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 ∂ t 2 u − div ( c 2 ( u ) ∇ u ) = 0 \partial _t^2u-\text {div}(c^2(u)\nabla u)=0 or ∂ t 2 u − c ( u ) d i v ( c ( u ) ∇ u ) = 0 \partial _t^2u-c(u)\mathrm {div}(c(u)\nabla u)=0 with small initial data ( u ( 0 , x ) , ∂ t u ( 0 , x ) ) = ( ε u 0 ( x ) , ε u 1 ( x ) ) (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x)) , where c ( u ) c(u) is a smooth function, c ( 0 ) ≠ 0 c(0)\not =0 , x ∈ R 2 x\in \mathbb R^2 , u 0 ( x ) , u 1 ( x ) ∈ C 0 ∞ ( R 2 ) u_0(x), u_1(x)\in C_0^{\infty }(\mathbb R^2) depend only on r = x 1 2 + x 2 2 r=\sqrt {x_1^2+x_2^2} , and ε > 0 \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 ) ≠ 0 c’(0)\not = 0 or c ′ ( 0 ) = 0 c’(0)=0 , c ( 0 ) ≠ 0 c(0)\not = 0 , we establish blowup and determine the lifespan of smooth solutions.