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

Abstract

<p style='text-indent:20px;'>We discuss how the higher-order term <inline-formula><tex-math id="M1">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in <inline-formula><tex-math id="M3">\begin{document}$ n\,(\geq 2) $\end{document}</tex-math></inline-formula> space dimensions. We show the existence of a certain "critical curve" in the <inline-formula><tex-math id="M4">\begin{document}$ pq $\end{document}</tex-math></inline-formula>-plane such that for any <inline-formula><tex-math id="M5">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M6">\begin{document}$ (p,q>1) $\end{document}</tex-math></inline-formula> lying below the curve, nonexistence of global solutions occurs, whereas for any <inline-formula><tex-math id="M7">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M8">\begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document}</tex-math></inline-formula> lying exactly on it, this system admits a unique global solution for small data. When <inline-formula><tex-math id="M9">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>, the discussion for the above system with <inline-formula><tex-math id="M10">\begin{document}$ (p,q) = (3,3) $\end{document}</tex-math></inline-formula>, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of <inline-formula><tex-math id="M11">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ p = 4 $\end{document}</tex-math></inline-formula> it is observed that no matter how large <inline-formula><tex-math id="M13">\begin{document}$ q $\end{document}</tex-math></inline-formula> is, the higher-order term <inline-formula><tex-math id="M14">\begin{document}$ |u|^q $\end{document}</tex-math></inline-formula> never becomes negligible and it essentially affects the lifespan of small solutions.</p>