Hostname: page-component-6b989bf9dc-mbg9n Total loading time: 0 Render date: 2024-04-14T05:56:17.428Z Has data issue: false hasContentIssue false

Comportement à l'infini pour certains systèmes dissipatifs non linéaires

Published online by Cambridge University Press:  14 November 2011

A. Haraux
Affiliation:
Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, Paris

Synopsis

In the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:

A is assumed to be a maximal monotone operator and F(t) is a periodic function.

When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:

with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1979

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.)

References

1Amério, L. et Prouse, G.. Uniqueness and almost-periodicity theorems for a non-linear wave equation. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 46 (1969), 18.Google Scholar
2Amério, L. et Prouse, G.. Almost periodic functions and functional equations Ch. 7 (New York: Van Nostrand, 1971).CrossRefGoogle Scholar
3Baillon, J. B. et Haraux, A.. Comportement à l'infini pour les équations paraboliques avec forcing périodique. Arch. Rational Mech. Anal. 67 (1977), 101109.CrossRefGoogle Scholar
4Benilan, Ph. et Brezis, H.. Solutions faibles d'équations d'évolution dans les espaces de Hilbert. Ann. Inst. Fourier (Grenoble) 22 (1972), 311329.CrossRefGoogle Scholar
5Biroli, M.. Sur les solutions bornées et presque périodiques des équations et inéquations d'évolution. Ann. Math. Pura Appl. 93 (1972), 179.CrossRefGoogle Scholar
6Brezis, H.. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (Amsterdam: North Holland, 1973).Google Scholar
7Brezis, H.. Problémes unilatéraux. J. Math. Pures Appl. 51 (1972), 1168.Google Scholar
8Brezis, H. et Lions, J. L.. Sur certains problèmes unilatéraux hyperboliques. C.R. Acad. Sci. Paris Sér. A–B 264 (1967), 928931.Google Scholar
9Browder, F. et Petryshyn, W.. The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Amer. Math. Soc. 72 (1966), 571575.CrossRefGoogle Scholar
10Cabannes, H.. Mouvement d'une corde vibrante soumise à un frottement solide. C.R. Acad. Sci. Paris Sér. A–B 287 (1978), 671673.Google Scholar
11Dafermos, CM.. Uniform processes and semicontinuous Liapunov functionals. J. Differential Equations 11 (1972), 401415.CrossRefGoogle Scholar
12Dafermos, C. M. et Slemrod, M.. Asymptotic behavior of non linear contraction semi-groups. J. Functional Analysis 12 (1973), 97106.CrossRefGoogle Scholar
13Dafermos, C. M.. Contraction semi-groups and trend to equilibrium in continuum mechanics. Lecture Notes in Mathematics 503, 295–306 (Berlin: Springer, 1968).Google Scholar
14Fink, A. M.. Almost periodic differential equations. Lecture Notes in Mathematics 377 (Berlin: Springer, 1974).Google Scholar
15Haraux, A.. Equations d'évolution non linéaires: solutions bornées et périodiques. Ann. Inst. Fourier (Grenoble) 28 (1978), 201220.CrossRefGoogle Scholar
16Haraux, A.. Opérateurs maximaux monotones et oscillations forcées non linéaires (Thése Univ. Paris VI, 1978).Google Scholar
17Haraux, A.. Comportement à l'infini pour une équation d'ondes non linéaire dissipative. C.R. Acad. Sci. Paris Sér. A–B 287 (1978), 507509.Google Scholar
18Haraux, A.. Comportement à l'infini pour une équation d'ondes non linéaire avec “forcing” periodique. C.R. Acad. Sci. Paris Sér. A–B 287 (1978), 619621.Google Scholar