The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations (Q1763990)

The sharp upper bound of the lifespan of the solution to the Cauchy problem for the following \(p-q\) systems of semilinear wave equations; \[ \partial_t^2 u - \Delta u = | v| ^p, \] \[ \partial_t^2 v - \Delta v = | u| ^p, \] in \(\mathbb R^n \times [0,\infty) \;for \;\;p,q>1,\) with smooth compactly supported initial data; \[ u| _{t=0} = \varepsilon f_1, \quad u_t| _{t=0} = \varepsilon g_1, \] \[ v| _{t=0} = \varepsilon f_2, \quad v_t| _{t=0} = \varepsilon g_2, \] where \(\varepsilon>0\) is a small parameter and \(f_i, g_i \in C_0^\infty(R^n)\) are given functions for \(i=1,2\) is studied. It is also proved the lower bound of the lifespan with a small error term. The present paper remove this error term and the give the sharp lifespan under the assumption of the radial symmetricity of the solution.