Abstract
We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or L1 data. The key difficulty consists of the 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-Hölder 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.
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