Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincar\'e inequalities

TL;DR

This paper proves global existence and smoothing effects for reaction-diffusion equations on manifolds and weighted Euclidean spaces, revealing infinite-time blowup phenomena on negatively curved manifolds and extending techniques via Sobolev and Poincaré inequalities.

Contribution

It introduces a new approach to analyze reaction-diffusion equations using Sobolev and Poincaré inequalities, establishing smoothing effects and blowup results in non-Euclidean settings.

Findings

Solutions become bounded at positive times with explicit bounds.

On negatively curved manifolds, large data lead to infinite-time blowup.

Methods extend to weighted Euclidean reaction-diffusion equations.

Abstract

We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with pow\-er-type nonlinearity and slow diffusion of porous medium time. We consider the particularly delicate case $p<m$ in problem (1.1), a case largely left open in [21] even when the initial datum is smooth and compactly supported. We prove global existence for L$^m$ data, and a smoothing effect for the evolution, i.e. that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L$^\infty$ norm. As a consequence of this fact and of a result of [21], it follows that on Cartan-Hadamard manifolds with curvature pinched between two strictly negative constants, solutions corresponding to sufficiently large L$^m$ data give rise to solutions that blow up pointwise everywhere in infinite time, a fact that has no Euclidean analogue. The methods of proof of the smoothing effect are functional analytic in character, as they depend solely on the validity of the Sobolev inequality and on the fact that the L$^2$ spectrum of $\Delta$ on $M$ is bounded away from zero (namely on the validity of a Poincar\'{e} inequality on $M$). As such, they are applicable to different situations, among which we single out the case of (mass) weighted reaction-diffusion equation in the Euclidean setting. In this latter setting, a modification of the methods of [37] allows to deal also, with stronger results for large times, with the case of globally integrable weights.