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

TL;DR

This paper introduces a new class of functional inequalities and applies them to analyze fourth-order nonlinear parabolic equations, including models like the thin-film and quantum drift-diffusion equations.

Contribution

It develops novel functional inequalities and demonstrates their application to the study of complex fourth-order nonlinear parabolic equations.

Findings

Established a new class of functional inequalities.

Applied inequalities to analyze specific nonlinear parabolic equations.

Provided insights into the behavior of solutions to these 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 \geq c\int_\Omega|\Delta u^\gamma |^2dx $, which seem to be of interest on their own right.