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On the control of the wave equation by memory-type boundary condition
1. | King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261 |
References:
[1] |
F. Alabau-Boussouira, On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.
doi: 10.1007/s00245. |
[2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary memory source term, Nonlinear Anal., 38 (1999), 281-294.
doi: 10.1016/S0362-546X(98)00195-3. |
[5] |
M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary conditions of memory type, Differential Integral Equations, 18 (2005), 583-600. |
[6] |
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. |
[7] |
I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64 (2006), 1757-1797.
doi: 10.1016/j.na.2005.07.024. |
[8] |
I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Anal., 69 (2008), 898-910.
doi: 10.1016/j.na.2008.02.069. |
[9] |
W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. |
[10] |
S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.
doi: 10.1016/j.jmaa.2007.11.048. |
[11] |
S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.
doi: 10.1016/j.na.2007.08.035. |
[12] |
S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal., 10 (2009), 3132-3140.
doi: 10.1016/j.nonrwa.2008.10.026. |
[13] |
S. A. Messaoudia and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlinear Anal., 11 (2010), 2896-2904.
doi: 10.1016/j.nonrwa.2009.10.013. |
[14] |
J. E. Munoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[15] |
J. E. Munoz Rivera and M. G. Naso, On the decay of the energy for systems with memory and indefinite dissipation, Asympt. Anal., 49 (2006), 189-204. |
[16] |
J. Y. Park, J. J. Bae and Hyo Jung, Uniform decay for wave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlinear Anal., 50 (2002), 871-884.
doi: 10.1016/S0362-546X(01)00781-7. |
[17] |
M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. J. Differ. Equ., 73 (2001), 1-11. |
[18] |
M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal., 54 (2003), 959-976.
doi: 10.1016/S0362-546X(03)00121-4. |
[19] |
M. L. Santos and F. Junior, A boundary condition with memory for Kirchhoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.
doi: 10.1016/S0096-3003(02)00915-3. |
[20] |
S. T. Wu, General decay for a wave equation of Kirchhoff type with a boundary control of memory type, Boundary Value Problems, 2011 (2011), 15pp.
doi: 10.1186/1687-2770-2011-55. |
[21] |
Q. Zhang, Global existence and exponential stability for a quasilinear wave equation with memory damping at the boundary, J. Optim. Theory Appl., 139 (2008), 617-634.
doi: 10.1007/s10957-008-9399-x. |
show all references
References:
[1] |
F. Alabau-Boussouira, On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.
doi: 10.1007/s00245. |
[2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.
doi: 10.1016/j.jde.2007.02.004. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary memory source term, Nonlinear Anal., 38 (1999), 281-294.
doi: 10.1016/S0362-546X(98)00195-3. |
[5] |
M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary conditions of memory type, Differential Integral Equations, 18 (2005), 583-600. |
[6] |
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. |
[7] |
I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64 (2006), 1757-1797.
doi: 10.1016/j.na.2005.07.024. |
[8] |
I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Anal., 69 (2008), 898-910.
doi: 10.1016/j.na.2008.02.069. |
[9] |
W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. |
[10] |
S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.
doi: 10.1016/j.jmaa.2007.11.048. |
[11] |
S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.
doi: 10.1016/j.na.2007.08.035. |
[12] |
S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal., 10 (2009), 3132-3140.
doi: 10.1016/j.nonrwa.2008.10.026. |
[13] |
S. A. Messaoudia and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlinear Anal., 11 (2010), 2896-2904.
doi: 10.1016/j.nonrwa.2009.10.013. |
[14] |
J. E. Munoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[15] |
J. E. Munoz Rivera and M. G. Naso, On the decay of the energy for systems with memory and indefinite dissipation, Asympt. Anal., 49 (2006), 189-204. |
[16] |
J. Y. Park, J. J. Bae and Hyo Jung, Uniform decay for wave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlinear Anal., 50 (2002), 871-884.
doi: 10.1016/S0362-546X(01)00781-7. |
[17] |
M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. J. Differ. Equ., 73 (2001), 1-11. |
[18] |
M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal., 54 (2003), 959-976.
doi: 10.1016/S0362-546X(03)00121-4. |
[19] |
M. L. Santos and F. Junior, A boundary condition with memory for Kirchhoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.
doi: 10.1016/S0096-3003(02)00915-3. |
[20] |
S. T. Wu, General decay for a wave equation of Kirchhoff type with a boundary control of memory type, Boundary Value Problems, 2011 (2011), 15pp.
doi: 10.1186/1687-2770-2011-55. |
[21] |
Q. Zhang, Global existence and exponential stability for a quasilinear wave equation with memory damping at the boundary, J. Optim. Theory Appl., 139 (2008), 617-634.
doi: 10.1007/s10957-008-9399-x. |
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