  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
$p$-$q$ 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.
