一维双曲型方程的一个反问题

A One-dimensional Inverse Problem of Hyperbolic Type Equation

本文讨论了确定双曲型方程:U_u(x,t)-U_(xx)(x,t)+P(x,t)U_x(x,t)+r(x,t)U_t(x,t)+q(x)U(x,t)=F(x,t)中系数q(x)的反问题,证明了此方程柯西问题古典解的存在唯一性,得到了与反问题等价的积分方程组,并由压缩映象原理证明了此反问题局部解的存在唯一性;推广了文[3]中的结论;最后给出了反问题整体解的唯一性定理.

This paper discusses how to determine the coefficient q(x) in the inverse problem of the hyperbolic type equation U_(tt)(x,t)-U_(xx)(x,t)+p(x,t)Ux(x,t)+r(x ,t)·U_t(x,t)+q(x)U(x,t)=F(x,t). It has been proved that the classical solution for the Cauchy problem of the equation is existent and unique. The nonliner integral equations that are equivalent to the inverse problems have been obtained and the existence-uniqueness of its local solutions is also proved. The result in [3] has been generalized. Finally ,the author gives a theorem on the uniqueness of the global solution for the inverse problem.