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: |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[4] |
M. M. Cavalcanti, A. D. D. Cavalcanti, I. 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[7] |
A. Favini, M. A. Horn, I. 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.
![]() ![]() |
[8] |
M. A. Jorge Silva, J. 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[11] |
D. Kim, J. 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.![]() ![]() ![]() |
[12] |
V. Komornik, On the nonlinear boundary stabilization of Kirchhoff plates, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 323-337.
doi: 10.1007/BF01194984.![]() ![]() ![]() |
[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.
![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[16] |
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Inte. Equa., 6 (1993), 507-533.
![]() ![]() |
[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.![]() ![]() ![]() |
[18] |
J. E. Muñoz Rivera, E. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.
doi: 10.1007/BF00042192.![]() ![]() ![]() |
[19] |
J. E. Muñoz Rivera, H. 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.![]() ![]() ![]() |
[20] |
J. E. Muñoz Rivera, A. 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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[26] |
S. H. Park, J. 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.
![]() ![]() |