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March  2015, 35(3): 1179-1192. doi: 10.3934/dcds.2015.35.1179

On the control of the wave equation by memory-type boundary condition

Muhammad I. Mustafa 1,

1. 

King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261

Received  March 2014 Revised  June 2014 Published  October 2014

In this paper we consider a wave equation with a viscoelastic boundary damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a larger class of relaxation functions and generalizes previous results existing in the literature.
Keywords: viscoelastic damping, relaxation function, General decay, convexity..
Mathematics Subject Classification: 35B37, 35L55, 74D05, 93D15, 93d2.
Citation: Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
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.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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