Yamna BOUKHATEM
,
Benyattou BENABDERRAHMANE
. GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS[J]. Acta mathematica scientia, Series B, 2017
, 37(5)
: 1453
-1471
.
DOI: 10.1016/S0252-9602(17)30084-X
[1] Abdallah C, Dorato P, Benitez-Read J, Byrne R. Delayed positive feedback can stabilize oscillatory system. ACC San Francisco, 1993:3106-3107
[2] Avalos G, Lasiecka I, Rebarber R. Uniform decay properties of a model in structural acoustics. J Math Pures Appl, 2000, 79(10):1057-1072
[3] Arnold V I. Mathematical Methods of Classical Mechanics. New York:Springer-Verlag, 1989
[4] Beale J T. Spectral properties of an acoustic boundary condition. Indiana Univ Math J, 1976, 25(9):895-917
[5] Beale J T, Rosencrans S I. Acoustic boundary conditions. Bull Amer Math Soc, 1974, 80:1276-1278
[6] Benaissa A, Benaissa A, Messaoudi S A. Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J Math Phys, 2012, 53:123514; doi:10.1063/1.4765046
[7] Boukhatem Y, Benabderrahmanne B. Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlinear Anal TMA, 2014, 97:191-209
[8] Boukhatem Y, Benabderrahmanne B. Polynomial decay and blow up of solutions for variable coeffcients viscoelastic wave equation with acoustic boundary conditions. Acta Math Sin, English Series, 2016, 32(2):153-174
[9] Bucci F, Lasiecka I. Exponential decay rates for structural acoustic model with an over damping on the interface and boundary layer dissipation. Appl Anal, 2002, 81(4):977-999
[10] Cavalcanti M M, Cavalcanti V D, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J Differ Equ, 2007, 236:407-459
[11] Cavalcanti M M, Domingos V N, Cavalcanti J S, Filho P, Soriano J A. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ Integral Equ, 2001, 14(1):85-116
[12] Cousin A T, Frota C L, Larkin N A. Global solvability and asymptotic behavior of a hyperbolic problem with acoustic boundary conditions. Funkcial Ekvac, 2001, 44(3):471-485
[13] Datko R, Lagnese J, Polis M P. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim, 1986, 24:152-156
[14] Frota C L, Goldstein J A. Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ, 2000, 164(1):92-109
[15] Frota C L, Larkin N A. Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Prog Nonlinear Differ Equ Their Appl, 2005, 66:297-312
[16] Li Gang, Zhu Biqing, Wang Danhua. General decay for a viscoelastic wave equation with dynamic boundary conditions and a time-varying delay. http://arxiv.org/abs/1505.07060v6
[17] Georgiev V, Todorov G. Existence of a solution of the wave equation with nonlinear damping and source term. J Differ Equ, 1994, 109:295-308
[18] Lasiecka I, Boundary stabilization of a 3-dimensional structural acoustic model. J Math Pures Appl, 1999, 78(2):203-232
[19] Lasiecka I, Triggiani R, Yao P F. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J Math Anal Appl, 1999, 235:13-57
[20] Messaoudi S A, Mustafa M I. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal TMA, 2010, 72:3602-3611
[21] Morse P M, Ingard K U. Theoretical Acoustics. New York:McGraw-Hill, 1968.
[22] Mugnolo D. Abstract wave equations with acoustic boundary conditions. Math Nachr, 2006, 279(3):299-318
[23] Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim, 2006, 45:561-1585
[24] Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differ Integral Equ, 2008, 21:935-958
[25] Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time dependent delay. Electron J Differ Equ, 2011, (41):1-20
[26] Nicaise S, Valein J, Fridman E. Stability of the heat and of the wave equations with boundary time-varying delays. Disc Cont Dyn Syst, Series 5, 2009, 2(3):559-581
[27] Ning Z H, Shen C X, Zhao X P. Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback. J Math Anal Appl, 2013, 405:148-155
[28] Ning Z H, Shen C X, Zhao X P, Li H, Lin C S, Zhang Y M. Nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay. Appl Math Comput, 2014, 232:511-520
[29] Park J Y, Ha T G. Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. J Math Phys, 2009, 50(1):013506
[30] Park J Y, Park S H. Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal TMA, 2011, 74(3):993-998
[31] Vicente A. Wave equation with acoustic/memory boundary conditions. Bol Soc Parana Mat, 2009, 27(1):29-39
[32] Xu C Q, Yung S P, Li L K. Stabilization of the wave system with input delay in the boundary control. ESAIM:Control Optim Calc Var, 2006, 12:770-785
[33] Liu W, Chen K. Existence and general decay for nondissipative distributed systems with boundary frictional and memory damping and acoustic boundary conditions. Z Angew Math Phys, 2015, 66(4):1595-1614
[34] Wu J Q. Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Nonlinear Anal TMA, 2012, 75(18):6562-6569
[35] Yao P F. Observability inequality for exact controllability of wave equations with variable coefficients. J Control Optim, 1999, 37:1568-1599