An optimal Liouville theorem for the linear heat equation with a nonlinear boundary condition

TL;DR

This paper establishes an optimal Liouville theorem for the linear heat equation in a halfspace with a nonlinear boundary condition, providing key insights into the nonexistence of positive bounded solutions over all time.

Contribution

It introduces a new optimal Liouville theorem for the linear heat equation with a nonlinear boundary condition in the halfspace, extending previous results to this specific setting.

Findings

Proves nonexistence of positive bounded solutions for the problem.

Establishes optimal estimates for solutions of related initial-boundary value problems.

Extends Liouville theorems to boundary conditions involving nonlinearities.

Abstract

Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times $t\in(-\infty,\infty)$) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition $\partial u/\partial\nu=u^q$, $q>1$.