On wave equations with supercritical nonlinearities (Q1973443)

The paper deals with solutions of the semilinear hyperbolic equation \[ \partial^2_t u - \Delta u + m^2 u + |u|^{\varrho - 1} u = 0,\;t > 0,\;x \in {\mathbb R}^n, \qquad u|_0 = \varphi , \quad \partial_t u|_0 = \psi, \] where \(n \geq 4, \varrho \in {\mathbb N}\) odd (conditions on \(\varphi\) and \(\psi\)), for \(t > 0\) small.