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

Research output: Contribution to journalArticle

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Abstract

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.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalAnalysis and PDE
Volume6
Issue number1
DOIs
Publication statusPublished - 2013 Aug 27

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Nonlinear Wave Equation
Wave equations
Wave equation
Scattering
Energy
Sobolev spaces
Scattering Theory
Second Order Equations
Logarithm
Semilinear
High Power
Global Existence
Sobolev Spaces
Logarithmic
Asymptotic Behavior
Regularity
Nonlinearity

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

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

In: Analysis and PDE, Vol. 6, No. 1, 27.08.2013, p. 1-24.

Research output: Contribution to journalArticle

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