A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine both partial differential operators together with the action of linear fractional transforms. The exotic geometry of the problem in the Borel plane, involving both sectorial regions and strip-like sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey asymptotic expansions. The main results lean on the appearance of domains in the complex plane which remain intimately related to Lambert W function, which turns out to be crucial in the construction of the analytic solutions. On the way, an accurate description of the deformation of the integration paths defining the analytic solutions and the knowledge of Lambert W function are needed in order to provide the asymptotic behavior of the solution near the origin, regarding the perturbation parameter. Such deformation varies depending on the analytic solution considered, which lies in two families with different geometric features.

研究了一类线性奇摄动柯西问题。定义问题的方程将两个偏微分算子与线性分数变换的作用结合在一起。在Borel平面中，该问题的奇异几何形状同时涉及扇形区域和条状集合，从而产生了与解析解和Gevrey渐近展开相关的形式解的渐近结果。主要结果取决于复杂平面中仍与Lambert W函数密切相关的域的出现，这在解析解的构建中至关重要。在此过程中，对定义解析解的积分路径的变形的准确描述和Lambert W的知识为了提供关于原点的扰动参数，解的渐近行为是必需的。这种变形根据所考虑的解析解而变化，解析解位于具有不同几何特征的两个族中。