The radial defocusing energy-supercritical cubic nonlinear wave equation in \(\mathbb{R}^{1+5}\) (Q2877445)

The author considers the energy-supercritical defocusing cubic nonlinear wave equation \(u_{tt}-\Delta u+|u|^2 u=0\) in spatial dimension \(d = 5\) for radially symmetric initial data. It is proved that an a priori bound \(L^\infty (I; \dot H^{3/2}\times \dot H^{1/2})\) (the scaling critical space, with maximal interval of existence \(I\)) implies global wellposedness and scattering, with global bound in \(L^6(\mathbb{R}\times \mathbb{R}^5)\). The main tool that the author used is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schrödinger equation.