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A Class of Functional Inequalities and their Applications to Fourth-Order Nonlinear Parabolic Equations

We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ \int_Ωu^{2γ-α-β}Δu^αΔu^βdx \geq c\int_Ω|Δu^γ|^2dx $, which seem to be of interest on their own right.