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 et al. in J Differ Equ 385:183–253, 2024), 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 \(1\le p\le p_c\), we will show that the smooth solution \(\phi \) may blow up and further the outgoing shock is formed in finite time.