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2013 Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations
Hsi-Wei Shih
Anal. PDE 6(1): 1-24 (2013). DOI: 10.2140/apde.2013.6.1

Abstract

We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, we consider the x1×Lx2 scattering theory for the energy log-subcritical wave equation

u = | u | 4 u g ( | u | )

in 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 log α ( 2 + | u | 2 )  for  0 < α 4 3

in 1+3. We prove the same results of global existence and (x1x2)×Hx1 scattering for this equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed in previous work by Tao (J. Hyperbolic Differ. Equ., 4:2 (2007), 259–265).

Citation

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Hsi-Wei Shih. "Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations." Anal. PDE 6 (1) 1 - 24, 2013. https://doi.org/10.2140/apde.2013.6.1

Information

Received: 27 May 2011; Revised: 22 November 2011; Accepted: 20 March 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1280.35079
MathSciNet: MR3068539
Digital Object Identifier: 10.2140/apde.2013.6.1

Subjects:
Primary: 35L15

Keywords: log-subcritical , radial Sobolev inequality , scattering

Rights: Copyright © 2013 Mathematical Sciences Publishers

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