The Cauchy Problem for Inhomogeneous Parabolic Systems in Anisotropic Zygmund Spaces

Abstract

This article deals with the Cauchy problem for a second-order parabolic system with constant coefficients and a non-zero right hand side which satisfy the condition of uniform parabolicity in the sense of Petrovsky. The initial condition can also be non-zero. Anisotropic Zygmund spaces which are analogous to parabolic Hölder spaces in the case of an integer smoothness index are used to construct a smoothness scale for solutions to such systems. The properties of the volume potential for a parabolic system were studied using their representation through the Poisson potential. Estimates of the operator given by the Poisson potential established estimates for the volume potential in weighted parabolic Zygmund spaces. The results are used to construct a smoothness scale for a bounded solution to the Cauchy problem for a second-order parabolic system in weighted anisotropic Zygmund spaces.