  We consider a family of linear singularly perturbed PDE relying on a complex
perturbation parameter $\epsilon$. As in a former study of the authors (A.
Lastra, S. Malek, Parametric Gevrey asymptotics for some nonlinear initial
value Cauchy problems, J. Differential Equations 259 (2015), no. 10,
5220--5270), our problem possesses an irregular singularity in time located at
the origin but, in the present work, it entangles also differential operators
of Fuchsian type acting on the time variable. As a new feature, a set of
sectorial holomorphic solutions are built up through iterated Laplace
transforms and Fourier inverse integrals following a classical multisummability
procedure introduced by W. Balser. This construction has a direct issue on the
Gevrey bounds of their asymptotic expansions w.r.t $\epsilon$ which are shown
to bank on the order of the leading term which combines both irregular and
Fuchsian types operators.
