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

We are concerned with a class of two-dimensional nonlinear wave equations $\partial _t^2u-\text {div}(c^2(u)\nabla u)=0$ or $\partial _t^2u-c(u)\mathrm {div}(c(u)\nabla u)=0$ with small initial data $(u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$, where $c(u)$ is a smooth function, $c(0)\not =0$, $x\in \mathbb R^2$, $u_0(x), u_1(x)\in C_0^{\infty }(\mathbb R^2)$ depend only on $r=\sqrt {x_1^2+x_2^2}$, and $\varepsilon >0$ is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, as well as in a liquid crystal model or other variational wave equations. When $c’(0)\not = 0$ or $c’(0)=0$, $c”(0)\not = 0$, we establish blowup and determine the lifespan of smooth solutions.