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June  2018, 7(2): 281-291. doi: 10.3934/eect.2018014

On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions

Tae Gab Ha

Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 54896, Republic of Korea

Received  October 2017 Revised  February 2018 Published  May 2018

Fund Project: This paper was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03932096).

In this paper, we consider the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. This work is devoted to prove, under suitable conditions on the initial data, the global existence and uniform decay rate of the solutions when the relaxation function is not necessarily of exponential or polynomial type.

Keywords: Viscoelastic equation, acoustic boundary conditions, Balakrishnan-Taylor damping, global existence, general decay rate.
Mathematics Subject Classification: 35L70, 35B40.
Citation: Tae Gab Ha. On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. Evolution Equations & Control Theory, 2018, 7 (2) : 281-291. doi: 10.3934/eect.2018014
References:
[1]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, in Proceedings "Damping 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. Google Scholar

[2]

R. W. Bass and D. Zes, Spillover, nonlinearity and flexible structures, in The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 (ed. L. W. Taylor), (1991), 1–14. doi: 10.1109/CDC.1991.261683.  Google Scholar

[3]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar

[4]

J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222.  doi: 10.1512/iumj.1977.26.26015.  Google Scholar

[5]

J. T. Beale and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 (1974), 1276-1278.  doi: 10.1090/S0002-9904-1974-13714-6.  Google Scholar

[6]

Y. Boukhatem and B. Benabderramane, Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions, Nonlinear Anal., 97 (2014), 191-209.  doi: 10.1016/j.na.2013.11.019.  Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

[8]

A. T. Cousin, C. L. Frota and N. A. Larkin, On a system of Klein-Gordon type equations with acoustic boundary conditions, J. Math. Anal. Appl., 293 (2004), 293-309.  doi: 10.1016/j.jmaa.2004.01.007.  Google Scholar

[9]

N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynaamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.  Google Scholar

[10]

C. L. Frota and J. A. Goldstein, Some Nonlinear wave equations with acoustic boundary conditins, J. Differential equations, 164 (2000), 92-109.  doi: 10.1006/jdeq.1999.3743.  Google Scholar

[11]

C. L. Frota and N. A. Larkin, Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, Progr. Nonlinear Differential Equations Appl., 66 (2006), 297-312.  doi: 10.1007/3-7643-7401-2_20.  Google Scholar

[12]

T. G. Ha, Asymptotic stability of the viscoelastic equation with variable coefficients and the Balakrishnan-Taylor damping, Taiwanese J. Math., To appear. doi: 10.11650/tjm/171203.  Google Scholar

[13]

T. G. Ha, General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett., 60 (2016), 43-49.  doi: 10.1016/j.aml.2016.04.006.  Google Scholar

[14]

T. G. Ha, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 67 (2016), Art. 32, 17 pp. doi: 10.1007/s00033-016-0625-3.  Google Scholar

[15]

T. G. Ha, Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 6899-6919.  doi: 10.3934/dcds.2016100.  Google Scholar

[16]

T. G. Ha, On viscoelastic wave equation with nonlinear boundary damping and source term, Commun. Pur. Appl. Anal., 9 (2010), 1543-1576.  doi: 10.3934/cpaa.2010.9.1543.  Google Scholar

[17]

T. G. Ha, Energy decay for the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett., 76 (2018), 201-207.  doi: 10.1016/j.aml.2017.09.005.  Google Scholar

[18]

T. G. Ha and J. Y. Park, Existence of solutions for the Kirchhoff-type wave equation with memory term and acoustic boundary conditions, Numer. Funct. Anal. Optim., 31 (2010), 921-935.  doi: 10.1080/01630563.2010.498301.  Google Scholar

[19]

T. G. Ha and J. Y. Park, On coupled Klein-Gordon-Schrödinger equations with acoustic boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 132751, 23pp. doi: 10.1155/2010/132751.  Google Scholar

[20]

T. G. Ha, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwanese J. Math., 21 (2017), 807-817.  doi: 10.11650/tjm/7828.  Google Scholar

[21]

P. Jameson Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, 88 (2014), 333-365.  doi: 10.1007/s00233-013-9534-3.  Google Scholar

[22]

P. Jameson Graber and B. Said-Houari, On the wave equation with semilinear porous acoustic boundary conditions, J. Differential Equations, 252 (2012), 4898-4941.  doi: 10.1016/j.jde.2012.01.042.  Google Scholar

[23]

P. Jameson Graber, Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions, Nonlinear Anal., 74 (2011), 3137-3148.  doi: 10.1016/j.na.2011.01.029.  Google Scholar

[24]

P. Jameson Graber, Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system, Nonlinear Anal., 73 (2010), 3058-3068.  doi: 10.1016/j.na.2010.06.075.  Google Scholar

[25]

W. Liu, Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions, Appl. Math. Lett., 38 (2014), 155-161.  doi: 10.1016/j.aml.2014.07.022.  Google Scholar

[26]

S. A. Messaoudi, General decay of 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

[27]

C. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2.  Google Scholar

[28]

J. Y. Park and T. G. Ha, Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys., 50 (2009), 013506, 18pp. doi: 10.1063/1.3040185.  Google Scholar

[29]

N.-e. Tatar and A. Zaraï, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstr. Math., 44 (2011), 67-90.   Google Scholar

[30]

J. Wu, Uniform energy decay of a variable coefficient wave equation with nonlinear acoustic boundary conditions, J. Math. Anal. Appl., 399 (2013), 369-377.  doi: 10.1016/j.jmaa.2012.09.056.  Google Scholar

[31]

Y. You, Inertial manifolds and stabilization of nonlinear beam equaitons with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.  Google Scholar

[32]

A. Zaraï and N.-e. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math.(BRNO), 46 (2010), 157-176.   Google Scholar

[33]

A. Zaraï, N.-e. Tatar and A. Abdelmalek, Elastic membrane equation with memory term and nonlinear boundary damping: golbal existence, decay and blowup of the solution, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 84-106.  doi: 10.1016/S0252-9602(12)60196-9.  Google Scholar

show all references

References:
[1]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, in Proceedings "Damping 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. Google Scholar

[2]

R. W. Bass and D. Zes, Spillover, nonlinearity and flexible structures, in The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 (ed. L. W. Taylor), (1991), 1–14. doi: 10.1109/CDC.1991.261683.  Google Scholar

[3]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar

[4]

J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222.  doi: 10.1512/iumj.1977.26.26015.  Google Scholar

[5]

J. T. Beale and S. I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 (1974), 1276-1278.  doi: 10.1090/S0002-9904-1974-13714-6.  Google Scholar

[6]

Y. Boukhatem and B. Benabderramane, Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions, Nonlinear Anal., 97 (2014), 191-209.  doi: 10.1016/j.na.2013.11.019.  Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040.  Google Scholar

[8]

A. T. Cousin, C. L. Frota and N. A. Larkin, On a system of Klein-Gordon type equations with acoustic boundary conditions, J. Math. Anal. Appl., 293 (2004), 293-309.  doi: 10.1016/j.jmaa.2004.01.007.  Google Scholar

[9]

N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynaamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.  Google Scholar

[10]

C. L. Frota and J. A. Goldstein, Some Nonlinear wave equations with acoustic boundary conditins, J. Differential equations, 164 (2000), 92-109.  doi: 10.1006/jdeq.1999.3743.  Google Scholar

[11]

C. L. Frota and N. A. Larkin, Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, Progr. Nonlinear Differential Equations Appl., 66 (2006), 297-312.  doi: 10.1007/3-7643-7401-2_20.  Google Scholar

[12]

T. G. Ha, Asymptotic stability of the viscoelastic equation with variable coefficients and the Balakrishnan-Taylor damping, Taiwanese J. Math., To appear. doi: 10.11650/tjm/171203.  Google Scholar

[13]

T. G. Ha, General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett., 60 (2016), 43-49.  doi: 10.1016/j.aml.2016.04.006.  Google Scholar

[14]

T. G. Ha, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 67 (2016), Art. 32, 17 pp. doi: 10.1007/s00033-016-0625-3.  Google Scholar

[15]

T. G. Ha, Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 6899-6919.  doi: 10.3934/dcds.2016100.  Google Scholar

[16]

T. G. Ha, On viscoelastic wave equation with nonlinear boundary damping and source term, Commun. Pur. Appl. Anal., 9 (2010), 1543-1576.  doi: 10.3934/cpaa.2010.9.1543.  Google Scholar

[17]

T. G. Ha, Energy decay for the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett., 76 (2018), 201-207.  doi: 10.1016/j.aml.2017.09.005.  Google Scholar

[18]

T. G. Ha and J. Y. Park, Existence of solutions for the Kirchhoff-type wave equation with memory term and acoustic boundary conditions, Numer. Funct. Anal. Optim., 31 (2010), 921-935.  doi: 10.1080/01630563.2010.498301.  Google Scholar

[19]

T. G. Ha and J. Y. Park, On coupled Klein-Gordon-Schrödinger equations with acoustic boundary conditions, Bound. Value Probl., 2010 (2010), Art. ID 132751, 23pp. doi: 10.1155/2010/132751.  Google Scholar

[20]

T. G. Ha, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwanese J. Math., 21 (2017), 807-817.  doi: 10.11650/tjm/7828.  Google Scholar

[21]

P. Jameson Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, 88 (2014), 333-365.  doi: 10.1007/s00233-013-9534-3.  Google Scholar

[22]

P. Jameson Graber and B. Said-Houari, On the wave equation with semilinear porous acoustic boundary conditions, J. Differential Equations, 252 (2012), 4898-4941.  doi: 10.1016/j.jde.2012.01.042.  Google Scholar

[23]

P. Jameson Graber, Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions, Nonlinear Anal., 74 (2011), 3137-3148.  doi: 10.1016/j.na.2011.01.029.  Google Scholar

[24]

P. Jameson Graber, Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system, Nonlinear Anal., 73 (2010), 3058-3068.  doi: 10.1016/j.na.2010.06.075.  Google Scholar

[25]

W. Liu, Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions, Appl. Math. Lett., 38 (2014), 155-161.  doi: 10.1016/j.aml.2014.07.022.  Google Scholar

[26]

S. A. Messaoudi, General decay of 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

[27]

C. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2.  Google Scholar

[28]

J. Y. Park and T. G. Ha, Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys., 50 (2009), 013506, 18pp. doi: 10.1063/1.3040185.  Google Scholar

[29]

N.-e. Tatar and A. Zaraï, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstr. Math., 44 (2011), 67-90.   Google Scholar

[30]

J. Wu, Uniform energy decay of a variable coefficient wave equation with nonlinear acoustic boundary conditions, J. Math. Anal. Appl., 399 (2013), 369-377.  doi: 10.1016/j.jmaa.2012.09.056.  Google Scholar

[31]

Y. You, Inertial manifolds and stabilization of nonlinear beam equaitons with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.  Google Scholar

[32]

A. Zaraï and N.-e. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math.(BRNO), 46 (2010), 157-176.   Google Scholar

[33]

A. Zaraï, N.-e. Tatar and A. Abdelmalek, Elastic membrane equation with memory term and nonlinear boundary damping: golbal existence, decay and blowup of the solution, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 84-106.  doi: 10.1016/S0252-9602(12)60196-9.  Google Scholar

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