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, Florianopolis, 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 and Continuous Dynamical Systems, 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), 417-426. doi: 10.1002/mma.1552.

[2]

M. A. Astaburuaga, C. Fernandez and G. Perla Menzala, Energy decay rates and the dynamical von Kármán equations, Appl. Math. Lett., 7 (1994), 7-10. doi: 10.1016/0893-9659(94)90021-3.

[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), viii+183 pp. doi: 10.1090/memo/0912.

[4]

P. G. Ciarlet, A justification of the von Kármán equations, Arch. Rational Mech. Anal., 73 (1980), 349-389. doi: 10.1007/BF00247674.

[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), 247-255. doi: 10.1016/j.jmaa.2013.06.016.

[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), 563-575. doi: 10.1142/S0219891613500203.

[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), 46-56. doi: 10.1016/j.jmaa.2011.02.075.

[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), 269-294. doi: 10.1142/S0219891609001824.

[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), 5017-5039. doi: 10.1016/j.jde.2015.06.012.

[10]

M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods Appl. Sci., 38 (2015), 1032-1045. doi: 10.1002/mma.3126.

[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), 2307-2336. doi: 10.1016/j.jde.2014.01.002.

[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. doi: 10.1002/mma.3713.

[13]

M. D'Abbicco and E. Jannelli, A damping term for higher-order hyperbolic equations, Ann. Mat. Pura ed Appl., (2015). DOI: 10.1007/s10231-015-0477-z. doi: 10.1007/s10231-015-0477-z.

[14]

M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping, Chin. Ann. Math. Ser. B, 34 (2013), 345-380. doi: 10.1007/s11401-013-0773-0.

[15]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet-Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353. doi: 10.1016/j.jde.2015.02.043.

[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), 237-246. doi: 10.1007/s11401-008-0185-8.

[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), 72-92. doi: 10.1016/j.na.2013.06.008.

[18]

R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping, Nonlinear Anal., 69 (2008), 1396-1401. doi: 10.1016/j.na.2007.06.039.

[19]

R. Ikehata and M. Soga, Asymptotic profiles for a strongly damped plate equation with lower order perturbation, Commun. Pure Appl. Anal., 14 (2015), 1759-1780. doi: 10.3934/cpaa.2015.14.1759.

[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), 183-236. doi: 10.2969/jmsj/06510183.

[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, Contemporary Mathematics, Amer. Math. Soc., 2015.

[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), 433-462.

[23]

G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.

[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), 197-216.

[25]

I. Lasiecka, Uniform stability of a full von Kármán system with nonlinear boundary feedbach}, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907.

[26]

I. Lasiecka and A. Benabdallah, Exponential decay rates for a full von Kármán thermoelasticity system with nonlinear thermal coupling, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2000), 13-38. doi: 10.1051/proc:2000002.

[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), 4307-4320. doi: 10.3934/dcds.2012.32.4307.

[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, Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 2003.

[29]

Q. Ma, Y. Yang and X. Zhang, Existence of exponential attractors for the plate equations with strong damping, Electron. J. Differential Equations, (2013), 1-10.

[30]

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.

[31]

S. Matthes and M. Reissig, Qualitative properties of structurally damped wave models, Eurasian Math. J., 4 (2013), 84-106.

[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), 73-94. doi: 10.1016/S0021-7824(00)00149-5.

[33]

J. P. Puel and M. Tucsnak, Global existence for full von Kármán system, Appl. Math. Optim., 34 (1996), 139-160. doi: 10.1007/BF01182621.

[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), 957-988.

[35]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.

[36]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[37]

Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Ph.D thesis, Osaka University, 2014.

[38]

Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, in Fourier Analysis, Trends in Mathematics, Springer, 2013, 375-390. doi: 10.1007/978-3-319-02550-6_19.

[39]

Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation, J. Appl. Math., (2013), Art. ID 353757, 8pp.

[40]

J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004.

[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), 126. DOI: 10.1186/s13661-015-0391-8. doi: 10.1186/s13661-015-0391-8.

[42]

K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach, Mathematical Topics, 12, Akademie Verlag, Berlin, 1997.

show all references

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), 417-426. doi: 10.1002/mma.1552.

[2]

M. A. Astaburuaga, C. Fernandez and G. Perla Menzala, Energy decay rates and the dynamical von Kármán equations, Appl. Math. Lett., 7 (1994), 7-10. doi: 10.1016/0893-9659(94)90021-3.

[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), viii+183 pp. doi: 10.1090/memo/0912.

[4]

P. G. Ciarlet, A justification of the von Kármán equations, Arch. Rational Mech. Anal., 73 (1980), 349-389. doi: 10.1007/BF00247674.

[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), 247-255. doi: 10.1016/j.jmaa.2013.06.016.

[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), 563-575. doi: 10.1142/S0219891613500203.

[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), 46-56. doi: 10.1016/j.jmaa.2011.02.075.

[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), 269-294. doi: 10.1142/S0219891609001824.

[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), 5017-5039. doi: 10.1016/j.jde.2015.06.012.

[10]

M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods Appl. Sci., 38 (2015), 1032-1045. doi: 10.1002/mma.3126.

[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), 2307-2336. doi: 10.1016/j.jde.2014.01.002.

[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. doi: 10.1002/mma.3713.

[13]

M. D'Abbicco and E. Jannelli, A damping term for higher-order hyperbolic equations, Ann. Mat. Pura ed Appl., (2015). DOI: 10.1007/s10231-015-0477-z. doi: 10.1007/s10231-015-0477-z.

[14]

M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping, Chin. Ann. Math. Ser. B, 34 (2013), 345-380. doi: 10.1007/s11401-013-0773-0.

[15]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet-Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353. doi: 10.1016/j.jde.2015.02.043.

[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), 237-246. doi: 10.1007/s11401-008-0185-8.

[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), 72-92. doi: 10.1016/j.na.2013.06.008.

[18]

R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping, Nonlinear Anal., 69 (2008), 1396-1401. doi: 10.1016/j.na.2007.06.039.

[19]

R. Ikehata and M. Soga, Asymptotic profiles for a strongly damped plate equation with lower order perturbation, Commun. Pure Appl. Anal., 14 (2015), 1759-1780. doi: 10.3934/cpaa.2015.14.1759.

[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), 183-236. doi: 10.2969/jmsj/06510183.

[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, Contemporary Mathematics, Amer. Math. Soc., 2015.

[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), 433-462.

[23]

G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.

[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), 197-216.

[25]

I. Lasiecka, Uniform stability of a full von Kármán system with nonlinear boundary feedbach}, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907.

[26]

I. Lasiecka and A. Benabdallah, Exponential decay rates for a full von Kármán thermoelasticity system with nonlinear thermal coupling, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2000), 13-38. doi: 10.1051/proc:2000002.

[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), 4307-4320. doi: 10.3934/dcds.2012.32.4307.

[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, Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 2003.

[29]

Q. Ma, Y. Yang and X. Zhang, Existence of exponential attractors for the plate equations with strong damping, Electron. J. Differential Equations, (2013), 1-10.

[30]

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.

[31]

S. Matthes and M. Reissig, Qualitative properties of structurally damped wave models, Eurasian Math. J., 4 (2013), 84-106.

[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), 73-94. doi: 10.1016/S0021-7824(00)00149-5.

[33]

J. P. Puel and M. Tucsnak, Global existence for full von Kármán system, Appl. Math. Optim., 34 (1996), 139-160. doi: 10.1007/BF01182621.

[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), 957-988.

[35]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.

[36]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[37]

Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Ph.D thesis, Osaka University, 2014.

[38]

Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, in Fourier Analysis, Trends in Mathematics, Springer, 2013, 375-390. doi: 10.1007/978-3-319-02550-6_19.

[39]

Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation, J. Appl. Math., (2013), Art. ID 353757, 8pp.

[40]

J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004.

[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), 126. DOI: 10.1186/s13661-015-0391-8. doi: 10.1186/s13661-015-0391-8.

[42]

K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach, Mathematical Topics, 12, Akademie Verlag, Berlin, 1997.

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