Asymptotic behaviors of radially symmetric solutions of \(\square u=| u| ^ p\) for super critical values \(p\) in odd space dimensions
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.
global radially symmetric small solutions
scientific article; zbMATH DE number 808299