On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

MIXED GEVREY ASYMPTOTICS

In this paper, we illustrate with two examples that the computation of Stokes multipliers in problems in which the asymptotic expansions are not of Gevrey order one, is much more complicated and that the singularities in the corresponding Borel plane can be essential singularities.