Cauchy Problem for Parabolic Systems with Convolution Operators in Periodic Spaces

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.