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December  2016, 36(12): 6899-6919. doi: 10.3934/dcds.2016100

Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions

Tae Gab Ha 1,

1. 

Department of Mathematics, Pusan National University, Busan, 609-735

Received  February 2016 Revised  July 2016 Published  October 2016

In this paper, we consider the viscoelastic equation with damping and source terms and acoustic boundary conditions. We prove a global existence of solutions and uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function. This paper is to improve the result of [4] by applying the method developed in [17].
Keywords: general decay rates., variable coefficients, acoustic boundary conditions, Viscoelastic equation, global existence.
Mathematics Subject Classification: Primary: 35L70; Secondary: 35B4.
Citation: Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100
References:
[1]

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.

[2]

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

[3]

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.

[4]

Y. Boukhatem and B. Benabderrahmane, 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.

[5]

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.

[6]

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.

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global solvability and asymptotic stability for the wave equation with nonlinear feedback and source term on the boundary, Adv. Math. Sci. Appl., 16 (2006), 661-696.

[8]

A. T. Cousin, C. L. Frota and N. A. Larkin, Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions, Funkcial. Ekvac., 44 (2001), 471-485.

[9]

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.

[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]

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.

[13]

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.

[14]

P. Jameson Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J. Evolution Equations, 12 (2012), 141-164. doi: 10.1007/s00028-011-0127-x.

[15]

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.

[16]

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.

[17]

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.

[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.

[20]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, John Wiley, Paris, 1994.

[21]

V. Komornik and E. Zuazua, A direct method for boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.

[22]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.

[23]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[24]

J. Li and S. G. Chai, Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback, Nonlinear Anal., 112 (2015), 105-117. doi: 10.1016/j.na.2014.08.021.

[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]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control, Optimisation Calc. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116.

[27]

J. Y. Park and T. G. Ha, Energy decay for nondissipative distributed systems with boundary damping and source term, Nonlinear Anal., 70 (2009), 2416-2434. doi: 10.1016/j.na.2008.03.026.

[28]

J. Y. Park and T. G. Ha, Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term, J. Math. Phys., 49 (2008), 053511, 26pp. doi: 10.1063/1.2919886.

[29]

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.

[30]

J. Y. Park, T. G. Ha and Y. H. Kang, Energy decay rates for solutions of the wave equation with boundary damping and source term, Z. Angew. Math. Phys., 61 (2010), 235-265. doi: 10.1007/s00033-009-0009-z.

[31]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.

[32]

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.

[33]

J. Wu, Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions, Nonlinear Anal., 75 (2012), 6562-6569. doi: 10.1016/j.na.2012.07.032.

[34]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-478. doi: 10.1137/0328025.

show all references

References:
[1]

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.

[2]

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

[3]

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.

[4]

Y. Boukhatem and B. Benabderrahmane, 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.

[5]

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.

[6]

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.

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global solvability and asymptotic stability for the wave equation with nonlinear feedback and source term on the boundary, Adv. Math. Sci. Appl., 16 (2006), 661-696.

[8]

A. T. Cousin, C. L. Frota and N. A. Larkin, Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions, Funkcial. Ekvac., 44 (2001), 471-485.

[9]

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.

[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]

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.

[13]

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.

[14]

P. Jameson Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J. Evolution Equations, 12 (2012), 141-164. doi: 10.1007/s00028-011-0127-x.

[15]

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.

[16]

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.

[17]

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.

[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.

[20]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, John Wiley, Paris, 1994.

[21]

V. Komornik and E. Zuazua, A direct method for boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.

[22]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507-533.

[23]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992), 189-224. doi: 10.1007/BF01182480.

[24]

J. Li and S. G. Chai, Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback, Nonlinear Anal., 112 (2015), 105-117. doi: 10.1016/j.na.2014.08.021.

[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]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control, Optimisation Calc. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116.

[27]

J. Y. Park and T. G. Ha, Energy decay for nondissipative distributed systems with boundary damping and source term, Nonlinear Anal., 70 (2009), 2416-2434. doi: 10.1016/j.na.2008.03.026.

[28]

J. Y. Park and T. G. Ha, Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term, J. Math. Phys., 49 (2008), 053511, 26pp. doi: 10.1063/1.2919886.

[29]

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.

[30]

J. Y. Park, T. G. Ha and Y. H. Kang, Energy decay rates for solutions of the wave equation with boundary damping and source term, Z. Angew. Math. Phys., 61 (2010), 235-265. doi: 10.1007/s00033-009-0009-z.

[31]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.

[32]

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.

[33]

J. Wu, Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions, Nonlinear Anal., 75 (2012), 6562-6569. doi: 10.1016/j.na.2012.07.032.

[34]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-478. doi: 10.1137/0328025.

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