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

`For the nonlinear wave equation in R-N x R+ (N greater than or equal to 2) : partial derivative(2)u(x,t)/partial derivative(t)(2) - a partial derivati ve/partial derivative(xi)(a/(x) partial derivative/partial derivative(xi)u) = \u\(p-1 u,) in 1980 Kato proved the solution of Cauchy problem may blow rtp infinite time if 1 < p less than or equal to N + 1/N - 1. In the presen t work his result allowing 1 < p less than or equal to N + 3/N - 1 is impro ved by using different estimates.