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

Abstract

We are concerned with a class of two-dimensional nonlinear wave equations $\p_t^2u-\div(c^2(u)\na u)=0$ or $\p_t^2u-c(u)\div(c(u)\na u)=0$ with small initial data $(u(0,x),\p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $c(u)$ is a smooth function, $c(0)\not =0$, $x\in\Bbb R^2$, $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$ depend only on $r=\sqrt{x_1^2+x_2^2}$, 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 $c'(0)\not= 0$ or $c'(0)=0$, $c"(0)\not= 0$, we establish blowup and determine the lifespan of smooth solutions.