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

This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu)$ $=0$, where $c_i(u)\in C^{\infty }(\mathbb {R}^n)$, $c_i(0)\neq 0$, and $(c_1’(0))^2+(c_2’(0))^2\neq 0$. This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty }(\mathbb {R}^2)$, and $\varepsilon >0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\varepsilon }$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\nabla _{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\varepsilon }$.