Boundary layer expansions for initial value problems with two complex time variables

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter $\epsilon$. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to $\epsilon$, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in[J. Mozo-Fern\'andez, R. Sch\"afke, Asymptotic expansions and summability with respect to an analytic germ, Publ. Math. 63 (2019), no. 1, 3--79.], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.