u=u5onR3+1 in the radial case, we establish the role of the “center stable” manifold $${\Sigma}$$Σ…

### Exotic blowup solutions for the u5 focusing wave equation in ℝ3

- Mathematics
- 2012

For the critical focusing wave equation $\Box u = u^5$ on $\mathbb{R}^{3+1}$ in the radial case, we construct a family of blowup solutions which are obtained from the stationary solutions $W(r)$ by…

### 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)…

### On the Energy Subcritical, Non-linear Wave Equation with Radial Data for $p\in (3,5)$

- Mathematics
- 2012

In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the…

### Scattering for radial, bounded solutions of focusing supercritical wave equations

- Mathematics
- 2012

In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the…

### 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)…

### Scattering for the cubic Klein–Gordon equation in two space dimensions

- Mathematics
- 2010

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and…

### Adapted Linear-Nonlinear Decomposition And Global Well-posedness For Solutions To The Defocusing Cubic Wave Equation On $\mathbb{R}^{3}$

- Mathematics
- 2007

We prove global well-posedness for the defocusing cubic wave equation with data in $H^{s} \times H^{s-1}$, $1>s>{13/18}$. The main task is to estimate the variation of an almost conserved quantity on…

### Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation

- Mathematics
- 2010

### On global solutions to a defocusing semi-linear wave equation

- Mathematics
- 2003

We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space Hs where s > 3/4. This result was obtained in [11] following Bourgain's method…