Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds

TL;DR

This paper investigates the conditions under which solutions to reaction-diffusion equations on certain Riemannian manifolds either blow up or exist globally, highlighting differences from Euclidean and hyperbolic spaces based on spectral properties.

Contribution

It introduces a new version of the Fujita phenomenon on manifolds with positive spectral bottom, extending previous results beyond hyperbolic space, and characterizes global existence for small initial data.

Findings

Fujita phenomenon differs on manifolds with positive spectral bottom.

Solutions with small initial data are globally existing on a broad class of manifolds.

The critical exponent is replaced by the spectral bottom bb; in hyperbolic space, the phenomenon does not occur.

Abstract

It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $\Lambda$ of the $L^2$ spectrum of $-\Delta$ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $\Lambda$. Such nonlinearities are time-independent, in contrast to the ones studied in [2]. As a consequence of our results we show that, on a class of manifolds much larger than the case $M=\mathbb{H}^N$ considered in [2], solutions to (1.1) with power nonlinearity $f(u)=u^p$, $p>1$, and corresponding to sufficiently small data, are global in time.