The lifespan of solutions to nonlinear systems of a high-dimensional wave equation

In this work we study the lifespan of solutions to a p-q   system in the higher-dimensional case n⩾4.n⩾4. A suitable local existence curve in the p-q   plane is found. The curve characterizes the local solutions in Sobolev space HsHs with s⩾0.s⩾0. Further, some lower and upper bounds of the lifespan of classical solutions are found too. This work is an extension of work [Geometric Optics and Related Topics, Progress in Nonlinear Differential Equations and their Applications, vol. 32, Birkhäuser, Basel, 1997, pp. 117–140], where a suitable global existence small data curve is studied. In the subcritical case, we give almost precise results for the lower bounds of the lifespan by using a suitable weighted Strichartz estimate for the higher-dimensional wave equation.