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}
}
  • Ruipeng Shen
  • Published 27 August 2018
  • Mathematics
  • arXiv: Analysis of PDEs
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… 
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References

SHOWING 1-10 OF 14 REFERENCES

Morawetz Estimates Method for Scattering of Radial Energy Sub-critical Wave Equation

In this short paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in (3,5)$. We prove that

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

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 the radial 3D cubic wave equation

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

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

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)

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

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

Scattering for radial, bounded solutions of focusing supercritical wave equations

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

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

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

Conformal invariance and time decay for nonlinear wave equations

We study the implications of the approximate conformal invariance of the non linear wave equation □φ+f(φ)=0(*) on the time decay of its solutions. We first prove the conformal conservation law in as

On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations

Abstract We prove existence and scattering results for semilinear wave equations with low regularity data. We also determine the minimal regularity that is needed to ensure local existence and

Conformal conservation law, time decay and scattering for nonlinear wave equations

We study the implications of the conformal conservation law for the time decay of solutions of nonlinear wave equations and present some improvements over previous work of Ginibre and Velo. We also