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

TL;DR

This paper proves the global existence of smooth solutions for a 3D quasilinear wave equation with short pulse initial data when the exponent p exceeds a critical value p_c, and discusses shock formation for p below or equal to p_c.

Contribution

It establishes the critical exponent p_c for global existence of solutions to a 3D quasilinear wave equation with short pulse data, a novel threshold analysis.

Findings

Global existence for p > p_c with p_c=1/(1-ε_0)

Shock formation occurs for p ≤ p_c before t=2

Critical exponent p_c depends on initial data parameter ε_0

Abstract

For the 3D quasilinear wave equation $-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with the 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 $p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $\omega= \frac{x}{r}\in\mathbb S^2$, and $\delta>0$ is sufficiently small, under the outgoing constraint condition $(\partial_t+\partial_r)^k\phi(1,x)=O(\delta^{2-\varepsilon_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $\phi$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent. In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $(\phi_0,\phi_1)$.