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March  2020, 40(3): 1757-1774. doi: 10.3934/dcds.2020092

New general decay results for a von Karman plate equation with memory-type boundary conditions

1. 

Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

2. 

Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, UAE

* Corresponding author: Abdelaziz Soufyane

Received  June 2019 Revised  October 2019 Published  December 2019

Fund Project: Baowei Feng has been supported by the National Natural Science Foundation of China grant 11701465. Abdelaziz Soufyane has been supported by University of Sharjah grant 1802144069

In this paper we consider a von Karman plate equation with memory-type boundary conditions. By assuming the relaxation function $ k_i $ $ (i = 1, 2) $ with minimal conditions on the $ L^1(0, \infty) $, we establish an optimal explicit and general energy decay result. In particular, the energy result holds for $ H(s) = s^p $ with the full admissible range $ [1, 2) $ instead of $ [1, 3/2) $. This result is new and substantially improves earlier results in the literature.

Citation: Baowei Feng, Abdelaziz Soufyane. New general decay results for a von Karman plate equation with memory-type boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1757-1774. doi: 10.3934/dcds.2020092
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 Ser. I, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

M. E. Bradley and I. Lasiecka, Global decay rates for the solutions to a von Karman plate without geometric conditions, J. Math. Anal. Appl., 181 (1994), 254-276.  doi: 10.1006/jmaa.1994.1019.  Google Scholar

[4]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. J. 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

[5]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differential Equations, 198 (2004), 196-231.  doi: 10.1016/j.jde.2003.08.008.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

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A. FaviniM. A. HornI. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Differ. Inte. Equa., 9 (1996), 267-294.   Google Scholar

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M. A. Jorge SilvaJ. E. Muñoz Rivera and R. Racke, On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates, Appl. Math. Optim., 73 (2016), 165-194.  doi: 10.1007/s00245-015-9298-0.  Google Scholar

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M. A. Jorge Silva, E. H. Gomes Tavares and T. F. Ma, Sharp decay rates for a class of nonlinear viscoelastic plate models, Comm. Contem. Math., 20 (2018), 1750010, 21 pp. doi: 10.1142/S0219199717500109.  Google Scholar

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J.-R. Kang, Exponential decay for a von Karman equation of memory type with acoustic boundary conditions, Math. Methods Appl. Sci., 38 (2015), 598-608.  doi: 10.1002/mma.3090.  Google Scholar

[11]

D. KimJ. Y. Park and Y. H. Kang, Energy decay rate for a von Karman system with a boundary nonlinear delay term, Comput. Math. Appl., 75 (2018), 3269-3282.  doi: 10.1016/j.camwa.2018.01.046.  Google Scholar

[12]

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

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J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 91 (1989), 211-236.   Google Scholar

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[15]

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

[16]

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

[17]

I. Lasiecka and X. J. 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 Series, Cham: Springer, 10 (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

[18]

J. E. Muñoz 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

[19]

J. E. Muñoz RiveraH. P. Oquendo and M. L. Santos, Asymptotic behavior to a von Karman plate with boundary memory conditions, Nonlinear Anal., 62 (2005), 1183-1205.  doi: 10.1016/j.na.2005.04.025.  Google Scholar

[20]

J. E. Muñoz RiveraA. Soufyane and M. L. Santos, General decay for full von Karman system with memory, Nonlinear Anal.: Real World Appl., 13 (2012), 2633-2647.  doi: 10.1016/j.nonrwa.2012.03.008.  Google Scholar

[21]

M. I. Mustafa, The control of Timoshenko beams by memory-type boundary conditions, Appl. Anal., (2019), 12 pp. doi: 10.1080/00036811.2019.1602724.  Google Scholar

[22]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.  Google Scholar

[23]

M. I. Mustafa and G. A. Abusharkh, Plate equations with viscoelastic boundary damping, Indag. Math. (N.S.), 26 (2015), 307-323.  doi: 10.1016/j.indag.2014.09.005.  Google Scholar

[24]

S.-H. Park, General decay of a von Karman plate equation with memory on the boundary, Comput. Math. Appl., 75 (2018), 3067-3080.  doi: 10.1016/j.camwa.2018.01.032.  Google Scholar

[25]

S.-H. Park, Arbitrary decay rates of energy for a von Karman equation of memory type, Comput. Math. Appl., 70 (2015), 1878-1886.  doi: 10.1016/j.camwa.2015.08.005.  Google Scholar

[26]

S. H. ParkJ. Y. Park and Y. H. Kang, General decay for a von Karman equation of memory type with acoustic boundary conditions, Z. Angew. Math. Phys., 63 (2012), 813-823.  doi: 10.1007/s00033-011-0188-2.  Google Scholar

[27]

J. Y. Park and S. H. Park, Uniform decay for a von Karman plate equation with a boundary memory condition, Math. Methods Appl. Sci., 28 (2005), 2225-2240.  doi: 10.1002/mma.663.  Google Scholar

[28]

C. A. Raposo and M. L. Santos, General decay to a von Karman system with memory, Nonlinear Anal., 74 (2011), 937-945.  doi: 10.1016/j.na.2010.09.047.  Google Scholar

[29]

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

[30]

M. L. Santos and A. Soufyane, General decay to a von Karman plate system with memory boundary conditions, Differ. Inte. Equa., 24 (2011), 69-81.   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 Ser. I, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

M. E. Bradley and I. Lasiecka, Global decay rates for the solutions to a von Karman plate without geometric conditions, J. Math. Anal. Appl., 181 (1994), 254-276.  doi: 10.1006/jmaa.1994.1019.  Google Scholar

[4]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. J. 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

[5]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differential Equations, 198 (2004), 196-231.  doi: 10.1016/j.jde.2003.08.008.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[7]

A. FaviniM. A. HornI. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Differ. Inte. Equa., 9 (1996), 267-294.   Google Scholar

[8]

M. A. Jorge SilvaJ. E. Muñoz Rivera and R. Racke, On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates, Appl. Math. Optim., 73 (2016), 165-194.  doi: 10.1007/s00245-015-9298-0.  Google Scholar

[9]

M. A. Jorge Silva, E. H. Gomes Tavares and T. F. Ma, Sharp decay rates for a class of nonlinear viscoelastic plate models, Comm. Contem. Math., 20 (2018), 1750010, 21 pp. doi: 10.1142/S0219199717500109.  Google Scholar

[10]

J.-R. Kang, Exponential decay for a von Karman equation of memory type with acoustic boundary conditions, Math. Methods Appl. Sci., 38 (2015), 598-608.  doi: 10.1002/mma.3090.  Google Scholar

[11]

D. KimJ. Y. Park and Y. H. Kang, Energy decay rate for a von Karman system with a boundary nonlinear delay term, Comput. Math. Appl., 75 (2018), 3269-3282.  doi: 10.1016/j.camwa.2018.01.046.  Google Scholar

[12]

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

[13]

J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 91 (1989), 211-236.   Google Scholar

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[15]

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

[16]

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

[17]

I. Lasiecka and X. J. 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 Series, Cham: Springer, 10 (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

[18]

J. E. Muñoz 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

[19]

J. E. Muñoz RiveraH. P. Oquendo and M. L. Santos, Asymptotic behavior to a von Karman plate with boundary memory conditions, Nonlinear Anal., 62 (2005), 1183-1205.  doi: 10.1016/j.na.2005.04.025.  Google Scholar

[20]

J. E. Muñoz RiveraA. Soufyane and M. L. Santos, General decay for full von Karman system with memory, Nonlinear Anal.: Real World Appl., 13 (2012), 2633-2647.  doi: 10.1016/j.nonrwa.2012.03.008.  Google Scholar

[21]

M. I. Mustafa, The control of Timoshenko beams by memory-type boundary conditions, Appl. Anal., (2019), 12 pp. doi: 10.1080/00036811.2019.1602724.  Google Scholar

[22]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.  Google Scholar

[23]

M. I. Mustafa and G. A. Abusharkh, Plate equations with viscoelastic boundary damping, Indag. Math. (N.S.), 26 (2015), 307-323.  doi: 10.1016/j.indag.2014.09.005.  Google Scholar

[24]

S.-H. Park, General decay of a von Karman plate equation with memory on the boundary, Comput. Math. Appl., 75 (2018), 3067-3080.  doi: 10.1016/j.camwa.2018.01.032.  Google Scholar

[25]

S.-H. Park, Arbitrary decay rates of energy for a von Karman equation of memory type, Comput. Math. Appl., 70 (2015), 1878-1886.  doi: 10.1016/j.camwa.2015.08.005.  Google Scholar

[26]

S. H. ParkJ. Y. Park and Y. H. Kang, General decay for a von Karman equation of memory type with acoustic boundary conditions, Z. Angew. Math. Phys., 63 (2012), 813-823.  doi: 10.1007/s00033-011-0188-2.  Google Scholar

[27]

J. Y. Park and S. H. Park, Uniform decay for a von Karman plate equation with a boundary memory condition, Math. Methods Appl. Sci., 28 (2005), 2225-2240.  doi: 10.1002/mma.663.  Google Scholar

[28]

C. A. Raposo and M. L. Santos, General decay to a von Karman system with memory, Nonlinear Anal., 74 (2011), 937-945.  doi: 10.1016/j.na.2010.09.047.  Google Scholar

[29]

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

[30]

M. L. Santos and A. Soufyane, General decay to a von Karman plate system with memory boundary conditions, Differ. Inte. Equa., 24 (2011), 69-81.   Google Scholar

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