# American Institute of Mathematical Sciences

May  2016, 36(5): 2419-2447. doi: 10.3934/dcds.2016.36.2419

## Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient

 1 Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes 3900, Ribeirão Preto, SP 14040-901, Brazil 2 Department of Mathematics, Federal University of Santa Catarina, Campus Trindade, Florianópolis, SC 88040-900, Brazil 3 Department of Mathematics, Federal University of Santa Catarina, Campus Trindade, Florianopolis, SC 88040-900, Brazil

Received  April 2015 Revised  September 2015 Published  October 2015

In this work we study decay rates for a hyperbolic plate equation under effects of an intermediate damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient. We obtain decay rates with very general conditions on the time-dependent coefficient (Theorem 2.1, Section 2), for the power fractional exponent of the Laplace operator $(-\Delta)^\theta$, in the damping term, $\theta \in [0,1]$. For the special time-dependent coefficient $b(t)=\mu (1+t)^{\alpha}$, $\alpha \in (0,1]$, we get optimal decay rates (Theorem 3.1, Section 3).
Citation: Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419
##### References:
 [1] D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms,, Math. Methods Appl. Sci., 35 (2012), 417. doi: 10.1002/mma.1552. Google Scholar [2] M. A. Astaburuaga, C. Fernandez and G. Perla Menzala, Energy decay rates and the dynamical von Kármán equations,, Appl. Math. Lett., 7 (1994), 7. doi: 10.1016/0893-9659(94)90021-3. Google Scholar [3] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of the American Mathematical Society, 195 (2008). doi: 10.1090/memo/0912. Google Scholar [4] P. G. Ciarlet, A justification of the von Kármán equations,, Arch. Rational Mech. Anal., 73 (1980), 349. doi: 10.1007/BF00247674. Google Scholar [5] R. Coimbra Charão, C. R. da Luz and R. Ikehata, Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space,, J. Math. Anal. Appl., 408 (2013), 247. doi: 10.1016/j.jmaa.2013.06.016. Google Scholar [6] R. Coimbra Charão, C. R. da Luz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping,, J. Hyperbolic Differ. Equ., 10 (2013), 563. doi: 10.1142/S0219891613500203. Google Scholar [7] R. Coimbra Charão and R. Ikehata, Energy decay rates of elastic waves in unbounded domain with potential type of damping,, J. Math. Anal. Appl., 380 (2011), 46. doi: 10.1016/j.jmaa.2011.02.075. Google Scholar [8] C. R. da Luz and R. Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains,, J. Hyperbolic Differ. Equ., 6 (2009), 269. doi: 10.1142/S0219891609001824. Google Scholar [9] C. R. da Luz, R. Ikehata and R. Coimbra Charão, Asymptotic behavior for abstract evolution differential equations of second order,, J. Differential Equations, 259 (2015), 5017. doi: 10.1016/j.jde.2015.06.012. Google Scholar [10] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, Math. Methods Appl. Sci., 38 (2015), 1032. doi: 10.1002/mma.3126. Google Scholar [11] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework,, J. Differential Equations, 256 (2014), 2307. doi: 10.1016/j.jde.2014.01.002. Google Scholar [12] M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators,, Math. Methods in the Appl. Sci., (2015). doi: 10.1002/mma.3713. Google Scholar [13] M. D'Abbicco and E. Jannelli, A damping term for higher-order hyperbolic equations,, Ann. Mat. Pura ed Appl., (2015), 10231. doi: 10.1007/s10231-015-0477-z. Google Scholar [14] M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping,, Chin. Ann. Math. Ser. B, 34 (2013), 345. doi: 10.1007/s11401-013-0773-0. Google Scholar [15] R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet-Neumann boundary conditions,, J. Differential Equations, 259 (2015), 1323. doi: 10.1016/j.jde.2015.02.043. Google Scholar [16] D. Fang, X. Lu and M. Reissig, High-order energy decay for structural damped systems in the electromagnetical field,, Chin. Ann. Math. Ser. B, 31 (2010), 237. doi: 10.1007/s11401-008-0185-8. Google Scholar [17] P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Kármán plates with geometrically localized dissipation and critical nonlinearity,, Nonlinear Analysis, 91 (2013), 72. doi: 10.1016/j.na.2013.06.008. Google Scholar [18] R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping,, Nonlinear Anal., 69 (2008), 1396. doi: 10.1016/j.na.2007.06.039. Google Scholar [19] R. Ikehata and M. Soga, Asymptotic profiles for a strongly damped plate equation with lower order perturbation,, Commun. Pure Appl. Anal., 14 (2015), 1759. doi: 10.3934/cpaa.2015.14.1759. Google Scholar [20] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential,, J. Math. Soc. Japan, 65 (2013), 183. doi: 10.2969/jmsj/06510183. Google Scholar [21] M. Kainane and M. Reissig, Qualitative properties of solution to structurally damped $\sigma-$evolution models with time decreasing coefficient in the dissipation,, in Complex Analysis and Dynamical Systems VI, (2015). Google Scholar [22] M. Kainane and M. Reissig, Qualitative properties of solution to structurally damped $\sigma-$evolution models with time increasing coefficient in the dissipation,, Adv. Differential Equations, 20 (2015), 433. Google Scholar [23] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations,, Studia Math., 143 (2000), 175. Google Scholar [24] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamical elasticity - full von Kármán systems,, Progress Nonlin. Diff. Eqs. Appl., 50 (2002), 197. Google Scholar [25] I. Lasiecka, Uniform stability of a full von Kármán system with nonlinear boundary feedbach},, SIAM J. Control Optim., 36 (1998), 1376. doi: 10.1137/S0363012996301907. Google Scholar [26] I. Lasiecka and A. Benabdallah, Exponential decay rates for a full von Kármán thermoelasticity system with nonlinear thermal coupling,, ESAIM: Control, 8 (2000), 13. doi: 10.1051/proc:2000002. Google Scholar [27] J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete Contin. Dyn. Syst., 32 (2012), 4307. doi: 10.3934/dcds.2012.32.4307. Google Scholar [28] J. R. Luyo Sánchez, O Sistema Dinâmico de Von Kármán em Domínios Não Limitados é Globalmente Bem Posto no Sentido de Hadamard: Análise do seu Limite Singular (Portuguese),, Ph.D thesis, (2003). Google Scholar [29] Q. Ma, Y. Yang and X. Zhang, Existence of exponential attractors for the plate equations with strong damping,, Electron. J. Differential Equations, (2013), 1. Google Scholar [30] P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIM Control Optim. Calc. Var., 4 (1999), 419. doi: 10.1051/cocv:1999116. Google Scholar [31] S. Matthes and M. Reissig, Qualitative properties of structurally damped wave models,, Eurasian Math. J., 4 (2013), 84. Google Scholar [32] G. P. Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dinamical von Kármán system,, J. Math. Pures Appl., 79 (2000), 73. doi: 10.1016/S0021-7824(00)00149-5. Google Scholar [33] J. P. Puel and M. Tucsnak, Global existence for full von Kármán system,, Appl. Math. Optim., 34 (1996), 139. doi: 10.1007/BF01182621. Google Scholar [34] R. Schnaubelt and M. Veraar, Structurally damped plate and wave equations with random point force in arbitrary space dimensions,, Differential Integral Equations, 23 (2010), 957. Google Scholar [35] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differ. Equ., 7 (2010), 471. doi: 10.1142/S0219891610002207. Google Scholar [36] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979). Google Scholar [37] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients,, Ph.D thesis, (2014). Google Scholar [38] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping,, in Fourier Analysis, (2013), 375. doi: 10.1007/978-3-319-02550-6_19. Google Scholar [39] Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation,, J. Appl. Math., (2013). Google Scholar [40] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations, 232 (2007), 74. doi: 10.1016/j.jde.2006.06.004. Google Scholar [41] L. Xu and Q. Ma, Existence of random attractors for the floating beam equation with strong damping and white noise,, Bound. Value Probl., 2015 (2015), 13661. doi: 10.1186/s13661-015-0391-8. Google Scholar [42] K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach,, Mathematical Topics, (1997). Google Scholar

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##### References:
 [1] D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms,, Math. Methods Appl. Sci., 35 (2012), 417. doi: 10.1002/mma.1552. Google Scholar [2] M. A. Astaburuaga, C. Fernandez and G. Perla Menzala, Energy decay rates and the dynamical von Kármán equations,, Appl. Math. Lett., 7 (1994), 7. doi: 10.1016/0893-9659(94)90021-3. Google Scholar [3] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Memoirs of the American Mathematical Society, 195 (2008). doi: 10.1090/memo/0912. Google Scholar [4] P. G. Ciarlet, A justification of the von Kármán equations,, Arch. Rational Mech. Anal., 73 (1980), 349. doi: 10.1007/BF00247674. Google Scholar [5] R. Coimbra Charão, C. R. da Luz and R. Ikehata, Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space,, J. Math. Anal. Appl., 408 (2013), 247. doi: 10.1016/j.jmaa.2013.06.016. Google Scholar [6] R. Coimbra Charão, C. R. da Luz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping,, J. Hyperbolic Differ. Equ., 10 (2013), 563. doi: 10.1142/S0219891613500203. Google Scholar [7] R. Coimbra Charão and R. Ikehata, Energy decay rates of elastic waves in unbounded domain with potential type of damping,, J. Math. Anal. Appl., 380 (2011), 46. doi: 10.1016/j.jmaa.2011.02.075. Google Scholar [8] C. R. da Luz and R. Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains,, J. Hyperbolic Differ. Equ., 6 (2009), 269. doi: 10.1142/S0219891609001824. Google Scholar [9] C. R. da Luz, R. Ikehata and R. Coimbra Charão, Asymptotic behavior for abstract evolution differential equations of second order,, J. Differential Equations, 259 (2015), 5017. doi: 10.1016/j.jde.2015.06.012. Google Scholar [10] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, Math. Methods Appl. Sci., 38 (2015), 1032. doi: 10.1002/mma.3126. Google Scholar [11] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework,, J. Differential Equations, 256 (2014), 2307. doi: 10.1016/j.jde.2014.01.002. Google Scholar [12] M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators,, Math. Methods in the Appl. Sci., (2015). doi: 10.1002/mma.3713. Google Scholar [13] M. D'Abbicco and E. Jannelli, A damping term for higher-order hyperbolic equations,, Ann. Mat. Pura ed Appl., (2015), 10231. doi: 10.1007/s10231-015-0477-z. Google Scholar [14] M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping,, Chin. Ann. Math. Ser. B, 34 (2013), 345. doi: 10.1007/s11401-013-0773-0. Google Scholar [15] R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet-Neumann boundary conditions,, J. Differential Equations, 259 (2015), 1323. doi: 10.1016/j.jde.2015.02.043. Google Scholar [16] D. Fang, X. Lu and M. Reissig, High-order energy decay for structural damped systems in the electromagnetical field,, Chin. Ann. Math. Ser. B, 31 (2010), 237. doi: 10.1007/s11401-008-0185-8. Google Scholar [17] P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Kármán plates with geometrically localized dissipation and critical nonlinearity,, Nonlinear Analysis, 91 (2013), 72. doi: 10.1016/j.na.2013.06.008. Google Scholar [18] R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping,, Nonlinear Anal., 69 (2008), 1396. doi: 10.1016/j.na.2007.06.039. Google Scholar [19] R. Ikehata and M. Soga, Asymptotic profiles for a strongly damped plate equation with lower order perturbation,, Commun. Pure Appl. Anal., 14 (2015), 1759. doi: 10.3934/cpaa.2015.14.1759. Google Scholar [20] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential,, J. Math. Soc. Japan, 65 (2013), 183. doi: 10.2969/jmsj/06510183. Google Scholar [21] M. Kainane and M. Reissig, Qualitative properties of solution to structurally damped $\sigma-$evolution models with time decreasing coefficient in the dissipation,, in Complex Analysis and Dynamical Systems VI, (2015). Google Scholar [22] M. Kainane and M. Reissig, Qualitative properties of solution to structurally damped $\sigma-$evolution models with time increasing coefficient in the dissipation,, Adv. Differential Equations, 20 (2015), 433. Google Scholar [23] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations,, Studia Math., 143 (2000), 175. Google Scholar [24] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamical elasticity - full von Kármán systems,, Progress Nonlin. Diff. Eqs. Appl., 50 (2002), 197. Google Scholar [25] I. Lasiecka, Uniform stability of a full von Kármán system with nonlinear boundary feedbach},, SIAM J. Control Optim., 36 (1998), 1376. doi: 10.1137/S0363012996301907. Google Scholar [26] I. Lasiecka and A. Benabdallah, Exponential decay rates for a full von Kármán thermoelasticity system with nonlinear thermal coupling,, ESAIM: Control, 8 (2000), 13. doi: 10.1051/proc:2000002. Google Scholar [27] J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete Contin. Dyn. Syst., 32 (2012), 4307. doi: 10.3934/dcds.2012.32.4307. Google Scholar [28] J. R. Luyo Sánchez, O Sistema Dinâmico de Von Kármán em Domínios Não Limitados é Globalmente Bem Posto no Sentido de Hadamard: Análise do seu Limite Singular (Portuguese),, Ph.D thesis, (2003). Google Scholar [29] Q. Ma, Y. Yang and X. Zhang, Existence of exponential attractors for the plate equations with strong damping,, Electron. J. Differential Equations, (2013), 1. Google Scholar [30] P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIM Control Optim. Calc. Var., 4 (1999), 419. doi: 10.1051/cocv:1999116. Google Scholar [31] S. Matthes and M. Reissig, Qualitative properties of structurally damped wave models,, Eurasian Math. J., 4 (2013), 84. Google Scholar [32] G. P. Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dinamical von Kármán system,, J. Math. Pures Appl., 79 (2000), 73. doi: 10.1016/S0021-7824(00)00149-5. Google Scholar [33] J. P. Puel and M. Tucsnak, Global existence for full von Kármán system,, Appl. Math. Optim., 34 (1996), 139. doi: 10.1007/BF01182621. Google Scholar [34] R. Schnaubelt and M. Veraar, Structurally damped plate and wave equations with random point force in arbitrary space dimensions,, Differential Integral Equations, 23 (2010), 957. Google Scholar [35] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation,, J. Hyperbolic Differ. Equ., 7 (2010), 471. doi: 10.1142/S0219891610002207. Google Scholar [36] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979). Google Scholar [37] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients,, Ph.D thesis, (2014). Google Scholar [38] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping,, in Fourier Analysis, (2013), 375. doi: 10.1007/978-3-319-02550-6_19. Google Scholar [39] Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation,, J. Appl. Math., (2013). Google Scholar [40] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations, 232 (2007), 74. doi: 10.1016/j.jde.2006.06.004. Google Scholar [41] L. Xu and Q. Ma, Existence of random attractors for the floating beam equation with strong damping and white noise,, Bound. Value Probl., 2015 (2015), 13661. doi: 10.1186/s13661-015-0391-8. Google Scholar [42] K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach,, Mathematical Topics, (1997). Google Scholar
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