Abstract:
We consider the Cauchy problem with respect to $z_2$ for a homogeneous linear partial differential equation with constant coefficients in two independent variables
$z_1,z_2 \in \mathbb C$. We show that the relative smoothness with respect to $z_1$
and $z_2$ of analytic and ultradifferentiable solutions of the Cauchy problem depends essentially on the value of $\rho_2$ and, as a rule, is completely determined by it. We also obtain rather general uniqueness theorems and find conditions which guarantee that the particular solution constructed depends both continuously and linearly on the initial functions.
Citation:
Yu. F. Korobeinik, “Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients”, Izv. Math., 61:3 (1997), 553–592