Long-time behavior for a nonlinear fourth-order parabolic equation

Abstract

We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.