Liouville property for solutions of the linearized degenerate thin film equation of fourth order in a halfspace

TL;DR

This paper proves a Liouville property for solutions of a linearized degenerate fourth-order thin film equation in a half-space, showing solutions with polynomial data have polynomial growth and establishing uniqueness results.

Contribution

It establishes a Liouville theorem for solutions of a degenerate linearized thin film equation, extending understanding of solution behavior and uniqueness in half-space boundary value problems.

Findings

Solutions with polynomial data have polynomial growth.

The Liouville property does not hold in the normal variable.

A uniqueness theorem for solutions of power growth is proved.

Abstract

We consider a boundary value problem in the half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.