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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407

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