Propagation and extinction of fronts for semilinear parabolic equations on Riemannian manifolds
type: Item
instance of: scholarly article
title: Propagation and extinction of fronts for semilinear parabolic equations on Riemannian manifolds
zbMATH Open document ID: 1332.58007
author: Fabio Punzo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2015-12-17T00:00:00Z
review text: In this article, the author considers the initial value problem for a semilinear parabolic equation of the form \(\frac{\partial u}{\partial t}= \Delta u + f(u)\) in \(M\times (0,+\infty), u = u_0 \) in \(M\times \{0\} \) in the following framework: \(M\) is an \(n\)-dimensional complete, non compact Riemanian manifold, \(f\in C^1([0,1])\), \(f(0) = f(1) =0,\) \(u_0\) is continuous on \(M\) and satifies \(0 \leq u_0(x) \leq 1.\) In this context, a unique global in time solution exists. The question discussed in this paper is then to give conditions on \(M,f\) in order that the solution satisfies either \(\lim_{t\to + \infty} u(t,x) = 0\) (extinction) or \(\lim_{t\to + \infty} u(t,x) = 1\) (propagation). The main results of the paper give several very technical sufficient conditions on the manifold (through the sectional curvature, the Ricci curvature\dots) for the above properties to hold.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35R01
zbMATH DE Number: 6521863
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: propagation and extinction of fronts
zbMATH Keywords: Riemannian manifolds
reviewed by: Claude Zuily
MaRDI profile type: Publication
full work available at URL: j.na.2015.09.010
OpenAlex ID: W1896212057
Wikidata QID: Q115342931
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DOI: 10.1016/J.NA.2015.09.010
description: scientific article; zbMATH DE number 6521863