Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T22:28:20.244Z Has data issue: false hasContentIssue false
  • English
  • Français

The Nonlinearly Damped Oscillator

Published online by Cambridge University Press:  15 September 2003

Juan Luis Vázquez
Juan Luis Vázquez*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain; juanluis.vazquez@uam.es.
Get access

Abstract

We study the large-time behaviour of thenonlinear oscillator \[\hskip-20mm m\,x'' + f(x') + k\,x=0\,,\] where m, k>0 and f is a monotone real function representingnonlinear friction. We are interested in understanding thelong-time effect of a nonlinear damping term, with specialattention to the model case   $f(x')= A\,|x'|^{\alpha-1}x'$  withα real, A>0. We characterize the existence and behaviourof fast orbits, i.e., orbits that stop in finite time.

Keywords

Nonlinear oscillatornonlinear dampingfast orbits.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003, pp. 231 - 246
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Angenent, S. and Aronson, D.G., The focusing problem for the radially symmetric porous medium equation. Comm. Partial Differential Equations 20 (1995) 1217-1240. CrossRef
D.G. Aronson, The Porous Medium Equation. Springer-Verlag, Berlin/New York, Lecture Notes in Math. 1224 (1985).
Aronson, D.G., Gil, O. and Vázquez, J.L., Limit behaviour of focusing solutions to nonlinear diffusions. Comm. Partial Differential Equations 23 (1998) 307-332. CrossRef
Aronson, D.G. and Graveleau, J., A selfsimilar solution to the focusing problem for the porous medium equation. Euro. J. Appl. Math. 4 (1992) 65-81.
Aronson, D.G. and Vázquez, J.L., The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 203-330. CrossRef
Brezis, H., Peletier, L.A. and Terman, D., A very singular solution of the heat equation with absorption. Arch. Rational Mech. Anal. 95 (1986) 185-209. CrossRef
J. Carr, Applications of centre manifold theory. Springer-Verlag, New York-Berlin, Appl. Math. Sci. 35 (1981) vi+142 pp.
M. Chaves and V. Galaktionov, On the focusing problem for the PME with absorption. A geometrical approach (in preparation).
J.I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Pitman (Advanced Publishing Program), Boston, MA, Res. Notes in Math. 106 (1985).
Díaz, J.I. and Li, A. nán, On the asymptotic behaviour for a damped oscillator under a sublinear friction. Rev. Acad. Cien. Ser. A Mat. 95 (2001) 155-160.
R. Ferreira and J.L. Vázquez, Self-similar solutions to a very fast diffusion equation. Adv. Differential Equations (to appear).
Galaktionov, V.A., Shmarev, S.I. and Vázquez, J.L., Second-order interface equations for nonlinear diffusion with very strong absorption. Commun. Contemp. Math. 1 (1999) 51-64. CrossRef
Galaktionov, V.A., Shmarev, S.I. and Vázquez, J.L., Behaviour of interfaces in a diffusion-absorption equation with critical exponents. Interfaces Free Bound. 2 (2000) 425-448. CrossRef
Galaktionov, V.A., Shmarev, S.I. and Vázquez, J.L., Regularity of interfaces in diffusion processes under the influence of strong absorption. Arch. Ration. Mech. Anal. 149 (1999) 183-212. CrossRef
J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Springer-Verlag, New York, Appl. Math. Sci. 42 (1990).
Haraux, A., Comportement à l'infini pour certains systèmes non linéaires. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 213-234. CrossRef
M.W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra. Academic Press, New York-London, Pure Appl. Math. 60 (1974).
S. Kamin, L.A. Peletier and J.L. Vázquez, A nonlinear diffusion-absorption equation with unbounded initial data, in Nonlinear diffusion equations and their equilibrium states, Vol. 3. Gregynog (1989) 243-263. Birkhäuser Boston, Boston, MA, Progr. Nonlinear Differential Equations Appl. 7 (1992).
E.B. Lee and L. Markus, Foundations of Optimal Control Theory. J. Wiley and Sons, New York, SIAM Ser. Appl. Math. (1967).
Oleinik, O.A., Kalashnikov, A.S. and Chzou, Y.-I., The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv. Akad. Nauk SSR Ser. Mat. 22 (1958) 667-704.
L. Perko, Differential equations and dynamical systems, Third edition. Springer-Verlag, New York, Texts in Appl. Math. 7 (2001).
J.L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation, in Shape Optimization and Free Boundaries, edited by M.C. Delfour. Kluwer Ac. Publ., Dordrecht, Boston and Leiden, Math. Phys. Sci. Ser. C 380 (1992) 347-389.