On the critical exponent $p_c$ of the 3D quasilinear wave equation $-(1+({\partial }_t\ddot{I}•{)}^p){\partial }_t^2\ddot{I}•+\hat{I}”\ddot{I}•=0$ with short pulse initial data. II, shock formation

Abstract

In the previous paper [Ding Bingbing, Lu Yu, Yin Huicheng, On the critical exponent $p_c$ of the 3D quasilinear wave equation $-(1+({\partial }_t\ddot{I}•{)}^p){\partial }_t^2\ddot{I}•+\hat{I}”\ddot{I}•=0$ with short pulse initial data. I, global existence, Preprint, 2022], for the 3D quasilinear wave equation $-(1+({\partial }_t\ddot{I}•{)}^p){\partial }_t^2\ddot{I}•+\hat{I}”\ddot{I}•=0$ with short pulse initial data $(\ddot{I}•,{\partial }_t\ddot{I}•)(1,x)=(\hat{I}{´}^{2-{\epsilon }_0}\ddot{I}{•}_0(\frac{r-1}{\hat{I}}´,\ddot{I}‰),\hat{I}{´}^{1-{\epsilon }_0}\ddot{I}{•}_1(\frac{r-1}{\hat{I}}´,\ddot{I}‰))$, where $p\in \mathbb{N}$, $0<{\epsilon }_0<1$, under the outgoing constraint condition $({\partial }_t+{\partial }_r{)}^k\ddot{I}•(1,x)=O(\hat{I}{´}^{2-{\epsilon }_0-k\max \{0,1-(1-{\epsilon }_0)p\}})$ for $k=1,2$, the authors establish the global existence of smooth large solution $\ddot{I}•$ when $p>p_c$ with $p_c=\frac{1}{1-{\epsilon }_0}$. In the present paper, under the same outgoing constraint condition, when $1\le p\le p_c$, we will show that the smooth solution $\ddot{I}•$ blows up and further the outgoing shock is formed in finite time.

arXiv admin note: text overlap with arXiv:2211.16153