On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

We study a family of singularly perturbed linear partial differential equations with irregular type inthe complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Boreltransform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near theorigin in and can be extended on a finite number of unbounded sectors with small opening and bisectingdirections, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can beapplied. In this paper, we introduce new conditions for the Borel transform to be analytically continuedin the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Boreltransform described by Fruchard and Schäfke (2011) and is based on a more accurate descriptionof the Stokes phenomenon for the sectorial solutions mentioned above