非线性扰动Klein-Gordon方程初值问题的渐近理论

Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations

在二维空间中研究一类非线性扰动Klein_Gordon方程初值问题解的渐近理论.首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein_Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用。

The asymptotic theory of initial value problems for a class of nonlinear perturbed Klein_Gordon equations in two space dimensions is considered. Firstly, using the contraction mapping principle, combining some priori estimates and the convergence of Bessel function, the well_posedness of solutions of the initial value problem in twice continuous differentiable space was obtained according to the equivalent integral equation of initial value problem for the Klein_Gordon equations. Next, formal approximations of initial value problem was constructed by perturbation method and the asymptotic validity of the formal approximation is got. Finally, an application of the asymptotic theory was given, the asymptotic approximation degree of solutions for the initial value problem of a specific nonlinear Klein_Gordon equation was analyzed by using the asymptotic approximation theorem.