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 $\partial_t^2 u-Δu=|v|^p, \qquad \partial_t^2 v-Δv=|\partial_t u|^{(n+1)/(n-1)} +|u|^q$ 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.

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