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

This paper investigates the 2D quasilinear wave equation ${\displaystyle \sum _{\alpha ,\beta =0}^2{g}^{\alpha \beta }(\mathrm{\partial }\phi ){\mathrm{\partial }}_{\alpha \beta }^2\phi =0}$ with the short pulse initial data $(\phi ,{\mathrm{\partial }}_t\phi ){\mathrm{\mid }}_{t=1}=({\delta }^{2-{\epsilon }_0}{\phi }_0(\frac{r-1}{\delta },\omega ),{\delta }^{1-{\epsilon }_0}{\phi }_1(\frac{r-1}{\delta },\omega ))$, where $0<{\epsilon }_0<1$ is a fixed constant and $\delta >0$ 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 $\phi$ under the $k$-th order null condition ($k\ge 2$) when the $(k+1)$-th order null condition fails, provided that $0<{\epsilon }_0<{\epsilon }_k^{\ast }=\frac{k}{k+1}$. As a natural continuation, the present paper demonstrates that for ${\epsilon }_0\ge {\epsilon }_k^{\ast }$, the smooth solution $\phi$ blows up in finite time. Consequently, the threshold ${\epsilon }_0={\epsilon }_k^{\ast }$ 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 $\delta$ 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.