Contents Online

# Communications in Mathematical Sciences

## Volume 18 (2020)

### Number 7

### A class of functional inequalities and their applications to fourth-order nonlinear parabolic equations

Pages: 1911 – 1948

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n7.a5

#### Authors

#### 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\[\int_{\Omega} u^{2 \gamma - \alpha - \beta} \Delta u^\alpha \Delta u^\beta dx \geq c \int_{\Omega} {\lvert \Delta u^\gamma \rvert}^2 dx \; \textrm{,}\]which seem to be of interest in their own right.

The research of J.L. was partially supported by KI-Net NSF RNMS grant No. 1107291, and by NSF grant DMS 1514826.

Received 3 September 2019

Accepted 7 May 2020

Published 11 December 2020