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

@article{Shen2018EnergyDO, title={Energy distribution of radial solutions to energy subcritical wave equation with an application on scattering theory}, author={Ruipeng Shen}, journal={arXiv: Analysis of PDEs}, year={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 radial and come with a finite energy. We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: "scattering energy" which concentrates around…

## 8 Citations

### 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…

### Scattering of solutions to NLW by Inward Energy Decay

- Mathematics
- 2019

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…

### 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…

### 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…

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

- MathematicsAdvances in Mathematics
- 2020

### Energy distribution of solutions to defocusing semi-linear wave equation in two dimensional space

- MathematicsMathematische Annalen
- 2022

We consider finite-energy solutions to the defocusing nonlinear wave equation in two dimensional space. We prove that almost all energy moves to the infinity at almost the light speed as time tends…

### Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space

- Mathematics
- 2021

The topic of this paper is a semi-linear, defocusing wave equation utt − ∆u = −|u| u in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite…

### Scattering theory for subcritical wave equation with inverse square potential.

- Mathematics
- 2020

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method of [37], [38] and [41], we establish…

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