A Triviality Result for Semilinear Parabolic Equations

TL;DR

This paper proves that under certain conditions, bounded eternal solutions of the semilinear heat equation on specific Riemannian manifolds must be trivial, extending understanding of solution behavior in geometric analysis.

Contribution

It establishes a triviality result for bounded eternal solutions of the semilinear heat equation on manifolds with nonnegative Ricci curvature, for subcritical exponents.

Findings

Bounded eternal solutions are trivial under specified conditions.

Results apply to manifolds with nonnegative Ricci tensor and dimension ≥ 5.

The triviality holds for exponents below the critical Sobolev exponent.

Abstract

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation \begin{equation*} u_{t}=\Delta u + |u|^{p} \end{equation*} on complete Riemannian manifolds of dimension $n \geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\frac{n+2}{n-2}$.