Alberto Lastra, Stéphane Malek. Source: Advances in Differential Equations, Volume 24, Number 1/2, 69--136.Abstract:
We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $\epsilon$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $\epsilon$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|\epsilon|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $\epsilon$ tends to 0 on appropriate sets of sectors that cover a neighborhood of the origin in $ \mathbb{C}^{\ast}$. We observe that their Gevrey orders are distinct in general.