一类拟线性双曲型方程的初边值问题和Cauchy问题

Initial Boundary Value Problem and Cauchy Problem for a Class of Quasilinear Hyperbolic Equation

首先讨论方程utt - uxx - M(∫+ l- lu2x dx)uttxx = f(x,t) 的初边值问题,用Galerkin 方法和紧性方法得到了其整体广义解和整体古典解的存在惟一性;然后用构造初边值问题序列并取极限的方法证明了方程utt - uxx - M(∫+ ∞- ∞u2x dx)uttx x = f(x,t) 的Cauchy 问题整体广义解和整体古典解的存在惟一性.

The initial boundary value problem of the equationu tt -u xx -M(∫ +l -l u 2 x d x)u ttxx =f(x,t)is studied. By using Galerkin method and compactness argument, the existence and uniqueness of the generalized global solution and the classical global solution of the problem are obtained. Then, by building the sequence of the initial boundary problems and taking limit, the existence and uniqueness of the generalized global solution and the classical global solution for the Cauchy problem of the equationu tt -u xx -M(∫ +∞ -∞ u 2 x d x)u ttxx =f(x,t)are proved.