General decay for second-order abstract viscoelastic equation in Hilbert spaces with time delay
Résumé
The paper is concerned with a second-order abstract viscoelastic equation with time delay and a relaxation function satisfying $ h^{\prime}(t)\leq -\zeta(t) G(h(t))$. Under a suitable conditions, we establish an explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some proprieties of the convex functions. Finally, some applications are given. This work generalizes the previous results without time delay term to those with delay.
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Références
Alabau-Boussouira F, Cannarsa P and Sforza D. Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254(2008), 1342-1372. DOI: https://doi.org/10.1016/j.jfa.2007.09.012
F. Alabau-Boussouira, P. Cannarsa. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. C. R. Math. 347(2009), no. 15-16, 867-872. DOI: https://doi.org/10.1016/j.crma.2009.05.011
K. Ammari, S. Nicaise and C. Pignotti. Feedback boundary stabilization of wave equations with interior delay. Syst. Control Lett. 59(2010), 623-628. DOI: https://doi.org/10.1016/j.sysconle.2010.07.007
V. Arnold. Mathematical methods of classical mechanics. New York: Springer, 1989. DOI: https://doi.org/10.1007/978-1-4757-2063-1
A. Batkai, S. Piazzera. Semigroups for delay equations. Research Notes in Mathematics, 10. AK Peters, Ltd., Wellesley, MA, (2005). DOI: https://doi.org/10.1201/9781439865682
F. Belhannache, M. M. Algharabli and S.A. Messaoudi. Asymptotic Stability for a Viscoelastic Equation with Nonlinear Damping and Very General Type of Relaxation Functions. J. Dyna. Control Syst. 26(2020), no. 1, 45-67. DOI: https://doi.org/10.1007/s10883-019-9429-z
A. Benaissa, A. K. Benaissa and S. A. Messaoudi. Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J. Math. Phys. 53(2012), no. 12, 123514 DOI: https://doi.org/10.1063/1.4765046
Y. Boukhatem, B. Benabderrahmane. General decay for a viscoelastic equation of variable coefficients with a timevarying delay in the boundary feedback and acoustic boundary conditions. ACTA Math. Sci. 37(2017), no. 5, 1453-1471. DOI: https://doi.org/10.1016/S0252-9602(17)30084-X
Y. Boukhatem, B. Benabderrahmane. General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions. ACTA Appl. Math. 154(2018), no. 1, 131-152. DOI: https://doi.org/10.1007/s10440-017-0137-y
Y. Boukhatem, B. Benabderrahmane, Existence and exponential decay of solutions for the variable coefficients wave equations. Analele Universitatii Oradea Fasc. Matematica, Tom XXIII (2), (2016), 93-108.
M. M. Cavalcanti,A. D. Cavalcanti, I. Lasiecka and X. Wang. Existence and sharp decay rate estimates for a von Karman system with long memory. NA: Real World Appl. 22(2015), 289-306. DOI: https://doi.org/10.1016/j.nonrwa.2014.09.016
H. Chellaoua, Y. Boukhatem. Optimal decay for second-order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay. Math. Meth. Appl. Sci. 44(2021), no. 2, 2071-2095. DOI: https://doi.org/10.1002/mma.6917
H. Chellaoua, Y. Boukhatem. Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay. Z. Angew. Math. Phys. 72(2021), no. 2, 1-18. DOI: https://doi.org/10.1007/s00033-021-01477-y
C. M. Dafermos. Asymptotic stability in viscoelasticity. Arch. Rational Mech. An. 37(1970), 297-308. DOI: https://doi.org/10.1007/BF00251609
R. Datko. Two questions concerning the boundary control of certain elastic systems, J. Differ. Equa. 92(1991), no. 1, 27-44. DOI: https://doi.org/10.1016/0022-0396(91)90062-E
M. Fabrizio, S. Polidoro, Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(2002), no. 6, 1245-1264. DOI: https://doi.org/10.1080/0003681021000035588
J. H. Hassan, S. A. Messaoudi. General decay rate for a class of weakly dissipative second-order systems with memory. Math. Meth. Appl. Sci. 42(2019), no. 8, 2842-2853. DOI: https://doi.org/10.1002/mma.5554
J. R. Kang. General decay for viscoelastic plate equation with p-Laplacian and time-varying delay. Bound. Value Probl. 2018(2018), no. 1, 1-11. DOI: https://doi.org/10.1186/s13661-018-0942-x
M. Kirane, B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 62(2011), 1065-1082. DOI: https://doi.org/10.1007/s00033-011-0145-0
I. Lasiecka, S. A. Messaoudi and M. I. Mustafa. Note on intrinsic decay rates for abstract wave equations with memory. J Math Phys, 54(2013), no. 3, 031504. DOI: https://doi.org/10.1063/1.4793988
Z. Liu, C.Gao and Z. B. Fang. A general decay estimate for the nonlinear transmission problem of weak viscoelastic equations with time-varying delay. J. Math. Phys. 60(2019), no. 10, 101507. DOI: https://doi.org/10.1063/1.5103177
S. A. Messaoudi, General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341(2008), no. 2, 1457-1467. DOI: https://doi.org/10.1016/j.jmaa.2007.11.048
Messaoudi S. General stability in viscoelasticity, viscoelastic and viscoplastic materials, Mohamed El-Amin. IntechOpen (2016). DOI: https://doi.org/10.5772/64217
J. E. Mu˜noz Rivera, A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials. Q. Appl. Math. 59(2001), no. 3, 557-578. DOI: https://doi.org/10.1090/qam/1848535
J. E. Mu˜noz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory. J. Math. Anal. Appl. 286(2003), no. 2, 692-704. DOI: https://doi.org/10.1016/S0022-247X(03)00511-0
M. I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete Conti. Dyn-A, 35(2014), no. 3, 1179. DOI: https://doi.org/10.3934/dcds.2015.35.1179
M. I. Mustafa. Memory-type plate system with nonlinear delay. Adv. Pure Appl. Math. 8(2017), no. 4, 227-240. DOI: https://doi.org/10.1515/apam-2016-0111
M. I. Mustafa. Asymptotic stability for the second order evolution equation with memory. J. Dyna. Control Syst. 25(2018), no. 2, 263-273. DOI: https://doi.org/10.1007/s10883-018-9410-2
M. I. Mustafa. Optimal decay rates for the abstract viscoelastic equation. J. Evol. Equ. (2019), 1-17. DOI: https://doi.org/10.1007/s00028-019-00488-7
M. I. Mustafa. Optimal decay rates for the viscoelastic wave equation. Math. Meth. Appl. Sci. (2017), 1-13. DOI: https://doi.org/10.1002/mma.4604
S. Nicaise, C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(2006), 1561-1585. DOI: https://doi.org/10.1137/060648891
S. Nicaise, J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM Control Optim. Calc. Var. 16(2010), no. 2, 420-456. DOI: https://doi.org/10.1051/cocv/2009007
V. Pata, Exponential stability in linear viscoelasticity. Q. Appl. Math. 64(2006), no. 3, 499-513. DOI: https://doi.org/10.1090/S0033-569X-06-01010-4
T. J. Xiao, J. Liang. Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equa. 254(2013), no. 5, 2128-2157. DOI: https://doi.org/10.1016/j.jde.2012.11.019
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