临界情形的拟线性二阶系统的边值问题

A Boundary Value Problem of the Second Order Quasilinear System in the Critical Case

本文研究了临界情形的拟线性二阶系统的边值问题ε(d^2x)/(dt^2)=A(t)(dx)/(dt)+B(x,t)x(o,ε)=a(ε),ε[a(dx)/(dt)(0,ε)+b(dx)/(dt)(1,ε]=β(e),利用改进的 Vasiléva 方法构造了具有任意精度的两端均具边界层且左端边界层有两个具有不同尺度 t/ε^(1/2),t/ε的边界层函数的形式渐近解,并证明了精确解的存在唯一性及所构造的渐近解的一致有效性,并给出了余项估计。

In this paper,we consider the boundary value problem εd^2x/d^t2=A(t)dx/dt+B(x,t),x(0,ε)=a(ε),ε[adx(0,ε)/dt+bdx(1,ε)/dt]=β(ε),for a m—dimensional systems of quasilineardifferential equations.The asymptotic solution is constructed by a modified Vasil'eva method wherethere are 3 boundary layer corrections,i.e.,The left boundary layer is of 2 different stretchedvariables t/ε and t/ε.The existence and uniqueness of the exact solution,the uniform validity of theformal asymptotic solution for the boundary value problem are proved.