Hodge-type Decomposition for Time-dependent First-order Parabolic Operators with Non-constant Coefficients: The Variable Exponent Case

In this paper we present a Hodge-type decomposition for variable exponent spaces. More concretely, we address some time-dependent parabolic firstorder partial differential operators with non-constant coefficients, where one of the components is the kernel of the parabolic-type Dirac operator. This decomposition is presented over different types of domains in the n-dimensional Euclidean space n-dimensional Euclidean space ({mathbb{R}^{n}}) . The case of the time-dependent Schrödinger operator is included as a special case within this context.