Parabolic equations with non-standard growth and measure or integrable data

We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-H\"{o}lder continuity of the variable exponent~$p$ with respect to the spatial variable. On the other hand, we do not assume any smoothness of~$p$ with respect to the time variable.