Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure

TL;DR

This paper introduces a straightforward approach to establish Liouville theorems for scaling invariant superlinear parabolic equations with gradient structure, leading to nonexistence results and optimal estimates for solutions.

Contribution

The authors develop a simple method to derive Liouville theorems for a class of superlinear parabolic problems with gradient structure, including boundary conditions, extending previous nonexistence results.

Findings

Proved Liouville theorems for scalar nonlinear heat equations.

Extended nonexistence results to vector-valued and boundary value problems.

Provided optimal universal estimates for solutions in various domains.

Abstract

We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-\Delta u=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\partial\nu=u^q$. Here $\nu$ denotes the outer unit normal on the boundary of the halfspace ${\mathbb R}^n_+$ and the exponents $p,q>1$ satisfy $p<n/(n-2)$ and $q<(n-1)/(n-2)$ if $n>2$ (or $p<(n+2)/(n-2)$ and $q<n/(n-2)$ if $n>2$ and some symmetry of the solutions is assumed). As a typical application of our nonexistence results we provide optimal universal estimates for positive solutions of related problems in bounded and unbounded domains.