# American Institute of Mathematical Sciences

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

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

 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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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