Ukrainian Mathematical Bulletin

Volume 4, Issue 1, 2007 pp. 21-55.

A Cauchy Problem for Singular Pseudo-differential Parabolic Type Systems

Authors: Vladislav Antonovych Litovchenko
Author institution: Chernivtsi National University, vul. Kotsyubyns'kogo 2, 58012, Chernivtsi Ukraine, vladlit@chnu.cv.ua

Summary: Using convex functions, we define a class of pseudo-differen\-ti\-al singular systems with integral analytic symbols. This class contains $\ora{2B}$-parabolic differential systems, i.e., parabolic systems, with a Bessel operator, in which each spatial variable has, generally speaking, its own weight with respect to the temporal variable. We study properties of the fundamental matrix for such systems and prove a theorem on correct solvability of such systems in the case where the initial conditions are generalized functions of Gevrey ultradistribution type. For systems in a special subclass, we describe maximal classes of initial conditions for which the Cauchy problem is correctly solvable, and the solution possesses the needed properties.

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