  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.
