Blow-up for semilinear wave equations with slowly decaying data in high dimensions (Q1343206)

Nonlinear wave equations are considered (for \(t, r\geq 0\)) \[ (1)\quad u_{tt}- u_{rr}- [(n- 1)/r] u_r= F(u),\quad \text{and} \quad(2)\quad u_{tt}- u_{rr}- [(n- 1)/r] u_r= F(u_r), \] with the initial conditions \(u(r, 0)= 0\), \(u_r(r, 0)= \varepsilon \psi\). It is assumed that \(F(s)\geq \text{const} |s|^p\), \(p> 1\), and that \(\psi\geq \text{const}(1+ r)^{- \kappa}\), where \(0< \kappa< (p+ 1)/(p- 1)\) in case (1) and \(0< \kappa< 1/(p- 1)\) in case (2). It is shown that the life span of the solution (or the blow up time) is of order \(\varepsilon^{-(p- 1)/[p+ 1 - (p- 1) \kappa]}\). The result depends on careful estimates of the spherical wave representation of the solution.