Remarks on weak stabilization of semilinear wave equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 553-560.

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 non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble.

Classification : 35B35, 35L55, 35L90
Mots-clés : weak stabilization, semilinear, wave equations 
@article{COCV_2001__6__553_0,
     author = {Haraux, Alain},
     title = {Remarks on weak stabilization of semilinear wave equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {553--560},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1849416},
     zbl = {0988.35029},
     language = {en},
     url = {http://archive.numdam.org/item/COCV_2001__6__553_0/}
}
TY  - JOUR
AU  - Haraux, Alain
TI  - Remarks on weak stabilization of semilinear wave equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2001
SP  - 553
EP  - 560
VL  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/COCV_2001__6__553_0/
LA  - en
ID  - COCV_2001__6__553_0
ER  - 
%0 Journal Article
%A Haraux, Alain
%T Remarks on weak stabilization of semilinear wave equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2001
%P 553-560
%V 6
%I EDP-Sciences
%U http://archive.numdam.org/item/COCV_2001__6__553_0/
%G en
%F COCV_2001__6__553_0
Haraux, Alain. Remarks on weak stabilization of semilinear wave equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 553-560. http://archive.numdam.org/item/COCV_2001__6__553_0/

[1] L. Amerio and G. Prouse, Abstract almost periodic functions and functional equations. Van Nostrand, New-York (1971). | MR | Zbl

[2] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169-179. | MR | Zbl

[3] M. Biroli, Sur les solutions bornées et presque périodiques des équations et inéquations d'évolution. Ann. Math. Pura Appl. 93 (1972) 1-79. | Zbl

[4] T. Cazenave and A. Haraux, Propriétés oscillatoires des solutions de certaines équations des ondes semi-linéaires. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 449-452. | MR | Zbl

[5] T. Cazenave and A. Haraux, Oscillatory phenomena associated to semilinear wave equations in one spatial dimension. Trans. Amer. Math. Soc. 300 (1987) 207-233. | MR | Zbl

[6] T. Cazenave and A. Haraux, Some oscillatory properties of the wave equation in several space dimensions. J. Funct. Anal. 76 (1988) 87-109. | MR | Zbl

[7] T. Cazenave, A. Haraux and F.B. Weissler, 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. | MR | Zbl

[8] T. Cazenave, A. Haraux and F.B. Weissler, A class of nonlinear completely integrable abstract wave equations. J. Dynam. Differential Equations 5 (1993) 129-154. | MR | Zbl

[9] T. Cazenave, A. Haraux and F.B. Weissler, Detailed asymptotics for a convex hamiltonian system with two degrees of freedom. J. Dynam. Differential Equations 5 (1993) 155-187. | MR | Zbl

[10] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedbacks. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 485-515. | Numdam | MR | Zbl

[11] A. Haraux, Comportement à l'infini pour une équation des ondes non linéaire dissipative. C. R. Acad. Sci. Paris Sér. I Math. 287 (1978) 507-509. | Zbl

[12] A. Haraux, Comportement à l'infini pour certains systèmes dissipatifs non linéaires. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 213-234. | Zbl

[13] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985) 145-154. | MR | Zbl

[14] A. Haraux and V. Komornik, Oscillations of anharmonic Fourier series and the wave equation. Rev. Mat. Iberoamericana 1 (1985) 57-77. | MR | Zbl

[15] 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). | MR | Zbl

[16] A. Haraux, Systèmes dynamiques dissipatifs et applications, R.M.A. 17, edited by Ph. Ciarlet and J.L. Lions. Masson, Paris (1990). | MR | Zbl

[17] A. Haraux, Strong oscillatory behavior of solutions to some second order evolution equations, Publication du Laboratoire d'Analyse Numérique 94033, 10 p.

[18] B.M. Levitan and V.V. Zhikov, Almost periodic functions and differential equations. Cambridge University Press, Cambridge (1982). | MR | Zbl

[19] M. Slemrod, 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. | MR | Zbl

[20] J. Vancostenoble, Weak asymptotic stability of second order evolution equations by nonlinear and nonmonotone feedbacks. SIAM J. Math. Anal. 30 (1998) 140-154. | MR | Zbl

[21] J. Vancostenoble, Weak asymptotic decay for a wave equation with weak nonmonotone damping, 17p (to appear).

[22] G.F. Webb, Compactness of trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Ser. A 84 (1979) 19-34. | MR | Zbl