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

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

$$\square u=|u{|}^{4}ug\left(\left|u\right|\right)$$

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

$$\square u=|u{|}^{4}u{log}^{\alpha }\left(2+|u{|}^2\right)\text{ for }0<\alpha \le \frac{4}{3}$$

in ${ℝ}^{1+3}$. We prove the same results of global existence and $\left({Ḣ}_x^1\cap {Ḣ}_x^2\right)\times H_x^1$ 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).