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

Abstract

In this paper we investigate the large-time behavior of weak solutions to the nonlinear fourth-order parabolic equation $n_t = -(n(\log n)_{xx})_{xx}$ modeling interface fluctuations in spin systems. We study here the case $x\in \Omega =(0,1)$ , with n = 1, n x = 0 on $\partial \Omega$ . In particular, we prove the exponential decay of u(x,t) towards the constant steady state $n_\infty =1$ in the L 1 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.