September  2014, 19(7): 1987-2011. doi: 10.3934/dcdsb.2014.19.1987

Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects

1. 

Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil, Brazil

2. 

University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152

3. 

Department of Mathematics, State University of Ceará- FAFIDAM, 62930-000 Limoeiro do Norte - CE

Received  April 2013 Revised  September 2013 Published  August 2014

Wave equation defined on a compact Riemannian manifold $(M, \mathfrak{g})$ subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of $a(x)$ while the frictional damping affects the portion of the manifold quantified by the support of $b(x)$ where both $a(x)$ and $b(x)$ are smooth functions. Assuming that $a(x) + b(x) \geq \delta >0 $ for all $x\in M$ and that the relaxation function satisfies certain nonlinear differential inequality, it is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. In the special case when the viscoelastic effect is active on the entire domain and the frictional dissipation is differentiable at the origin, then the overall decay rates are dictated by the viscoelasticity. The obtained decay estimates are intrinsic without any prior quantification of decay rates of both viscoelastic and frictional dissipative effects. This particular topic has been motivated by influential paper of Fabrizio-Polidoro [15] where it was shown that viscoelasticity with poorly behaving relaxation kernel destroys exponential decay rates generated by linear frictional dissipation. In this paper we extend these considerations to: (i) nonlinear dissipation with unquantified growth at the origin (frictional) and infinity (viscoelastic) , (ii) more general geometric settings that accommodate competing nature of frictional and viscoelastic damping.
Citation: Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987
References:
[1]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, Journal of Functional Analysis, 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[2]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,, Applied Mathematics and Optimization, 51 (2005), 61. doi: 10.1007/s00245. Google Scholar

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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, 347 (2009), 867. doi: 10.1016/j.crma.2009.05.011. Google Scholar

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M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Anal., 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar

[8]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping,, Methods Appl. Anal., 15 (2008), 405. doi: 10.4310/MAA.2008.v15.n4.a1. Google Scholar

[9]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping,, Transactions of AMS, 361 (2009), 4561. doi: 10.1090/S0002-9947-09-04763-1. Google Scholar

[10]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result,, Arch. Ration. Mech. Anal., 197 (2010), 925. doi: 10.1007/s00205-009-0284-z. Google Scholar

[11]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits,, J. Funct. Anal., 246 (2007), 145. doi: 10.1016/j.jfa.2006.09.012. Google Scholar

[12]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations,, Nonlinear evolution equations, 40 (1977), 103. Google Scholar

[13]

M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partialy supported nonlinear boundary dissipation without growth restrictions,, DCDS-S, 2 (2009), 67. doi: 10.3934/dcdss.2009.2.67. Google Scholar

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B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Anna. Sci. Ec. Norm. Super., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1. Google Scholar

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory,, Appl. Anal., 81 (2002), 1245. doi: 10.1080/0003681021000035588. Google Scholar

[16]

M. Hitrik, Expansions and eigenfrequencies for damped wave equations,, Journées équations aux Dérivées Partielles, (2001). Google Scholar

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping,, Differential and integral Equations, 6 (1993), 507. Google Scholar

[18]

I. Lasiecka, S. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory,, Journal Mathematical Physics, 54 (2013). doi: 10.1063/1.4793988. Google Scholar

[19]

I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source,, Nonlinear Anal., 69 (2008), 898. doi: 10.1016/j.na.2008.02.069. Google Scholar

[20]

G. Lebeau, Equations des ondes amorties,, Algebraic Geometric Methods in Maths. Physics, (1996), 73. Google Scholar

[21]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping,, Rev. Mat. Complutense, 12 (1999), 251. Google Scholar

[22]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554. doi: 10.1137/S036301290139107X. Google Scholar

[23]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations,, Journal of Mathematical Physics, 53 (2012). doi: 10.1063/1.4711830. Google Scholar

[24]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials,, Quart. Appl. Math., 59 (2001), 557. Google Scholar

[25]

M. Nakao, Decay and global existence for nonlinear wave equations with localized dissipations in general exterior domains,, New trends in the theory of hyperbolic equations, 159 (2005), 213. doi: 10.1007/3-7643-7386-5_3. Google Scholar

[26]

M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains,, Math. Nachr., 278 (2005), 771. doi: 10.1002/mana.200310271. Google Scholar

[27]

J. Rauch and M. Taylor, Decay of solutions to n ondissipative hyperbolic systems on compact manifolds,, Comm. Pure Appl. Math., 28 (1975), 501. doi: 10.1002/cpa.3160280405. Google Scholar

[28]

T. Qin, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications,, J. Math. Anal. Appl., 233 (1999), 130. doi: 10.1006/jmaa.1999.6271. Google Scholar

[29]

D. Toundykov, Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary,, Nonlinear Analysis T. M. A., 67 (2007), 512. doi: 10.1016/j.na.2006.06.007. Google Scholar

[30]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-Order terms for general Riemannian wave equations. Global uniqueness and observability in one shot,, Appl. Math. and Optim, 46 (2002), 331. doi: 10.1007/s00245-002-0751-5. Google Scholar

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205. doi: 10.1080/03605309908820684. Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, Journal of Functional Analysis, 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[2]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,, Applied Mathematics and Optimization, 51 (2005), 61. doi: 10.1007/s00245. Google Scholar

[3]

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, 347 (2009), 867. doi: 10.1016/j.crma.2009.05.011. Google Scholar

[4]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055. Google Scholar

[5]

M. Bellassoued, Decay of solutions of the elastic wave equation with a localized dissipation,, Annales de la Faculté des Sciences de Toulouse, 12 (2003), 267. doi: 10.5802/afst.1049. Google Scholar

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Anal., 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar

[8]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping,, Methods Appl. Anal., 15 (2008), 405. doi: 10.4310/MAA.2008.v15.n4.a1. Google Scholar

[9]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping,, Transactions of AMS, 361 (2009), 4561. doi: 10.1090/S0002-9947-09-04763-1. Google Scholar

[10]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result,, Arch. Ration. Mech. Anal., 197 (2010), 925. doi: 10.1007/s00205-009-0284-z. Google Scholar

[11]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits,, J. Funct. Anal., 246 (2007), 145. doi: 10.1016/j.jfa.2006.09.012. Google Scholar

[12]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations,, Nonlinear evolution equations, 40 (1977), 103. Google Scholar

[13]

M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partialy supported nonlinear boundary dissipation without growth restrictions,, DCDS-S, 2 (2009), 67. doi: 10.3934/dcdss.2009.2.67. Google Scholar

[14]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Anna. Sci. Ec. Norm. Super., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1. Google Scholar

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory,, Appl. Anal., 81 (2002), 1245. doi: 10.1080/0003681021000035588. Google Scholar

[16]

M. Hitrik, Expansions and eigenfrequencies for damped wave equations,, Journées équations aux Dérivées Partielles, (2001). Google Scholar

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping,, Differential and integral Equations, 6 (1993), 507. Google Scholar

[18]

I. Lasiecka, S. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory,, Journal Mathematical Physics, 54 (2013). doi: 10.1063/1.4793988. Google Scholar

[19]

I. Lasiecka and D. Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source,, Nonlinear Anal., 69 (2008), 898. doi: 10.1016/j.na.2008.02.069. Google Scholar

[20]

G. Lebeau, Equations des ondes amorties,, Algebraic Geometric Methods in Maths. Physics, (1996), 73. Google Scholar

[21]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping,, Rev. Mat. Complutense, 12 (1999), 251. Google Scholar

[22]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554. doi: 10.1137/S036301290139107X. Google Scholar

[23]

S. Messaoudi and M. Mustafa, General stability result for viscoelastic wave equations,, Journal of Mathematical Physics, 53 (2012). doi: 10.1063/1.4711830. Google Scholar

[24]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials,, Quart. Appl. Math., 59 (2001), 557. Google Scholar

[25]

M. Nakao, Decay and global existence for nonlinear wave equations with localized dissipations in general exterior domains,, New trends in the theory of hyperbolic equations, 159 (2005), 213. doi: 10.1007/3-7643-7386-5_3. Google Scholar

[26]

M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains,, Math. Nachr., 278 (2005), 771. doi: 10.1002/mana.200310271. Google Scholar

[27]

J. Rauch and M. Taylor, Decay of solutions to n ondissipative hyperbolic systems on compact manifolds,, Comm. Pure Appl. Math., 28 (1975), 501. doi: 10.1002/cpa.3160280405. Google Scholar

[28]

T. Qin, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications,, J. Math. Anal. Appl., 233 (1999), 130. doi: 10.1006/jmaa.1999.6271. Google Scholar

[29]

D. Toundykov, Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary,, Nonlinear Analysis T. M. A., 67 (2007), 512. doi: 10.1016/j.na.2006.06.007. Google Scholar

[30]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-Order terms for general Riemannian wave equations. Global uniqueness and observability in one shot,, Appl. Math. and Optim, 46 (2002), 331. doi: 10.1007/s00245-002-0751-5. Google Scholar

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205. doi: 10.1080/03605309908820684. Google Scholar

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