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

Abstract

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 ${\int }_{\Omega }{u}^{2\gamma -\alpha -\beta }\Delta u^{\alpha }\Delta u^{\beta }dx\ge c{\int }_{\Omega }|\Delta u^{\gamma }{|}^{2}dx$, which seem to be of interest on their own right.

References in corpus (1)

• The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness