Vol. 6, No. 1, 2013

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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

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|4ug(|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|4ulogα(2 + |u|2) for 0 < α 4 3

in 1+3. We prove the same results of global existence and (x1 x2) × 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).

scattering, log-subcritical, radial Sobolev inequality
Mathematical Subject Classification 2010
Primary: 35L15
Received: 27 May 2011
Revised: 22 November 2011
Accepted: 20 March 2012
Published: 1 June 2013
Hsi-Wei Shih
School of Mathematics
University of Minnesota
127 Vincent Hall, 206 Church St. SE
Minneapolis, MN 55455
United States