A triviality result for semilinear parabolic equations

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} $.

Keywords

• ancient solution
• eternal solution
• superlinear heat equation