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

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 $\partial u/\partial \nu =u^q$, $q>1$.