  We study the asymptotic behavior of the solutions related to a family of
singularly perturbed linear partial differential equations in the complex
domain. The analytic solutions obtained by means of a Borel-Laplace summation
procedure are represented by a formal power series in the perturbation
parameter. Indeed, the geometry of the problem gives rise to a decomposition of
the formal and analytic solutions so that a multi-level Gevrey order phenomenon
appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey
levels.
