Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Firstly, we study the equation $\mathrm{\square }u=\mathrm{\mid }u{\mathrm{\mid }}^{q_c}+\mathrm{\mid }\mathrm{\partial }u{\mathrm{\mid }}^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\geq q_c.$Weobtaintheoptimallifespan$\ln({T_\varepsilon})\approx\varepsilon^{-q_c(q_c-1)}$in$n=3$, and improve the lower-bound of $T_{\epsilon }$ from $\exp (c{\epsilon }^{-(q_c-1)})$ to $\exp (c{\epsilon }^{-(q_c-1{)}^2\mathrm{/}2})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\mathrm{\square }u=\mathrm{\mid }v{\mathrm{\mid }}^q,$ $\mathrm{\square }v=\mathrm{\mid }{\mathrm{\partial }}_{t}u{\mathrm{\mid }}^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.