Global existence and blow up for systems of nonlinear wave equations related to the weak null condition

We discuss how the higher-order term \(|u|^q\) \((q>1+2/(n-1))\) has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations $$ \partial_t^2 u-\Delta u=|v|^p, \qquad \partial_t^2 v-\Delta v=|\partial_t u|^{(n+1)/(n-1)} +|u|^q $$ in \(n\,(\geq 2)\) space dimensions. We show the existence of a certain "critical curve" on the \(pq\)-plane such that for any \((p,q)\) \((p,q>1)\) lying below the curve, nonexistence of global solutions occurs, whereas for any \((p,q)\) \((p>1+3/(n-1),\,q>1+2/(n-1))\) lying exactly on it, this system admits a unique global solution for small data. When \(n=3\), the discussion for the above system with \((p,q)=(3,3)\), which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of \(n=2\) and \(p=4\) it is observed that no matter how large \(q\) is, the higher-order term \(|u|^q\) never becomes negligible and it essentially affects the lifespan of small solutions.