Asymptotic behaviors of radially symmetric solutions of \(\square u=| u| ^ p\) for super critical values \(p\) in odd space dimensions (Q1900731)

Semilinear wave equations \(u_{tt}- \Delta u= F(u)\) in \(x\in \mathbb{R}^n\), \(t\in \mathbb{R}\), where typically \(F(u)= |u|^p\) or \(F(u)= |u|^{p- 1}u\) with \(p> 1\), \(n\geq 2\) are considered. If \(p\) is larger than the positive root of \({n- 1\over 2} p^2- {n+ 1\over 2} p- 1= 0\) and \(p\leq {n+ 1\over n- 3}\) if \(n\geq 7\), global radially symmetric small solutions are shown to exist as well as the scattering operator, provided \(n\geq 5\) and \(n\) is odd.