Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients
Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 553-592.

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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.
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Yu. F. Korobeinik. Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients. Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 553-592. http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a3/

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