Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients

TL;DR

This paper proves that certain semi-linear PDEs with random coefficients, modeling hypersurface motion through random obstacles, do not admit global stationary solutions, indicating interfaces cannot remain stationary in such random environments.

Contribution

It establishes the non-existence of global stationary solutions for a class of semi-linear PDEs with random coefficients, advancing understanding of interface dynamics in random media.

Findings

No global nonnegative stationary solutions exist.

Stationary solutions grow unbounded on large domains.

Random lower order terms cannot be uniformly bounded.

Abstract

We consider a so-called random obstacle model for the motion of a hypersurface through a field of random obstacles, driven by a constant driving field. The resulting semi-linear parabolic PDE with random coefficients does not admit a global nonnegative stationary solution, which implies that an interface that was flat originally cannot get stationary. The absence of global stationary solutions is shown by proving lower bounds on the growth of stationary solutions on large domains with Dirichlet boundary conditions. Difficulties arise because the random lower order part of the equation cannot be bounded uniformly.