Global existence and blow up for systems of nonlinear wave equations related to the weak null condition
Global existence and blow up for systems of nonlinear wave equations related to the weak null condition
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DOI：
10.3934/dcds.2022058
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发表时间：
2021-03
影响因子：
1.1
通讯作者：
K. Hidano;K. Yokoyama
中科院分区：
文献类型：
--
作者：
K. Hidano;K. Yokoyama
We discuss how the higher-order term \begin{document}$ |u|^q $\end{document} \begin{document}$ (q>1+2/(n-1)) $\end{document} has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations \begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document} in \begin{document}$ n\,(\geq 2) $\end{document} space dimensions. We show the existence of a certain "critical curve" in the \begin{document}$ pq $\end{document}-plane such that for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p,q>1) $\end{document} lying below the curve, nonexistence of global solutions occurs, whereas for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document} lying exactly on it, this system admits a unique global solution for small data. When \begin{document}$ n = 3 $\end{document}, the discussion for the above system with \begin{document}$ (p,q) = (3,3) $\end{document}, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of \begin{document}$ n = 2 $\end{document} and \begin{document}$ p = 4 $\end{document} it is observed that no matter how large \begin{document}$ q $\end{document} is, the higher-order term \begin{document}$ |u|^q $\end{document} never becomes negligible and it essentially affects the lifespan of small solutions.
We discuss how the higher-order term \begin{document}$ |u|^q $\end{document} \begin{document}$ (q>1+2/(n-1)) $\end{document} has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations \begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document} in \begin{document}$ n\,(\geq 2) $\end{document} space dimensions. We show the existence of a certain "critical curve" in the \begin{document}$ pq $\end{document}-plane such that for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p,q>1) $\end{document} lying below the curve, nonexistence of global solutions occurs, whereas for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document} lying exactly on it, this system admits a unique global solution for small data. When \begin{document}$ n = 3 $\end{document}, the discussion for the above system with \begin{document}$ (p,q) = (3,3) $\end{document}, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of \begin{document}$ n = 2 $\end{document} and \begin{document}$ p = 4 $\end{document} it is observed that no matter how large \begin{document}$ q $\end{document} is, the higher-order term \begin{document}$ |u|^q $\end{document} never becomes negligible and it essentially affects the lifespan of small solutions.