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In this monograph we study nonlinear elliptic and parabolic problems in the framework of the Sobolev spaces with variable exponents. They are: the problems considered in perforated domains with Dirichlet and Neumann boundary conditions and the problems with rapidly oscillating coefficients. Using the method of Khruslov’s mesoscopic characteristics, the classical Gamma-convergence and two-scale convergence methods we derive the corresponding homogenized models.