Uniqueness of the Solution of the Cauchy Problem for Parabolic Systems

Abstract

We consider the Cauchy problem for a second-order Petrovskii parabolic system with bounded continuous coefficients under the condition that the leading coefficients are Dini continuous in the spatial variables. We prove the uniqueness of the classical solution of this problem in the space of functions increasing with respect to the spatial variables, belonging to the Tikhonov class, and having derivatives that may be unbounded when approaching the initial data plane.