On the critical exponent $p_c$ of the 3D quasilinear wave equation $-(1+({\partial }_t\varphi {)}^p){\partial }_t^2\varphi +\Delta \varphi =0$ with short pulse initial data. I, global existence

Abstract

For the 3D quasilinear wave equation $-(1+({\partial }_t\varphi {)}^p){\partial }_t^2\varphi +\Delta \varphi =0$ with the short pulse initial data $(\varphi ,{\partial }_t\varphi )(1,x)=({\delta }^{2-{\epsilon }_0}{\varphi }_0(\frac{r-1}{\delta },\omega ),{\delta }^{1-{\epsilon }_0}{\varphi }_1(\frac{r-1}{\delta },\omega ))$, where $p\in \mathbb{N}$, $p\ge 2$, $0<{\epsilon }_0<1$, $r=|x|$, $\omega =\frac{x}{r}\in {\mathbb{S}}^2$, and $\delta >0$ is sufficiently small, under the outgoing constraint condition $({\partial }_t+{\partial }_r{)}^k\varphi (1,x)=O({\delta }^{2-{\epsilon }_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $\varphi$ when $p>p_c$ with $p_c=\frac{1}{1-{\epsilon }_0}$ being the critical exponent. In the forthcoming paper, when $1\le p\le p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $({\varphi }_0,{\varphi }_1)$.