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 ${\text{p}}_t^{2}u-÷(c^2(u)\text{na}u)=0$ or ${\text{p}}_t^{2}u-c(u)÷(c(u)\text{na}u)=0$ with small initial data $(u(0,x),{\text{p}}_{t}u(0,x))=(\text{ve}{u}_0(x),\text{ve}{u}_1(x))$, where $c(u)$ is a smooth function, $c(0)≠0$, $x\in {\mathbb{R}}^2$, $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$dependonlyon$r=\sqrt{x_1^2+x_2^2}$,and$\ve>0$ is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, also in a liquid crystal model or other variational wave equations. When $c^{\mathrm{\prime }}(0)≠0$ or $c^{\mathrm{\prime }}(0)=0$, $c\mathrm{"}(0)≠0$, we establish blowup and determine the lifespan of smooth solutions.