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Optimal decay rate estimates of a nonlinear viscoelastic Kirchhoff plate. (English) Zbl 1441.35050
Summary: This paper is concerned with a nonlinear viscoelastic Kirchhoff plate $$u_{t t}\left( t\right)-\sigma\Delta u_{t t}\left( t\right)+ \Delta^2u\left( t\right)-{\displaystyle \int_0^t g \left( t - s\right) \Delta^2 u \left( s\right) \text{d} s}=\text{div}F\left( \nabla u \left( t\right)\right).$$ By assuming the minimal conditions on the relaxation function $$g$$: $$g^\prime\left( t\right)\leq\xi\left( t\right)G\left( g \left( t\right)\right)$$, where $$G$$ is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for $$G\left( t\right)=tp$$ with the range $$p\in\left[ 1, 2\right)$$, which improves earlier decay results with the range $$p\in\left[ 1, 3 / 2\right)$$. At last, we give some numerical illustrations and related comparisons.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35L76 Higher-order semilinear hyperbolic equations 74K20 Plates 35R09 Integral partial differential equations
##### Keywords:
optimal energy decay estimate
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##### References:
 [1] Cabanillas, E. L.; Muñoz Rivera, J. E., Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Communications in Mathematical Physics, 177, 3, 583-602 (1996) · Zbl 0852.73026 [2] Muñoz Rivera, J. E., Asymptotic behaviour in linear viscoelasticity, Quarterly of Applied Mathematics, 52, 4, 628-648 (1994) · Zbl 0814.35009 [3] Muñoz Rivera, J. E.; Peres Salvatierra, A., Asymptotic behaviour of the energy in partially viscoelastic materials, Quarterly of Applied Mathematics, 59, 3, 557-578 (2001) · Zbl 1028.35025 [4] Santos, M. L., Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electronic Journal of Differential Equations, 73, 1-11 (2001) [5] Cavalcanti, M. M.; Oquendo, H. P., Frictional versus viscoelastic damping in a semilinear wave equation, SIAM Journal on Control and Optimization, 42, 4, 1310-1324 (2003) · Zbl 1053.35101 [6] Messaoudi, S. A., General decay of solutions of a viscoelastic equation, Journal of Mathematical Analysis and Applications, 341, 2, 1457-1467 (2008) · Zbl 1145.35078 [7] Messaoudi, S. A., General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis: Theory, Methods & Applications, 69, 8, 2589-2598 (2008) · Zbl 1154.35066 [8] Han, X.; Wang, M., General decay of energy for a viscoelastic equation with nonlinear damping, Mathematical Methods in the Applied Sciences, 32, 3, 346-358 (2009) · Zbl 1161.35319 [9] Liu, W. J., General decay of solutions to a viscoelastic wave equation with nonlinear localized damping, Annales Academiae Scientiarum Fennicae Mathematica, 34, 291-302 (2009) · Zbl 1200.35029 [10] Liu, W., General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, Journal of Mathematical Physics, 50, 11, 113506 (2009) · Zbl 1304.35438 [11] Messaoud, S. A.; Mustafa, M. I., On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Analysis: Real World Applications, 10, 5, 3132-3140 (2009) · Zbl 1162.74030 [12] Mustafa, M. I., Uniform decay rates for viscoelastic dissipative systems, Journal of Dynamical and Control Systems, 22, 1, 101-116 (2016) · Zbl 1336.35063 [13] Park, J.; Park, S., General decay for quasilinear viscoelastic equations with nonlinear weak damping, Journal of Mathematical Physics, 50 (2009) · Zbl 1298.35221 [14] Lasiecka, I.; Tataru, D., Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 8, 507-533 (1993) · Zbl 0803.35088 [15] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Lasiecka, I.; Nascimento, F. A. F., Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete & Continuous Dynamical Systems—B, 19, 7, 1987-2011 (2014) · Zbl 1326.35041 [16] Cavalcanti, M. M.; Cavalcanti, A. D. D.; Lasiecka, I.; Wang, X., Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Analysis: Real World Applications, 22, 289-306 (2015) · Zbl 1326.35040 [17] Lasiecka, I.; Messaoudi, S. A.; Mustafa, M. I., Note on intrinsic decay rates for abstract wave equations with memory, Journal of Mathematical Physics, 54, 3 (2013) · Zbl 1282.74018 [18] Lasiecka, I.; Wang, X., Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, 10, 271-303 (2014), Cham, Switzerland: Springer, Cham, Switzerland · Zbl 06848482 [19] Mustafa, M. I., On the control of the wave equation by memory-type boundary condition, Discrete & Continuous Dynamical Systems—A, 35, 3, 1179-1192 (2015) · Zbl 1304.35719 [20] Mustafa, M. I.; Messaoudi, S. A., General stability result for viscoelastic wave equations, Journal of Mathematical Physics, 53 (2012) · Zbl 1276.76009 [21] Xiao, T.-J.; Liang, J., Coupled second order semilinear evolution equations indirectly damped via memory effects, Journal of Differential Equations, 254, 5, 2128-2157 (2013) · Zbl 1264.34121 [22] Mustafa, M. I., Optimal decay rates for the viscoelastic wave equation, Mathematical Methods in the Applied Sciences, 41, 1, 192-204 (2018) · Zbl 1391.35058 [23] Mustafa, M. I., General decay result for nonlinear viscoelastic equations, Journal of Mathematical Analysis and Applications, 457, 1, 134-152 (2018) · Zbl 1379.35028 [24] Rivera, J. E. M. o.; Lapa, E. C.; Barreto, R., Decay rates for viscoelastic plates with memory, Journal of Elasticity, 44, 1, 61-87 (1996) · Zbl 0876.73037 [25] Alabau-Boussouira, F.; Cannarsa, P.; Sforza, D., Decay estimates for second order evolution equations with memory, Journal of Functional Analysis, 254, 5, 1342-1372 (2008) · Zbl 1145.35025 [26] Cavalcanti, M. M., Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation, Discrete & Continuous Dynamical Systems—A, 8, 3, 675-695 (2012) · Zbl 1009.74034 [27] Andrade, D.; Jorge Silva, M. A.; Ma, T. F., Exponential stability for a plate equation with p-Laplacian and memory terms, Mathematical Methods in the Applied Sciences, 35, 4, 417-426 (2012) · Zbl 1235.35198 [28] Ferreira, J.; Messaoudi, S. A., On the general decay of a nonlinear viscoelastic plate equation with a strong damping and -Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 104, 40-49 (2014) · Zbl 1396.35062 [29] Feng, B., Global well-posedness and stability for a viscoelastic plate equation with a time delay, Mathematical Problems in Engineering, 2015 (2015) · Zbl 1394.35535 [30] Gomes Tavares, E. H.; Jorge Silva, M. A.; Ma, T. F., Sharp decay rates for a class of nonlinear viscoelastic plate models, Communications in Contemporary Mathematics, 20, 02, 1750010 (2018) · Zbl 1375.35541 [31] Silva, M. A. J.; Rivera, J. E. M.; Racke, R., On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates, Applied Mathematics & Optimization, 73, 1, 165-194 (2016) · Zbl 1339.35044 [32] Feng, B., Well-posedness and exponential stability for a plate equation with time-varying delay and past history, Zeitschrift für angewandte Mathematik und Physik, 68, 1, 26 (2017) · Zbl 1408.35183 [33] Feng, B., Long-time dynamics of a plate equation with memory and time delay, Bulletin of the Brazilian Mathematical Society, New Series, 49, 2, 395-418 (2018) · Zbl 1400.35034 [34] Feng, B.; Liu, G., Well-posedness and stability of two classes of plate equations with memory and strong time-dependent delay, Taiwanese Journal of Mathematics, 23, 1, 159-192 (2019) · Zbl 1415.35210 [35] Feng, B.; Jorge Silva, M. A.; Caixeta, A. H., Long-time behavior for a class of semi-linear viscoelastic Kirchhoff beams/plates, Applied Mathematics & Optimization (2018) [36] Gomes Tavares, E. H.; Jorge Silva, M. A.; Narciso, V., On a decay rate for nonlinear extensible viscoelastic beams with history setting, Applicable Analysis, 97, 11, 1916-1932 (2018) · Zbl 1395.35031 [37] Jorge Silva, M. A.; Ma, T. F., On a viscoelastic plate equation with history setting and pertubation of p-Laplacian type, IMA Journal of Applied Mathematics, 78, 6, 1130-1146 (2013) · Zbl 1282.35372 [38] Jorge Silva, M. A.; Ma, T. F., Long-time dynamics for a class of Kirchhoff models with memory, Journal of Mathematical Physics, 54 (2013) · Zbl 1290.35154 [39] Alabau-Boussouira, F.; Cannarsa, P., A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Mathematique, 347, 15-16, 867-872 (2009) · Zbl 1179.35058 [40] Arnold, V. I., Mathematical Methods of Classical Mechanics (1989), New York, NY, USA: Springer-Verlag, New York, NY, USA [41] Afilal, M.; Guesmia, A.; Soufyane, A.; Zahri, M., On the exponential and polynomial stability for a linear Bresse system, to appear in Math, Mathematical Methods in the Applied Sciences, 43, 5, 2626-2645 (2019) [42] Hassan, J. H.; Messaoudi, S. A.; Zahri, M., Existence and New General Decay Results for a Viscoelastic-type Timoshenko System, to appear in Zeitchrift für Analysis und ihre Anwendungen (2019)
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