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

Abstract

Firstly, we study the equation $\square u=|u{|}^{q_c}+|\partial u{|}^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\ge q_c.$ We obtain the optimal lifespan $\ln (T_{\epsilon })\approx {\epsilon }^{-q_c(q_c-1)}$ in $n=3$, and improve the lower-bound of $T_{\epsilon }$ from $\exp (c{\epsilon }^{-(q_c-1)})$ to $\exp (c{\epsilon }^{-(q_c-1{)}^2/2})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u=|v{|}^q,$ $\square v=|{\partial }_{t}u{|}^p$ 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.

Final version, to appear in Journal of Differential Equations. 22 pages, 1 figure