On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ 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 $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with short pulse initial data. I, global existence, Preprint, 2022], for the 3D quasilinear wave equation $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with short pulse initial data $(\phi,\partial_t\phi)(1,x)=\big(\delta^{2-\varepsilon_0}\phi_0(\frac{r-1}{\delta},\omega),\delta^{1-\varepsilon_0}\phi_1(\frac{r-1}{\delta},\omega)\big)$, where $p\in\mathbb{N}$, $0<\varepsilon_0<1$, under the outgoing constraint condition $(\partial_t+\partial_r)^k\phi(1,x)=O(\delta^{2-\varepsilon_0-k\max\{0,1-(1-\varepsilon_0)p\}})$ for $k=1,2$, the authors establish the global existence of smooth large solution $\phi$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$. In the present paper, under the same outgoing constraint condition, when $1\leq p\leq p_c$, we will show that the smooth solution $\phi$ blows up and further the outgoing shock is formed in finite time.