Abstract
We consider the Cauchy problem with respect to for a homogeneous linear partial differential equation with constant coefficients in two independent variables . We show that the relative smoothness with respect to and of analytic and ultradifferentiable solutions of the Cauchy problem depends essentially on the value of 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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