ON THE BLOWUP AND LIFESPAN OF SMOOTH SOLUTIONS TO A CLASS OF 2-D NONLINEAR WAVE EQUATIONS WITH SMALL INITIAL DATA

摘要

We are concerned with a class of two-dimensional nonlinear wave equations partial derivative(2)(t)u - div(c(2)(u)del u) = 0 or partial derivative(2)(t)u - c(u) div(c(u)del u) = 0 with small initial data (u(0, x), partial derivative(t)u(0, x)) = (epsilon u(0)(x), epsilon u(1)(x)), where c(u) is a smooth function, c(0) not equal 0, x is an element of R-2, u(0)(x), u(1)(x) is an element of C-0(infinity) (R-2) depend only on r = root x(1)(2) + x(2)(2), and epsilon > 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 equal 0 or c' (0) = 0, c '' (0) not equal 0, we establish blowup and determine the lifespan of smooth solutions.