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.