A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities

TL;DR

This paper investigates a nonlinear fourth-order parabolic equation relevant to nonequilibrium interfaces and quantum semiconductors, establishing existence, uniqueness criteria, and exponential convergence to equilibrium via new logarithmic Sobolev inequalities.

Contribution

It introduces a new optimal logarithmic Sobolev inequality for higher derivatives and applies it to analyze the equation's long-term behavior.

Findings

Existence of global non-negative weak solutions

A criterion for solution uniqueness

Exponential convergence to mean value in entropy norm

Abstract

A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.