Topological Methods in Nonlinear Analysis

Heteroclinics for non autonomous third order differential equations

Denis Bonheure, José Ángel Cid, Colette De Coster, and Luís Sanchez

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the existence of heteroclinics connecting the two equilibria $\pm 1$ of the third order differential equation $$u'''=f(u)+p(t)u'$$ where $f$ is a continuous function such that $f(u)(u^2-1)> 0$ if $u\neq\pm 1$ and $p$ is a bounded non negative function. Uniqueness is also addressed.

Article information

Topol. Methods Nonlinear Anal. Volume 43, Number 1 (2014), 53-68.

First available in Project Euclid: 11 April 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)


Bonheure, Denis; Cid, José Ángel; De Coster, Colette; Sanchez, Luís. Heteroclinics for non autonomous third order differential equations. Topol. Methods Nonlinear Anal. 43 (2014), no. 1, 53--68.

Export citation


  • A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20 , Oxford University Press (2000) \ref\key 2
  • D. Bonheure and L. Sanchez, Heteroclinic Orbits for Some Classes of Second and Fourth Order Differential Equations, Handbook of Differential Equations: Ordinary Differential Equations, . 3 (A. Cañada, P. Drabek, A. Fonda, eds.), Elsevier (2006) \ref\key 3
  • C. Conley, Isolated Invariant Sets and the Morse Index, C.B.M.S., 38 , Amer. Math. Soc., Providence (1978) \ref\key 4
  • I.M. Gel'fand, Some problems in the theory of quasi-linear equations , Uspehi Mat. Nauk, 14 (1959), 87–158 \ref\key 5
  • N. Kopell and L.N. Howard, Bifurcations and trajectories joining critical points , Adv. Math., 18 (1975), 306–358 \ref\key 6
  • V. Manukian and S. Schecter, Travelling waves for a thin liquid film with surfactant on an inclined plane , Nonlinearity, 22 (2009), 85–122 \ref\key 7
  • Ch.K. McCord, Uniqueness of connecting orbits in the equation $Y^{(3)}=Y^2-1$ , J. Math. Anal. Appl., 114 (1986), 584–592 \ref\key 8
  • M.S. Mock, On fourth-order dissipation and single conservation laws , Comm. Pure Appl. Math., 29 (1976), 383–388 \ref\key 9 ––––, The half-line boundary value problem for $u_{xxx}=f(u)$ , J. Differential Equations, 32 (1979), 258–273 \ref\key 10
  • L. Nirenberg, On elliptic partial differential equations , Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115–162 \ref\key 11
  • J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer–Verlag, New York (1983) \ref\key 12
  • J.F. Toland, Existence and uniqueness of heteroclinic orbits for the equation $\lambda u'''+u'=f(u)$ , Proc. Royal Soc. Edinburgh, 109A (1988), 23–36 \ref\key 13
  • W. Walter, Ordinary Differential Equations, Springer–Verlag, New York (1998)