### Abstract

We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, we consider the Ḣ^{1}_{x} × L^{2}_{x} scattering theory for the energy log-subcritical wave equation u= |u|^{4}ug(|u|) in R^{1+3}, where g has logarithmic growth at 0. We discuss the solution with general (respectively spherically symmetric) initial data in the logarithmically weighted (respectively lower regularity) Sobolev space. We also include some observation about scattering in the energy subcritical case. The second problem studied involves the energy log-supercritical wave equation u = |u|^{4}u logα(2+|u|^{2}) for 0 < α ≤ 43 in R^{1+3}. We prove the same results of global existence and (Ḣ^{1}_{x}) Ḣ^{1}_{x})× H^{1}_{x} scattering for this equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed in previous work by Tao.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Analysis and PDE |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Aug 27 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

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**Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations.** / Shih, Hsi-Wei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations

AU - Shih, Hsi-Wei

PY - 2013/8/27

Y1 - 2013/8/27

N2 - We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, we consider the Ḣ1x × L2x scattering theory for the energy log-subcritical wave equation u= |u|4ug(|u|) in R1+3, where g has logarithmic growth at 0. We discuss the solution with general (respectively spherically symmetric) initial data in the logarithmically weighted (respectively lower regularity) Sobolev space. We also include some observation about scattering in the energy subcritical case. The second problem studied involves the energy log-supercritical wave equation u = |u|4u logα(2+|u|2) for 0 < α ≤ 43 in R1+3. We prove the same results of global existence and (Ḣ1x) Ḣ1x)× H1x scattering for this equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed in previous work by Tao.

AB - We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, we consider the Ḣ1x × L2x scattering theory for the energy log-subcritical wave equation u= |u|4ug(|u|) in R1+3, where g has logarithmic growth at 0. We discuss the solution with general (respectively spherically symmetric) initial data in the logarithmically weighted (respectively lower regularity) Sobolev space. We also include some observation about scattering in the energy subcritical case. The second problem studied involves the energy log-supercritical wave equation u = |u|4u logα(2+|u|2) for 0 < α ≤ 43 in R1+3. We prove the same results of global existence and (Ḣ1x) Ḣ1x)× H1x scattering for this equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed in previous work by Tao.

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U2 - 10.2140/apde.2013.6.1

DO - 10.2140/apde.2013.6.1

M3 - Article

VL - 6

SP - 1

EP - 24

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 1

ER -