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

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