About a condition for blow up of solutions of cauchy problem for a wave equation

Abstract

For the nonlinear wave equation in$R^N \times R^ + (N \geqslant 2):\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} - \frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{\partial }{{\partial x_j }}u} \right) = \left| {u^{p - 1} } \right| \cdot u$, in 1980 Kato proved the solution of Cauchy problem may blow up in finite time$1< p \leqslant \frac{{N + 1}}{{N - 1}}$. In the present work his result allowing$1< p \leqslant \frac{{N + 3}}{{N - 1}}$ is improved by using different estimates