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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