A Triviality Result for Semilinear Parabolic Equations

Giovanni Catino; Daniele Castorina; Carlo Mantegazza

A Triviality Result for Semilinear Parabolic Equations

Analysis of PDEs 2020-07-16 v1 Differential Geometry

Authors: Giovanni Catino , Daniele Castorina , Carlo Mantegazza

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

semilinear heat equation eternal solutions triviality result ricci tensor sobolev exponent riemannian manifolds

Cite

@article{arxiv.2007.07655,
  title  = {A Triviality Result for Semilinear Parabolic Equations},
  author = {Giovanni Catino and Daniele Castorina and Carlo Mantegazza},
  journal= {arXiv preprint arXiv:2007.07655},
  year   = {2020}
}

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