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

Authors:	Cao Zhenchao Wang Bixiang

Institution:	1. Department of Mathematics, Xiamen University, Xiamen 361005, P. R. China; 2. Department of Applied Mathematics, Tsinghua University, Beijing 100084, P. R. China

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 Foundation item: the National Natural Science Foundation of China (19771069)

Keywords:	condition for blow up wave equation Cauchy problem
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