2020/06/18 by Gabriele Grillo, Grillo, Gabriele, Giulia Meglioli +3 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.10354
openalex publication_date 2020/06/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider reaction-diffusion equations either posed on Riemannian manifolds\nor in the Euclidean weighted setting, with pow -er-type nonlinearity and slow\ndiffusion of porous medium time. We consider the particularly delicate case\np<m in problem (1.1), a case largely left open in [21] even when the initial\ndatum is smooth and compactly supported. We prove global existence for Lm\ndata, and a smoothing effect for the evolution, i.e. that solutions\ncorresponding to such data are bounded at all positive times with a\nquantitative bound on their L^\∞ norm. As a consequence of this fact and\nof a result of [21], it follows that on Cartan-Hadamard manifolds with\ncurvature pinched between two strictly negative constants, solutions\ncorresponding to sufficiently large Lm data give rise to solutions that blow\nup pointwise everywhere in infinite time, a fact that has no Euclidean\nanalogue. The methods of proof of the smoothing effect are functional analytic\nin character, as they depend solely on the validity of the Sobolev inequality\nand on the fact that the L2 spectrum of \Δ on M is bounded away from\nzero (namely on the validity of a Poincar 'e inequality on M). As such,\nthey are applicable to different situations, among which we single out the case\nof (mass) weighted reaction-diffusion equation in the Euclidean setting. In\nthis latter setting, a modification of the methods of [37] allows to deal also,\nwith stronger results for large times, with the case of globally integrable\nweights.\n