Gevrey well posedness of Goursat-Darboux problems and asymptotic solutions (Q1728587)

The well posedness of generalized Goursat-Darboux problem for a third order operator is considered in Zbl 0862.35020. It turns out that the conditions for the problem to be $C^\infty$ well-posed are very strong. In this paper, the authors investigate the local solvability of this problem in Gevrey classes. In particular, they formulate the appropriate notion of well-posedness, and proceed by showing that the problem at hand can be reduced to appropriate Cauchy problem. Then the authors provide asymptotic representations of the solution of the Cauchy problem by using asymptotic analysis of the initial data in order to have only one exponential function as dominant term. Under the assumption of Gevrey type well-posedness, the corresponding asymptotic estimates are derived. For the entire collection see [Zbl 1398.35005].