Abstract
We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t → ±∞, to two positive limits l * and l *, and never goes below min(l *, l *). Furthermore, the convergence to max(l *, l *) is supposed to be not too fast (slower than a suitable exponential term).
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References
Amann H.: Ordinary Differential Equations, de Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter, Berlin (1990)
Bonheure D., Sanchez L.: Heteroclinic orbits for some classes of second and fourth order differential equations. In: Cañada, A., Drabek, P., Fonda, A. (eds) Handbook of Differential Equations, vol. III, Elsevier, Amsterdam (2006)
Brézis H.: Analyse Fonctionnelle, Collection mathématiques appliquées pour la mait̂rise. Masson, Paris (1983)
Chen C.-N., Tzeng S.-Y.: Existence and multiplicity results for heteroclinic orbits of second order Hamiltonian systems. J. Differ. Equ. 158, 211–250 (1999)
Gavioli A.: Existence of heteroclinic trajectories for asymptotically autonomous equations. Topol. Methods Nonlinear Anal. 34, 251–266 (2009)
Gavioli, A.: Heteroclinic solutions to asymptotically autonomous equations via continuation methods. Adv. Nonlinear Stud. (to appear)
Gavioli A., Sanchez L.: On a class of bounded trajectories for some non-autonomous systems. Math. Nachr. 281(11), 1557–1565 (2008)
Gavioli, A., Sanchez, L.: On Bounded Trajectories for Some Non-Autonomous Systems. In: Differential Equations, Chaos and Variational Problems, vol. 75, pp. 393–404. Birkhauser, Basel (2007)
Hale, J.: Ordinary Differential Equations. Pure and Applied Mathematics, Wiley Interscience, New York (1969)
Rabinowitz P.H.: Homoclinic and heteroclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 1, 1–36 (1993)
Rabinowitz, P.H., Coti Zelati, V.: Multichain type solutions for Hamiltonian systems. In: Proceedings of the Conference on Nonlinear Differential Equations, vol. 5, pp. 223–235. Southwest Texas State Univ., San Marcos (2000) (electronic)
Spradlin, G.S.: Heteroclinic solutions to an asymptotically autonomous second order equation. Electron. J. Differ. Equ. (to appear)
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This study was supported by M.I.U.R., Italy.
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Gavioli, A. Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis. Nonlinear Differ. Equ. Appl. 18, 79–100 (2011). https://doi.org/10.1007/s00030-010-0085-y
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DOI: https://doi.org/10.1007/s00030-010-0085-y