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 and 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), 1342-1372. doi: 10.1016/j.jfa.2007.09.012.

[2]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Applied Mathematics and Optimization, 51 (2005), 61-105. doi: 10.1007/s00245.

[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, Ser. I, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011.

[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), 1024-1065. doi: 10.1137/0330055.

[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), 267-301. doi: 10.5802/afst.1049.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040.

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324. doi: 10.1137/S0363012902408010.

[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), 405-426. doi: 10.4310/MAA.2008.v15.n4.a1.

[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), 4561-4580. doi: 10.1090/S0002-9947-09-04763-1.

[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), 925-964. doi: 10.1007/s00205-009-0284-z.

[11]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., 246 (2007), 145-195. doi: 10.1016/j.jfa.2006.09.012.

[12]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, Nonlinear evolution equations, (Proc. Sympos., Univ. Wisconsin, Madison, Wis., Publ. Math. Res. Center Univ. Wisconsin, Academic Press, New York-London, 40 (1977), (1978), 103-123.

[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), 67-94. doi: 10.3934/dcdss.2009.2.67.

[14]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[16]

M. Hitrik, Expansions and eigenfrequencies for damped wave equations, Journées équations aux Dérivées Partielles" (Plestin-les-Grèves, 2001), Exp. No. VI, Univ. Nantes, Nantes, (2001), 10 pp.

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and integral Equations, 6 (1993), 507-533.

[18]

I. Lasiecka, S. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, Journal Mathematical Physics, 54 (2013), 031504. doi: 10.1063/1.4793988.

[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), 898-910. doi: 10.1016/j.na.2008.02.069.

[20]

G. Lebeau, Equations des ondes amorties, Algebraic Geometric Methods in Maths. Physics, (1996), 73-109.

[21]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complutense, 12 (1999), 251-283.

[22]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation, SIAM J. Control Optim., 41 (2002), 1554-1566. doi: 10.1137/S036301290139107X.

[23]

S. Messaoudi and M. Mustafa , General stability result for viscoelastic wave equations, Journal of Mathematical Physics, 53 (2012), 053702. doi: 10.1063/1.4711830.

[24]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578.

[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, Oper. Theory Adv. Appl., Birkhäuser, Basel, 159 (2005), 213-299. doi: 10.1007/3-7643-7386-5_3.

[26]

M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr., 278 (2005), 771-783. doi: 10.1002/mana.200310271.

[27]

J. Rauch and M. Taylor, Decay of solutions to n ondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math., 28 (1975), 501-523. doi: 10.1002/cpa.3160280405.

[28]

T. Qin, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications, J. Math. Anal. Appl., 233 (1999), 130-147. doi: 10.1006/jmaa.1999.6271.

[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), 512-544. doi: 10.1016/j.na.2006.06.007.

[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), 331-375. Special issue dedicated to J. L. Lions. doi: 10.1007/s00245-002-0751-5.

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235. doi: 10.1080/03605309908820684.

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), 1342-1372. doi: 10.1016/j.jfa.2007.09.012.

[2]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Applied Mathematics and Optimization, 51 (2005), 61-105. doi: 10.1007/s00245.

[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, Ser. I, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011.

[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), 1024-1065. doi: 10.1137/0330055.

[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), 267-301. doi: 10.5802/afst.1049.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040.

[7]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324. doi: 10.1137/S0363012902408010.

[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), 405-426. doi: 10.4310/MAA.2008.v15.n4.a1.

[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), 4561-4580. doi: 10.1090/S0002-9947-09-04763-1.

[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), 925-964. doi: 10.1007/s00205-009-0284-z.

[11]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., 246 (2007), 145-195. doi: 10.1016/j.jfa.2006.09.012.

[12]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, Nonlinear evolution equations, (Proc. Sympos., Univ. Wisconsin, Madison, Wis., Publ. Math. Res. Center Univ. Wisconsin, Academic Press, New York-London, 40 (1977), (1978), 103-123.

[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), 67-94. doi: 10.3934/dcdss.2009.2.67.

[14]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.

[15]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[16]

M. Hitrik, Expansions and eigenfrequencies for damped wave equations, Journées équations aux Dérivées Partielles" (Plestin-les-Grèves, 2001), Exp. No. VI, Univ. Nantes, Nantes, (2001), 10 pp.

[17]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and integral Equations, 6 (1993), 507-533.

[18]

I. Lasiecka, S. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, Journal Mathematical Physics, 54 (2013), 031504. doi: 10.1063/1.4793988.

[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), 898-910. doi: 10.1016/j.na.2008.02.069.

[20]

G. Lebeau, Equations des ondes amorties, Algebraic Geometric Methods in Maths. Physics, (1996), 73-109.

[21]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complutense, 12 (1999), 251-283.

[22]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation, SIAM J. Control Optim., 41 (2002), 1554-1566. doi: 10.1137/S036301290139107X.

[23]

S. Messaoudi and M. Mustafa , General stability result for viscoelastic wave equations, Journal of Mathematical Physics, 53 (2012), 053702. doi: 10.1063/1.4711830.

[24]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578.

[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, Oper. Theory Adv. Appl., Birkhäuser, Basel, 159 (2005), 213-299. doi: 10.1007/3-7643-7386-5_3.

[26]

M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr., 278 (2005), 771-783. doi: 10.1002/mana.200310271.

[27]

J. Rauch and M. Taylor, Decay of solutions to n ondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math., 28 (1975), 501-523. doi: 10.1002/cpa.3160280405.

[28]

T. Qin, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications, J. Math. Anal. Appl., 233 (1999), 130-147. doi: 10.1006/jmaa.1999.6271.

[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), 512-544. doi: 10.1016/j.na.2006.06.007.

[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), 331-375. Special issue dedicated to J. L. Lions. doi: 10.1007/s00245-002-0751-5.

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235. doi: 10.1080/03605309908820684.

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