Long Time Behavior of Stochastic Thin Film Equation

TL;DR

This paper studies the long-term behavior of solutions to a stochastic thin-film equation, showing convergence of the solution's supremum norm to a random factor times the initial mean.

Contribution

It establishes existence of nonnegative weak solutions and analyzes their asymptotic behavior under stochastic perturbations.

Findings

Solutions' supremum norm converges to initial mean times a stochastic factor

Existence of nonnegative weak martingale solutions is proven

Asymptotic behavior characterized by convergence in square mean

Abstract

We consider the stochastic thin-film equation with linear deterministic and stochastic It\^o 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.