Long Time Behavior of Stochastic Thin Film Equation

Abstract

We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t\to \infty$ is investigated. It is shown that in square mean the $L^{\infty }$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.