Homoclinic solutions to periodic motions in a class of reversible equations
type: Item
instance of: scholarly article
title: Homoclinic solutions to periodic motions in a class of reversible equations
zbMATH Open document ID: 0967.34041
author: M. Calanchi
author: Enrico Serra
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-08-20T00:00:00Z
review text: The authors deal with the equation \[ \ddot u(t) + W_u(t, u(t)) = h(t), \] where \(W \in C^2 (\mathbb{R}\times \mathbb{R}; \mathbb{R})\) is 1-periodic in \(t\) and \(S\)-periodic in \(u\), \(h \in C (\mathbb{R}; \mathbb{R})\) is 1-periodic and has zero mean, and \(V(t,x) = W(t,x) - h(t)x\) is even in \(t\). They study the existence of homoclinic and heteroclinic solutions to consecutive global minimizers \(u_1\), \(u_2\) of the action functional associated to the 1-periodic solutions. The authors prove: 1. The existence of heteroclinics connecting \(u_1\) and \(u_2\). 2. The existence of infinitely many homoclinic solutions to \(u_1\) and \(u_2\) provided an additional assumption is met. This additional assumption constrains the abundance of heteroclinic solutions, excluding in particular the case of autonomous systems. 3. The nonexistence of homoclinic solutions if there are no consecutive minimizers. 4. Under nondegeneracy of the minimizers, nonexistence of homoclinic solutions when the additional condition in 2 does not hold. As mentioned by the authors, this paper is strongly influenced by the work of P. Rabinowitz, in particular by investigations in [Ergodic Theory Dyn. Syst. 14, No. 4, 817-829 (1994; Zbl 0818.34025) and Differ. Integral Equ. 7, No. 5-6, 1557-1572 (1994; Zbl 0835.34050)].
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1398356
zbMATH Keywords: homoclinic and heteroclinic solutions
zbMATH Keywords: variational methods
MaRDI profile type: Publication
full work available at URL: s005260050136
OpenAlex ID: W2027399004
DOI: 10.1007/S005260050136
description: scientific article; zbMATH DE number 1398356