An optimal Liouville theorem for the porous medium equation

TL;DR

This paper establishes a sharp Liouville theorem for the porous medium equation under specific growth conditions, demonstrating optimality and extending classical results to inhomogeneous and homogeneous cases.

Contribution

It introduces a new optimal Liouville theorem for the porous medium equation, considering inhomogeneous and homogeneous cases with minimal growth assumptions.

Findings

Proves a Liouville theorem under sharp growth conditions for inhomogeneous equations.

Shows the theorem's optimality by demonstrating growth condition cannot be weakened.

Extends classical Liouville results to broader classes of porous medium equations.

Abstract

Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened.