On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation ${\text{p}}_t^{2}u-\text{ds}{\sum }_{i=1}^2{\text{p}}_i(c_i^2(u){\text{p}}_{i}u)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$,$c_i(0)\neq 0$,and$(c_1’(0))^2+(c_2’(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$with$u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$,and$\ve>0$issmall,wewillshowthattheclassicalsolution$u(t,x)$stopstobesmoothatsomefinitetime$T_{\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives ${\text{na}}_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\text{ve}}$.