Cauchy Problem for Parabolic Systems with Convolution Operators in Periodic Spaces

V. A. Litovchenko

Chernivtsi National University named after Yuriy Fedkovych

Abstract: For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{\vec p,\vec h\}$-parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.

Keywords: Cauchy problem, parabolic system, Gevrey ultradistribution, convolution operator, periodic space, Weyl operator, trigonometric Fourier series, Banach space.

UDC: 517.928

Received: 23.06.2006

DOI: 10.4213/mzm4185

 English version: Mathematical Notes, 2007, 82:6, 766–786

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