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    General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term
June  2017, 6(2): 261-276. doi: 10.3934/eect.2017014

Viscoelastic plate equation with boundary feedback

Department of Mathematics -University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates

Received  March 2016 Revised  October 2016 Published  April 2017

In this paper we consider a viscoelastic plate equation with a nonlinear weakly dissipative feedback localized on a part of the boundary. Without imposing restrictive assumptions on the boundary frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

Citation: Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations and Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014
References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for the second order evalution equation with memory, J. Funct. Anal., 245 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

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.

[4]

K. Ammari and M. Tucsnak, Stabilization of Bernoulli -Euler beam by means of a pointwise feedback force, Siam J. Control Optim., 39 (2000), 1160-1181.  doi: 10.1137/S0363012998349315.

[5]

N. E. Amroun and A. Benaissa, Global existence and energy decay of solutions to a Petrovsky equation with general nonlinear dissipation and source term, Georgian Math. J., 13 (2006), 397-410. 

[6]

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

[7]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal. Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.

[8]

M. M. CavalcantiV. N. Domingos CavalcantiI. Lasiecka and F. A. Falco Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.

[9]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705-731.  doi: 10.1142/S0219199704001483.

[10]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[11]

J. Ferreira and S. A. Messaoudi, On the general decay of a nonlinear viscoelastic plate equation with a strong damping and $\overrightarrow{p}(x, t)$-Laplacian, Nonlinear Anal., 104 (2014), 40-49.  doi: 10.1016/j.na.2014.03.010.

[12]

A. Guesmia and S. A. Messaoudi, General energy decay estemates of Timoshenko system with frictional versus vescolastic damping, Math. Methods Appl. Sci., 32 (2009), 2102-2122.  doi: 10.1002/mma.1125.

[13]

R. B. Guzman and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation, Systems Control Lett., 48 (2003), 191-197.  doi: 10.1016/S0167-6911(02)00264-5.

[14]

X. Han and M. Wang, Energy decay rate for a coupled hyperbolic system with nonlinear damping, Nonlinear Anal., 70 (2009), 3264-3272.  doi: 10.1016/j.na.2008.04.029.

[15]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equation with memory, Acta Appl. Math., 110 (2010), 195-207.  doi: 10.1007/s10440-008-9397-x.

[16]

M. A. Horn, Uniform decay rates for the solution to the Euler Bernoulli plate equation with boundary feedback via benoling moments, Differential Integral Equation, 6 (1992), 1121-1150. 

[17]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plates with dissipation acting only via moments-limiting behavior, J. Math. Anal. Appl., 229 (1999), 452-479.  doi: 10.1006/jmaa.1998.6170.

[18]

J. R. Kang, Energy decay rates for von Karman system with memory and boundary feedback, Appl. Math. Comput., 218 (2012), 9085-9094.  doi: 10.1016/j.amc.2012.02.053.

[19]

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

[20]

V. Komornik, Decay estimates for a petrovski system with a nonlinear distributed feedback, IMA Preprints Series, 1992.

[21]

V. Komornik, On the nonlinear boundary stabilization of Kirchhoff plates, Nonlinear Differential Equation Appl., 1 (1994), 323-337.  doi: 10.1007/BF01194984.

[22]

J. E. Lagnese, Asymptotic energy estimates for Kirchoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, birkhcauser-Verlag, Bassel, 91 (1989), 211-236.

[23]

J. E. Lagnese, Boundary Stabilization of Thin Plates SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[24]

I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pures Appl., 78 (1999), 203-232.  doi: 10.1016/S0021-7824(01)80009-X.

[25]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R.

[26]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory J. Math. Phys. 54 (2013), 031504, 18pp. doi: 10.1063/1.4793988.

[27]

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

[28]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 271-303. doi: 10.1007/978-3-319-11406-4_1.

[29]

W. J. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms J. Math. Phys. 50 (2009), 113506, 17pp. doi: 10.1063/1.3254323.

[30]

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

[31]

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

[32]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.

[33]

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.

[34]

S. A. Messaoudi, On the control of solutions of a viscoelastic equations, J. Franklin Inst., 334 (2007), 765-776.  doi: 10.1016/j.jfranklin.2006.02.029.

[35]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dynam. Systems Appl., 18 (2009), 457-468. 

[36]

S. A. Messaoudi and M. I. Mustafa, On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Anal., 72 (2010), 3602-3611.  doi: 10.1016/j.na.2009.12.040.

[37]

J. E. Munoz RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[38]

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. 

[39]

J. E. Munoz RiveraM. 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. 

[40]

M. I. Mustafa, Uniform decay for wave equations with weakly dissipative boundary feedback, Dyn. Syst., 30 (2015), 241-250.  doi: 10.1080/14689367.2014.1002455.

[41]

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

[42]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. 

[43]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscolastic wave equatios J. Math. Phys. 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830.

[44]

A. F. PazotoL. Colho and R. C. Charao, Uniform stabilization of a plate equation with nonlinear localized dissipation, Proyecciones (Universidad Catolica del Norte), 23 (2004), 205-234. 

[45]

M. L. Santos and F. junior, A boundary condition with memory for Kirchoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.  doi: 10.1016/S0096-3003(02)00915-3.

[46]

C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with with nonlinear damping, Annales de Toulouse, 8 (1999), 173-193.  doi: 10.5802/afst.928.

show all references

References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for the second order evalution equation with memory, J. Funct. Anal., 245 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

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.

[4]

K. Ammari and M. Tucsnak, Stabilization of Bernoulli -Euler beam by means of a pointwise feedback force, Siam J. Control Optim., 39 (2000), 1160-1181.  doi: 10.1137/S0363012998349315.

[5]

N. E. Amroun and A. Benaissa, Global existence and energy decay of solutions to a Petrovsky equation with general nonlinear dissipation and source term, Georgian Math. J., 13 (2006), 397-410. 

[6]

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

[7]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal. Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.

[8]

M. M. CavalcantiV. N. Domingos CavalcantiI. Lasiecka and F. A. Falco Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.

[9]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705-731.  doi: 10.1142/S0219199704001483.

[10]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[11]

J. Ferreira and S. A. Messaoudi, On the general decay of a nonlinear viscoelastic plate equation with a strong damping and $\overrightarrow{p}(x, t)$-Laplacian, Nonlinear Anal., 104 (2014), 40-49.  doi: 10.1016/j.na.2014.03.010.

[12]

A. Guesmia and S. A. Messaoudi, General energy decay estemates of Timoshenko system with frictional versus vescolastic damping, Math. Methods Appl. Sci., 32 (2009), 2102-2122.  doi: 10.1002/mma.1125.

[13]

R. B. Guzman and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation, Systems Control Lett., 48 (2003), 191-197.  doi: 10.1016/S0167-6911(02)00264-5.

[14]

X. Han and M. Wang, Energy decay rate for a coupled hyperbolic system with nonlinear damping, Nonlinear Anal., 70 (2009), 3264-3272.  doi: 10.1016/j.na.2008.04.029.

[15]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equation with memory, Acta Appl. Math., 110 (2010), 195-207.  doi: 10.1007/s10440-008-9397-x.

[16]

M. A. Horn, Uniform decay rates for the solution to the Euler Bernoulli plate equation with boundary feedback via benoling moments, Differential Integral Equation, 6 (1992), 1121-1150. 

[17]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plates with dissipation acting only via moments-limiting behavior, J. Math. Anal. Appl., 229 (1999), 452-479.  doi: 10.1006/jmaa.1998.6170.

[18]

J. R. Kang, Energy decay rates for von Karman system with memory and boundary feedback, Appl. Math. Comput., 218 (2012), 9085-9094.  doi: 10.1016/j.amc.2012.02.053.

[19]

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

[20]

V. Komornik, Decay estimates for a petrovski system with a nonlinear distributed feedback, IMA Preprints Series, 1992.

[21]

V. Komornik, On the nonlinear boundary stabilization of Kirchhoff plates, Nonlinear Differential Equation Appl., 1 (1994), 323-337.  doi: 10.1007/BF01194984.

[22]

J. E. Lagnese, Asymptotic energy estimates for Kirchoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, birkhcauser-Verlag, Bassel, 91 (1989), 211-236.

[23]

J. E. Lagnese, Boundary Stabilization of Thin Plates SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[24]

I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pures Appl., 78 (1999), 203-232.  doi: 10.1016/S0021-7824(01)80009-X.

[25]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R.

[26]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory J. Math. Phys. 54 (2013), 031504, 18pp. doi: 10.1063/1.4793988.

[27]

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

[28]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 271-303. doi: 10.1007/978-3-319-11406-4_1.

[29]

W. J. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms J. Math. Phys. 50 (2009), 113506, 17pp. doi: 10.1063/1.3254323.

[30]

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

[31]

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

[32]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.

[33]

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.

[34]

S. A. Messaoudi, On the control of solutions of a viscoelastic equations, J. Franklin Inst., 334 (2007), 765-776.  doi: 10.1016/j.jfranklin.2006.02.029.

[35]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dynam. Systems Appl., 18 (2009), 457-468. 

[36]

S. A. Messaoudi and M. I. Mustafa, On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Anal., 72 (2010), 3602-3611.  doi: 10.1016/j.na.2009.12.040.

[37]

J. E. Munoz RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[38]

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. 

[39]

J. E. Munoz RiveraM. 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. 

[40]

M. I. Mustafa, Uniform decay for wave equations with weakly dissipative boundary feedback, Dyn. Syst., 30 (2015), 241-250.  doi: 10.1080/14689367.2014.1002455.

[41]

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

[42]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. 

[43]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscolastic wave equatios J. Math. Phys. 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830.

[44]

A. F. PazotoL. Colho and R. C. Charao, Uniform stabilization of a plate equation with nonlinear localized dissipation, Proyecciones (Universidad Catolica del Norte), 23 (2004), 205-234. 

[45]

M. L. Santos and F. junior, A boundary condition with memory for Kirchoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.  doi: 10.1016/S0096-3003(02)00915-3.

[46]

C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with with nonlinear damping, Annales de Toulouse, 8 (1999), 173-193.  doi: 10.5802/afst.928.

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