Gevrey Well-Posedness of the Generalized Goursat-Darboux Problem for a Linear PDE

摘要

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 so such that if the Goursat-Darboux problem is wellposed in Γ~s for s > so, 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.