Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T09:45:42.527Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on weak stabilization of semilinear wave equations

Published online by Cambridge University Press:  15 August 2002

Alain Haraux
Alain Haraux*
Affiliation:
Université Pierre et Marie Curie, Analyse Numérique, Tour 55-65 5 étage, 4 place Jussieu, 75252 Paris Cedex 05, France; [email protected].
Get access

Abstract

If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a nonnegligible sub-region for at least one sign of the velocity, all solutions of theperturbed system converge weakly to 0 as time tends to infinity. We present here asimple and natural method of proof of this kind of property, implying as a consequencesome recent very general results of Judith Vancostenoble.

Keywords

Weak stabilizationsemilinearwave equations.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 553 - 560
Copyright
© EDP Sciences, SMAI, 2001

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

L. Amerio and G. Prouse, Abstract almost periodic functions and functional equations. Van Nostrand, New-York (1971).
Ball, J.M. and Slemrod, M., Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169-179. CrossRef
Biroli, M., Sur les solutions bornées et presque périodiques des équations et inéquations d'évolution. Ann. Math. Pura Appl. 93 (1972) 1-79. CrossRef
Cazenave, T. and Haraux, A., Propriétés oscillatoires des solutions de certaines équations des ondes semi-linéaires. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 449-452.
Cazenave, T. and Haraux, A., Oscillatory phenomena associated to semilinear wave equations in one spatial dimension. Trans. Amer. Math. Soc. 300 (1987) 207-233. CrossRef
Cazenave, T. and Haraux, A., Some oscillatory properties of the wave equation in several space dimensions. J. Funct. Anal. 76 (1988) 87-109. CrossRef
Cazenave, T., Haraux, A. and Weissler, F.B., Une équation des ondes complètement intégrable avec non-linéarité homogène de degré 3. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 237-241.
Cazenave, T., Haraux, A. and Weissler, F.B., A class of nonlinear completely integrable abstract wave equations. J. Dynam. Differential Equations 5 (1993) 129-154. CrossRef
Cazenave, T., Haraux, A. and Weissler, F.B., Detailed asymptotics for a convex hamiltonian system with two degrees of freedom. J. Dynam. Differential Equations 5 (1993) 155-187. CrossRef
Conrad, F. and Pierre, M., Stabilization of second order evolution equations by unbounded nonlinear feedbacks. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 485-515. CrossRef
Haraux, A., Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. I Math. 287 (1978) 507-509.
Haraux, A., Comportement à l'infini pour certains systèmes dissipatifs non linéaires. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 213-234. CrossRef
Haraux, A., Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985) 145-154. CrossRef
Haraux, A. and Komornik, V., Oscillations of anharmonic Fourier series and the wave equation. Rev. Mat. Iberoamericana 1 (1985) 57-77. CrossRef
A. Haraux, Semi-linear hyperbolic problems in bounded domains, Mathematical Reports Vol. 3, Part 1 , edited by J. Dieudonné. Harwood Academic Publishers, Gordon & Breach (1987).
A. Haraux, Systèmes dynamiques dissipatifs et applications, R.M.A. 17, edited by Ph. Ciarlet and J.L. Lions. Masson, Paris (1990).
A. Haraux, Strong oscillatory behavior of solutions to some second order evolution equations, Publication du Laboratoire d'Analyse Numérique 94033, 10 p.
B.M. Levitan and V.V. Zhikov, Almost periodic functions and differential equations. Cambridge University Press, Cambridge (1982).
Slemrod, M., Weak asymptotic decay via a relaxed invariance principle for a wave equation with nonlinear, nonmonotone damping. Proc. Roy. Soc. Edinburgh Ser. A 113 (1989) 87-97. CrossRef
Vancostenoble, J., Weak asymptotic stability of second order evolution equations by nonlinear and nonmonotone feedbacks. SIAM J. Math. Anal. 30 (1998) 140-154. CrossRef
J. Vancostenoble, Weak asymptotic decay for a wave equation with weak nonmonotone damping, 17p (to appear).
Webb, G.F., Compactness of trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 19-34. CrossRef