Initial‐Value Problem for the Equation$(∂∂t + a(t,x,y)∂∂x + b(t,x,y)∂∂y + c(t,x,y) + d(t,x,y)∂2∂x∂y)u=f(t,x,y)$ in the Complex Domain

The initial‐value problem is first considered in a complex polycylinder Q whose center is the origin. All functions appearing are holomorphic in Q. On the one hand, this problem has at most one holomorphic solution in Q, while on the other hand, the strong assumptions of holomorphy do not in general guarantee even the existence of a local holomorphic solution. We then treat the special initial‐value problem of Lambropoulos [J. Math. Phys. 8, 11 (1967)],where a, b, and c are complex constants. We are able to derive an infinite series as the formal solution, which is easy to examine. Moreover, some statements on the existence of a local holomorphic solution are given.