We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and an asymptotic representation for the solution are to be determined. The existence theorem is proved by the method of matching asymptotic expansions. To this end, we use theorems on existence of solutions of boundary value problems for singularly perturbed equations and methods for constructing asymptotic approximations to these solutions.

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右侧不连续的奇异微扰方程的内层

我们考虑一个普通奇异摄动二阶微分方程的边值问题，它的右手边是一个非线性函数，沿着一些与小参数无关的曲线不连续。对于这个问题，我们研究在位于这条曲线上的某个点的邻域中是否存在具有陡峭梯度的平滑解。点本身和解的渐近表示将被确定。存在定理是通过匹配渐近展开式的方法证明的。为此，我们使用关于奇异摄动方程的边值问题解的存在性的定理和构造这些解的渐近近似的方法。