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



Jian-Guo Liu (Department of Physics and Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Xiangsheng Xu (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Miss., U.S.A.)


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