Pr´ e-Publica¸ c˜ oes do Departamento de Matem´ atica Universidade de Coimbra Preprint Number 18–13 GEVREY WELL POSEDNESS OF THE GENERALIZED GOURSAT-DARBOUX PROBLEM FOR A LINEAR PDE JORGE MARQUES AND JAIME CARVALHO E SILVA Abstract: We consider the generalized Goursat-Darboux problem for a third order linear PDE with real coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes Γ s with s> 1. It is proved that there exists some critical index s 0 such that if the Goursat-Darboux problem is well posed in Γ s for s>s 0 then some conditions should be imposed on the coefficients of the derivatives with respect to one of the variables. In order to prove our results, we first construct an explicit solution of a family of problems with data depending on a parameter η> 0 and then we obtain an asymptotic representation of a solution as η tends to infinity. Keywords: Goursat-Darboux problems, Gevrey classes, asymptotic solutions. Math. Subject Classification (2000): 35G10 (35A07 35L30). 1. Introduction The simplest generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients in the classes of Gevrey functions was studied in [8]. Given an open set Ω ⊆ R 3+m , neighborhood of origin, the problem is defined on Ω by            ∂ t ∂ x ∂ y u(t, x, y, z )=  0≤|j |≤3 A j ∂ j z u(t, x, y, z ) u(0,x,y,z )= f 1 (x, y, z ) u(t, 0,y,z )= f 2 (t, y, z ) u(t, x, 0,z )= f 3 (t, x, z ) (1.1) where initial data satisfy necessary compatibility conditions        f 1 (0,y,z )= f 2 (0,y,z ) f 1 (x, 0,z )= f 3 (0, x, z ) f 2 (t, 0,z )= f 3 (t, 0,z ) f 1 (0, 0,z )= f 2 (0, 0,z )= f 3 (0, 0,z ) . (1.2) Received April 19, 2018. This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. 1