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

Shih, Hsi-Wei

doi:10.2140/apde.2013.6.1

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

Hsi-Wei Shih

Vol. 6 (2013), No. 1, 1–24

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 $Ḣ_{x}^{1} \times L_{x}^{2}$ scattering theory for the energy log-subcritical wave equation

□ u = | u |^{4} u g (| 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 {log}^{α} (2 + | u |^{2}) for 0 < α \leq \frac{4}{3}

in $ℝ^{1 + 3}$ . We prove the same results of global existence and $(Ḣ_{x}^{1} \cap Ḣ_{x}^{2}) \times H_{x}^{1}$ 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).

Keywords

scattering, log-subcritical, radial Sobolev inequality

Mathematical Subject Classification 2010

Primary: 35L15

Milestones

Received: 27 May 2011

Revised: 22 November 2011

Accepted: 20 March 2012

Published: 1 June 2013

Authors

Hsi-Wei Shih
School of Mathematics University of Minnesota 127 Vincent Hall, 206 Church St. SE Minneapolis, MN 55455 United States
http://math.umn.edu/~shihx029/

Vol. 6, No. 1, 2013