Exponential time decay of solutions to a nonlinear fourth-order parabolic equation

In this paper we investigate the large-time behavior of weak solutions to the nonlinear fourth-order parabolic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n_t = -(n(\log n)_{xx})_{xx}$\end{document} modeling interface fluctuations in spin systems. We study here the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in \Omega =(0,1)$\end{document}, with n = 1, nx = 0 on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial \Omega$\end{document}. In particular, we prove the exponential decay of u(x,t) towards the constant steady state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n_\infty =1$\end{document} in the L1 norm for long times and we give the explicit rate of decay. The result is based on classical entropy estimates, and on detailed lower bounds for the entropy production.