Long time behaviour of finite-energy radial solutions to energy subcritical wave equation in higher dimensions

Abstract

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional equation of $w$ and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as $x\rightarrow +\infty$. This generalize the 3-dimensional results in Shen[27] to higher dimensions.