A functional inequality and its applications to a class of nonlinear fourth-order parabolic equations

In this article we study the initial-boundary value problem for a family of nonlinear fourth-order parabolic equations. The classical quantum drift-diffusion model is a member of the family. Two new existence theorems are established. Our approach is based upon a semi-discretization scheme, which generates a sequence of positive approximate solutions, and a functional inequality of the type\[I_{\alpha} (u) \equiv \int_{\Omega} \Delta uu^{\alpha-1} \Delta u^{\alpha} dx \geq c \int_{\Omega} {\lvert \nabla^2 u^{\alpha} \rvert}^2 dx \textrm{ .}\]We show that a priori estimates that hold for the continuous model under the assumption that solutions are classical are mostly valid for the discretized problems. That is sufficient to justify passing to the limit in the approximation.

Keywords

existence, nonlinear fourth-order parabolic equations, Lipschitz boundaries, quantum drift-diffusion model, functional inequalities

2010 Mathematics Subject Classification

35A01, 35D30, 35Q99

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Published 30 June 2016