On the Goursat problem in the Gevrey class of some second order equations. (Q1411756)

The author considers the following operator in \(\mathbb{R}^3_{t,x,y}\): \[ Pu= \partial_t\partial_x- a(t) b(x)\partial^2_y+ Q_1(t, x,y,\partial_t, \partial_x, \partial_y), \] where \(a(t)\), \(b(x)\) are real-valued \(C^\infty\) functions, \(a(t)\geq 0\), \(b(x)\geq 0\), and \(Q_1\) is a first-order operator with \(C^\infty\) coefficients. The following problem of Goursat type is considered: \[ Pu= f\quad\text{in }\mathbb{R}^+_t\times \mathbb{R}^+_x\times \mathbb{R}_y,\quad u(0,x,y)= 0,\quad u(t,0,y)= 0. \] In the case \(Q_1=0\), well-posedness of the problem in \(C^\infty\) was studied by the author in a preceding paper [ibid. 50, 639--662 (1998; Zbl 0913.35093)]. Here the author treats the case of an arbitrary first-order perturbation and gives results of well-posedness for Gevrey functions, of order \(s\) with respect to the variable \(y\). In particular, the Goursat problem above is proved to be well-posed for \(1\leq s< 3/2\).