  We prove by means of advanced pseudo-monotonicity methods an abstract
existence result for parabolic partial differential equations with
$\log$-H\"older continuous variable exponent nonlinearity governed by the
symmetric part of a gradient only. To this end, we introduce the notions
Bochner pseudo-monotonicity and Bochner coercivity, which are appropriate
extensions of the concepts of pseudo-monotonicity and coercivity to unsteady
problems in variable exponent spaces. In this context, we apply the so-called
Hirano-Landes approach, which enables us to give general and easily verifiable
conditions for these new notions. Moreover, we prove essential parabolic
embedding and compactness results involving only the symmetric part of the
gradient.
