Variable exponent Bochner–Lebesgue spaces with symmetric gradient structure

We introduce function spaces for the treatment of non-linear parabolic partial differential equations with variable log–Hölder continuous exponents that only incorporate information symmetric part a gradient. As an analogue Korn's inequality these functions is not available, construction appropriate smoothing method proves to be difficult. Using point-wise Poincaré near boundary bounded Lipschitz domain involving gradient, we construct operator convenient properties. In particular, this leads several density results and, therefore, generalized formula integration by parts respect time. and theory maximal monotone operators, prove abstract existence result.