Pseudo Monotone Operator Theory for Unsteady Problems with Variable Exponents This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(. .) Navier–Stokes equations describing the motion of an incompressible electro rheological fluid. Due to the variable dependence of the power law index p(. .) in this system the classical weak existence analysis based on the pseudo monotone operator theory in the framework of Bochner–Lebesgue spaces is not applicable. As a substitute for Bochner–Lebesgue spaces variable Bochner–Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute the theory of pseudo monotone operators is extended to unsteady problems with variable exponents leading to the weak solvability of the unsteady p(. .) Navier–Stokes equations under general assumptions. Aimed primarily at graduate readers the book develops the material step by step starting with the basics of PDE theory and non linear functional analysis. The concise introductions at the beginning of each chapter together with illustrative examples graphics detailed derivations of all results and a short summary of the functional analytic prerequisites will ease newcomers into the subject.