Monotone operator theory for unsteady problems in variable exponent spaces

We introduce function spaces for the treatment of parabolic equations with variable exponents by means of the theory of monotone operators. We generalize classical results such as density of smooth functions and a formula for integration by parts to prove existence, uniqueness and L-2-continuity of weak solutions to parabolic equations involving elliptic operators A with p(tau, x)-structure, where p is a globally log-Holder continuous variable exponent satisfying 1 < p(-) <= p(+) < infinity.