Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data

Abstract

This paper investigates the 2D quasilinear wave equation $\displaystyle\sum_{α,β=0}^{2}g^{αβ}(\partialϕ)\partial_{αβ}^{2}ϕ=0$ with the short pulse initial data $(ϕ, \partial_tϕ)|_{t=1}=(δ^{2-\varepsilon_0}ϕ_0(\frac{r-1}δ,ω), δ^{1-\varepsilon_0}ϕ_1(\frac{r-1}δ,ω))$, where $0<\varepsilon_0<1$ is a fixed constant and $δ>0$ is a small parameter. In our previous work [Ding Bingbing, Xin Zhouping, Yin Huicheng, Sci. China Math., Vol. 69 (2026), No. 3, 667-736], we have established the global existence of a smooth solution $ϕ$ under the $k$-th order null condition ($k\ge 2$) when the $(k+1)$-th order null condition fails, provided that $0<\varepsilon_0<\varepsilon_k^*=\frac{k}{k+1}$. As a natural continuation, the present paper demonstrates that for $\varepsilon_0\ge \varepsilon_k^*$, the smooth solution $ϕ$ blows up in finite time. Consequently, the threshold $\varepsilon_0=\varepsilon_k^*$ is shown to be the critical exponent that precisely characterizes the global existence and finite-time blowup for smooth short pulse solutions to the aforementioned 2D quasilinear wave equation, under the assumption that the $k$-th order null condition holds while the $(k+1)$-th order null condition does not. A key novelty of our approach lies in deriving sharp orders of $δ$ for almost all quantities in both energy and pointwise estimates. This is achieved by utilizing the higher-order null structure in conjunction with a new good unknown recently introduced by us and the transport equation for the difference between this good unknown and its corrector.