September  2017, 37(9): 4857-4876. doi: 10.3934/dcds.2017209

General decay of solutions of a Bresse system with viscoelastic boundary conditions

Faculty of Mathematics, University of Sciences and Technology Houari Boumedienne, P.O. Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

* Corresponding author: Ammar Khemmoudj

Received  June 2016 Revised  April 2017 Published  June 2017

In this paper we are concerned with a multi-dimensional Bresse system, in a bounded domain, where the memory-type damping is acting on a portion of the boundary. We establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

Citation: Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209
References:
[1]

F. Alabau BoussouiraD. S. Almeida Júnior and J. E. M. Rivera, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. doi: 10.1016/j.jmaa.2010.07.046. Google Scholar

[2]

D. S. Almeida JúniorM. L. Santos and A. Soufyane, Asymptotic behavior to Bresse system with past history, Quarterly of Applied Mathematics, 73 (2015), 23-54. doi: 10.1090/S0033-569X-2014-01382-4. Google Scholar

[3]

J. J. Bae, On uniform decay of coupled wave equation of Kirchhoff type subject to memory condition on the boundary, Nonlin. Anal., 61 (2005), 351-372. doi: 10.1016/j.na.2004.11.014. Google Scholar

[4]

J. A. C. Bresse, Cours mécanique Appliquée, Mallet Bachelier, Paris, 1859.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]

M. M. CavalcantiV. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465. doi: 10.1016/S0096-3003(03)00284-4. Google Scholar

[7]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604. doi: 10.1016/j.aml.2011.09.067. Google Scholar

[8]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Jour. Appl. Math, 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. Google Scholar

[9]

L. H. FatoriJ. E. M. Rivera and J. A. S. Soriano, Bresse system with indefinite damping, Jour. Math. Anal. Appl., 387 (2012), 284-290. doi: 10.1016/j.jmaa.2011.08.072. Google Scholar

[10]

A. J. R. FeitosaM. Milla Miranda and M. L. Oliveira, Nonlinear boundary stabilization for Timoshenko beam system, Jour. Math. Anal. Appl., 428 (2015), 194-216. doi: 10.1016/j.jmaa.2015.02.019. Google Scholar

[11]

J. FerreiraD. C. PereiraC. A. Rapaso and M. L. Santos, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlin. Anal., 54 (2003), 959-976. doi: 10.1016/S0362-546X(03)00121-4. Google Scholar

[12]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Meth. Appl. Sci., 38 (2015), 2389-2402. doi: 10.1002/mma.3228. Google Scholar

[13]

J. R. Kang, General decay for Kirchhoff plates with a boundary condition of memory type, Boundary Value Problems, 2012 (2012), 1-11. doi: 10.1186/1687-2770-2012-129. Google Scholar

[14]

A. Khemmoudj and T. Hamadouche, Boundary stabilization of a Bresse-type system, Math. Meth. Appl. Sci., 39 (2016), 3282-3293. doi: 10.1002/mma.3773. Google Scholar

[15]

J. E. Lagnese, G. Leugering and J. P. G. Schmidt, Modelling Analysis and Control of Dynamic Elastic Multi-Link Structures Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[16]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[17]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of solutions of a nonlinear system of Timoshenko type, Nonlin. Anal., 67 (2007), 2107-2121. doi: 10.1016/j.na.2006.08.039. Google Scholar

[18]

S. A. Messaoudi and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlin. Anal., RWA, 11 (2010), 2896-2904. doi: 10.1016/j.nonrwa.2009.10.013. Google Scholar

[19]

S. A. Messaoudi and M. I. Mustafa, Energy decay rates for a Timoshenko system with viscoelastic boundary conditions, Appl. Math. Comput., 218 (2012), 9125-9131. doi: 10.1016/j.amc.2012.02.065. Google Scholar

[20]

N. Noun and A. Wehbe, Stabilisation faible interne locale de systéme élastique de Bresse, English, with English and French summaries, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498. doi: 10.1016/j.crma.2012.04.003. Google Scholar

[21]

P. PeiM. A. Rammaha and D. Toundykov, Local and global well-posedness of semilinear Reissner Mindlin Timoshenko plate equations, Nonlinear Analysis: Theory, Methods and Applications, 105 (2014), 62-85. doi: 10.1016/j.na.2014.03.024. Google Scholar

[22]

H. Portillo OquendoJ. E. Munoz Rivera and M. L. Santos, Asymptotic behavior to a von Kármán plate with boundary memory conditions, Nonlinear Analysis, 62 (2005), 1183-1205. doi: 10.1016/j.na.2005.04.025. Google Scholar

[23]

M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. Jour. Differ. Equ., 73 (2001), 1-11. Google Scholar

[24]

M. L. Santos, Decay rates for solutions of a Timoshenko system with a memory condition at the boundary, Abstr. Appl. Anal., 7 (2002), 531-546. doi: 10.1155/S1085337502204133. Google Scholar

[25]

M. L. Santos and A. Soufyane, General decay to a von Karman plate system with memory boundary conditions, Diff. Int. Equ., 24 (2011), 69-81. Google Scholar

[26]

M. L. Santos and C. C. S. Tavares, On the Kirchhoff plates equations with thermal effects and memory boundary conditions, Appl. Math. Comput., 213 (2009), 25-38. doi: 10.1016/j.amc.2009.03.009. Google Scholar

show all references

References:
[1]

F. Alabau BoussouiraD. S. Almeida Júnior and J. E. M. Rivera, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. doi: 10.1016/j.jmaa.2010.07.046. Google Scholar

[2]

D. S. Almeida JúniorM. L. Santos and A. Soufyane, Asymptotic behavior to Bresse system with past history, Quarterly of Applied Mathematics, 73 (2015), 23-54. doi: 10.1090/S0033-569X-2014-01382-4. Google Scholar

[3]

J. J. Bae, On uniform decay of coupled wave equation of Kirchhoff type subject to memory condition on the boundary, Nonlin. Anal., 61 (2005), 351-372. doi: 10.1016/j.na.2004.11.014. Google Scholar

[4]

J. A. C. Bresse, Cours mécanique Appliquée, Mallet Bachelier, Paris, 1859.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]

M. M. CavalcantiV. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465. doi: 10.1016/S0096-3003(03)00284-4. Google Scholar

[7]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604. doi: 10.1016/j.aml.2011.09.067. Google Scholar

[8]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Jour. Appl. Math, 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. Google Scholar

[9]

L. H. FatoriJ. E. M. Rivera and J. A. S. Soriano, Bresse system with indefinite damping, Jour. Math. Anal. Appl., 387 (2012), 284-290. doi: 10.1016/j.jmaa.2011.08.072. Google Scholar

[10]

A. J. R. FeitosaM. Milla Miranda and M. L. Oliveira, Nonlinear boundary stabilization for Timoshenko beam system, Jour. Math. Anal. Appl., 428 (2015), 194-216. doi: 10.1016/j.jmaa.2015.02.019. Google Scholar

[11]

J. FerreiraD. C. PereiraC. A. Rapaso and M. L. Santos, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlin. Anal., 54 (2003), 959-976. doi: 10.1016/S0362-546X(03)00121-4. Google Scholar

[12]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Meth. Appl. Sci., 38 (2015), 2389-2402. doi: 10.1002/mma.3228. Google Scholar

[13]

J. R. Kang, General decay for Kirchhoff plates with a boundary condition of memory type, Boundary Value Problems, 2012 (2012), 1-11. doi: 10.1186/1687-2770-2012-129. Google Scholar

[14]

A. Khemmoudj and T. Hamadouche, Boundary stabilization of a Bresse-type system, Math. Meth. Appl. Sci., 39 (2016), 3282-3293. doi: 10.1002/mma.3773. Google Scholar

[15]

J. E. Lagnese, G. Leugering and J. P. G. Schmidt, Modelling Analysis and Control of Dynamic Elastic Multi-Link Structures Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[16]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[17]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of solutions of a nonlinear system of Timoshenko type, Nonlin. Anal., 67 (2007), 2107-2121. doi: 10.1016/j.na.2006.08.039. Google Scholar

[18]

S. A. Messaoudi and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlin. Anal., RWA, 11 (2010), 2896-2904. doi: 10.1016/j.nonrwa.2009.10.013. Google Scholar

[19]

S. A. Messaoudi and M. I. Mustafa, Energy decay rates for a Timoshenko system with viscoelastic boundary conditions, Appl. Math. Comput., 218 (2012), 9125-9131. doi: 10.1016/j.amc.2012.02.065. Google Scholar

[20]

N. Noun and A. Wehbe, Stabilisation faible interne locale de systéme élastique de Bresse, English, with English and French summaries, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498. doi: 10.1016/j.crma.2012.04.003. Google Scholar

[21]

P. PeiM. A. Rammaha and D. Toundykov, Local and global well-posedness of semilinear Reissner Mindlin Timoshenko plate equations, Nonlinear Analysis: Theory, Methods and Applications, 105 (2014), 62-85. doi: 10.1016/j.na.2014.03.024. Google Scholar

[22]

H. Portillo OquendoJ. E. Munoz Rivera and M. L. Santos, Asymptotic behavior to a von Kármán plate with boundary memory conditions, Nonlinear Analysis, 62 (2005), 1183-1205. doi: 10.1016/j.na.2005.04.025. Google Scholar

[23]

M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. Jour. Differ. Equ., 73 (2001), 1-11. Google Scholar

[24]

M. L. Santos, Decay rates for solutions of a Timoshenko system with a memory condition at the boundary, Abstr. Appl. Anal., 7 (2002), 531-546. doi: 10.1155/S1085337502204133. Google Scholar

[25]

M. L. Santos and A. Soufyane, General decay to a von Karman plate system with memory boundary conditions, Diff. Int. Equ., 24 (2011), 69-81. Google Scholar

[26]

M. L. Santos and C. C. S. Tavares, On the Kirchhoff plates equations with thermal effects and memory boundary conditions, Appl. Math. Comput., 213 (2009), 25-38. doi: 10.1016/j.amc.2009.03.009. Google Scholar

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