General Decay Result for a Weakly Damped Thermo-Viscoelastic System with Second Sound

У статті розглядається $n$-вимірна термов'язкопружна система з другим звуком зі слабким демпфуванням тертя. Використовуючи деякі властивості опуклих функцій, встановлено явний та загальний результат згасання. Наш результат одержано без будь яких обмежувальних припущень щодо зростання демпфування тертя.

Mathematics Subject Classification 2010: 35B37, 35L55, 74D05, 93D15, 93D20
Ключові слова: загальне згасання, слабке демпфування тертя, термов'язкопружна система з другим звуком, опуклість

References

[1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (2005), 61–105 . CrossRef

[2] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989. CrossRef

[3] A. Boudiaf, S. Drabla, and F. Boulanouar, General decay rate for nonlinear thermoviscoelastic system with a weak damping and nonlinear source terme, Mediterr. j. Math., DIO 10.1007 (2015) CrossRef

[4] F. Boulanouar, and S. Drabla, General boundary stabilization result of memory-type thermo-elasticity with second sound, Electron. J. Diff. Equ. 202 (2014),1–18.

[5] M.M. Cavalcanti, V.N.D. Cavalcanti, and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Equ. 44 (2002), 1–14.

[6] M.M. Cavalcanti, V.N.D. Cavalcanti, and I. Lasiecka. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equ. 236 (2007), 407–459. CrossRef

[7] S. Drabla, S.A. Messaoudi, and F. Boulanouar, A general decay result for a multidimensional weakly damped thermoelastic system with second sound, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1329–1339. CrossRef

[8] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations 8 (1993), 507–533.

[9] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. 64 (2006), 1757–1797. CrossRef

[10] 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. CrossRef

[11] W.J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ric. Math. 48 (1999), 61–75.

[12] S.A. Messaoudi, Local Existence and blow up in thermoelasticity with second sound, Comm. Partial Diff. Equ. 26 (2002), 1681–1693. CrossRef

[13] S.A. Messaoudi and B. Madani, A general decay result for a memory-type thermoelasticity with second sound, Appl. Anal. 93 (2014), 1663–1673. CrossRef

[14] S.A. Messaoudi and M.I. Mustafa, On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Anal. 72 (2010), 3602–3611. CrossRef

[15] S.A. Messaoudi and B. Said-Houari, Blow up of solutions with positive energy in nonlinear thermo-elasticity with second sound,. J. Appl. Math. 2004 (2004), 201– 211. CrossRef

[16] S.A. Messaoudi and A. Al-Shehri, General boundary stabilization of memory- type thermo-elasticity with second sound, Z. Anal. Anwend. 31 (2012), 441–461. CrossRef

[17] M.I. Mustafa, Boundary stabilization of memory-type thermoelasticity with second sound, Z. Angew. Math. Phys. 63 (2012), 777–792. CrossRef

[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied mathematical Sciences, 44, Springer-Verlag, 1983. CrossRef

[19] R. Racke, Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d. Math. Meth. Appl. Sci. 25 (2002), 409-441. CrossRef

[20] F. Zhao, Z. Li and Y. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal. Real World Appl. 12 (2011), 1759–1773. CrossRef