The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations

The sharp upper bound of the lifespan of the solution to p-q systems in high space dimensions has been obtained in The lifespan of solutions to nonlinear systems of high dimensional wave equation, preprint for the sub-critical case. They also proved the lower bound of the lifespan with a small error term. The present paper remove this error term and give the sharp lifespan under the assumption of the radial symmetricity of the solution. (C) 2004 Elsevier Ltd. All rights reserved.