Smoothing effects and infinite time blowup for reaction-diffusion equations: An approach via Sobolev and Poincare inequalities

We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with power-type nonlinearity and slow diffusion of porous medium type. We consider the particularly delicate case p < m in problem (1.1), a case presently largely open even when the initial datum is smooth and compactly supported. We prove global existence for L-m data, and that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L-infinity norm. We also show 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 are functional analytic in character, as they depend solely on the validity of the Sobolev and of the Poincare inequalities. 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 we also consider, with stronger results for large times, the case of globally integrable weights. (C) 2021 Elsevier Masson SAS. All rights reserved.