# American Institute of Mathematical Sciences

March  2015, 14(2): 457-491. doi: 10.3934/cpaa.2015.14.457

## Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

 1 Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O. Box 5005, Dhahran 31261 2 Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261, Saudi Arabia

Received  August 2014 Revised  October 2014 Published  December 2014

In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.
Citation: Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure and Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
##### References:
 [1] M. Aassila, M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations, 15 (2002), 155-180. doi: 10.1007/s005260100096. [2] M. Aassila, M. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602. doi: 10.1137/S0363012998344981. [3] K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems Cont. Letters, 59 (2010), 623-628. doi: 10.1016/j.sysconle.2010.07.007. [4] T. A. Apalara, S. A. Messaoudi and M. I. Mustafa, Energy decay in thermoelasticity type III with viscoelastic damping and delay term, Elect. J. Diff. Equa., 2012 (2012), 1-15. [5] 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), 1-19. [6] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis T. M. A., 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. [7] M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis T. M. A., 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040. [8] M. M. Cavalcanti, V. N. Domingos and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diff. Equa., 44 (2002), 1-14. [9] 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. [10] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Anal., 46 (2006), 251-273. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] R. Datko, Two questions concerning the boundary control of certain elastic systems, J. Diff. Equa., 1 (1991), 27-44. doi: 10.1016/0022-0396(91)90062-E. [13] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 1 (1986), 152-156. doi: 10.1137/0324007. [14] M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589. [15] E. Fridman, S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 8 (2010), 5028-5052. doi: 10.1137/090762105. [16] C. Giorgi, J. E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99. doi: 10.1006/jmaa.2001.7437. [17] A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760. doi: 10.1016/j.jmaa.2011.04.079. [18] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Mathematical Control and Information, 30 (2013), 507-526. doi: 10.1093/imamci/dns039. [19] A. Guesmia, Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay,, \emph{J. Math. Phys.}, (). [20] A. Guesmia and S. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122. doi: 10.1002/mma.1125. [21] A. Guesmia and S. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Analysis T. M. A., 13 (2012), 476-485. doi: 10.1016/j.nonrwa.2011.08.004. [22] A. Guesmia, S. Messaoudi and B. Said-Houari, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonl. Diff. Equa. Appl., 18 (2011), 659-684. doi: 10.1007/s00030-011-0112-7. [23] A. Guesmia, S. Messaoudi and A. Soufyane, On the stabilization for a linear Timoshenko system with infinite history and applications to the coupled Timoshenko-heat systems, Elect. J. Diff. Equa., 2012 (2012), 1-45. [24] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082. doi: 10.1007/s00033-011-0145-0. [25] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. [26] Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. [27] S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048. [28] S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis T. M. A., 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. [29] S. A. Messaoudi and N. E. Tatar, Global existence and asymptotic behavior for a nonlinear viscoelastic problem, Math. Meth. Sci. Res. J., 4 (2003), 136-149. [30] S. A. Messaoudi and N. E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Meth. Appl. Sci., 30 (2007), 665-680. doi: 10.1002/mma.804. [31] J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707. doi: 10.1016/j.jmaa.2006.03.022. [32] M. I. Mustafa, Exponential decay in thermoelastic systems with boundary delay, J. Abst. Diff. Equa. Appl., 2 (2011), 1-13. [33] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 5 (2006), 1561-1585. doi: 10.1137/060648891. [34] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Diff. Integral Equa., 9-10 (2008), 935-958. [35] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Elect. J. Diff. Equa., 41 (2011), 1-20. [36] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Disc. Cont. Dyna. Syst. Series S, 3 (2011), 693-722. doi: 10.3934/dcdss.2011.4.693. [37] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim., 2 (2010), 420-456. doi: 10.1051/cocv/2009007. [38] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Disc. Cont. Dyna. Syst. Series S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559. [39] V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 3 (2006), 499-513. doi: 10.1007/s00032-009-0098-3. [40] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure. Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721. [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [42] B. Said-Houari, A stability result for a Timoshenko system with past history and a delay term in the internal feedback, Dynamic Systems and Applications, 20 (2011), 327-354. [43] B. Said-Houari and F. Falcão Nascimento, Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction, Comm. Pure Appl. Anal., 12 (2013), 375-403. doi: 10.3934/cpaa.2013.12.375. [44] N. E. Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Z. angew. Math. Phys., 60 (2009), 640-650. doi: 10.1007/s00033-008-8030-1. [45] N. E. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp., 215 (2009), 2298-2306. doi: 10.1016/j.amc.2009.08.034. [46] N. E. Tatar, How far can relaxation functions be increasing in viscoelastic problems? Appl. Math. Letters, 22 (2009), 336-340. doi: 10.1016/j.aml.2008.04.005. [47] N. E. Tatar, A new class of kernels leading to an arbitrary decay in viscoelasticity, Mediterr. J. Math., 6 (2010), 139-150. doi: 10.1007/s00009-012-0177-5. [48] N. E. Tatar, On a perturbed kernel in viscoelasticity, Appl. Math. Letters, 24 (2011), 766-770. doi: 10.1016/j.aml.2010.12.035. [49] N. E. Tatar, Arbitrary decays in linear viscoelasticity, J. Math. Phys., 52 (2011), 1-12. doi: 10.1063/1.3533766. [50] N. E. Tatar, Uniform decay in viscoelasticity for kernels with small non-decreasingness zones, Appl. Math. Comp., 218 (2012), 7939-7946. doi: 10.1016/j.amc.2012.02.012. [51] N. E. Tatar, Oscillating kernels and arbitrary decays in viscoelasticity, Math. Nachr., 285 (2012), 1130-1143. doi: 10.1002/mana.201000053. [52] A. Vicente, Wave equation with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat., 27 (2009), 29-39. doi: 10.5269/bspm.v27i1.9066.

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##### References:
 [1] M. Aassila, M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations, 15 (2002), 155-180. doi: 10.1007/s005260100096. [2] M. Aassila, M. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602. doi: 10.1137/S0363012998344981. [3] K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems Cont. Letters, 59 (2010), 623-628. doi: 10.1016/j.sysconle.2010.07.007. [4] T. A. Apalara, S. A. Messaoudi and M. I. Mustafa, Energy decay in thermoelasticity type III with viscoelastic damping and delay term, Elect. J. Diff. Equa., 2012 (2012), 1-15. [5] 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), 1-19. [6] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis T. M. A., 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. [7] M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis T. M. A., 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040. [8] M. M. Cavalcanti, V. N. Domingos and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diff. Equa., 44 (2002), 1-14. [9] 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. [10] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Anal., 46 (2006), 251-273. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] R. Datko, Two questions concerning the boundary control of certain elastic systems, J. Diff. Equa., 1 (1991), 27-44. doi: 10.1016/0022-0396(91)90062-E. [13] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 1 (1986), 152-156. doi: 10.1137/0324007. [14] M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589. [15] E. Fridman, S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 8 (2010), 5028-5052. doi: 10.1137/090762105. [16] C. Giorgi, J. E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99. doi: 10.1006/jmaa.2001.7437. [17] A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760. doi: 10.1016/j.jmaa.2011.04.079. [18] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Mathematical Control and Information, 30 (2013), 507-526. doi: 10.1093/imamci/dns039. [19] A. Guesmia, Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay,, \emph{J. Math. Phys.}, (). [20] A. Guesmia and S. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122. doi: 10.1002/mma.1125. [21] A. Guesmia and S. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Analysis T. M. A., 13 (2012), 476-485. doi: 10.1016/j.nonrwa.2011.08.004. [22] A. Guesmia, S. Messaoudi and B. Said-Houari, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonl. Diff. Equa. Appl., 18 (2011), 659-684. doi: 10.1007/s00030-011-0112-7. [23] A. Guesmia, S. Messaoudi and A. Soufyane, On the stabilization for a linear Timoshenko system with infinite history and applications to the coupled Timoshenko-heat systems, Elect. J. Diff. Equa., 2012 (2012), 1-45. [24] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082. doi: 10.1007/s00033-011-0145-0. [25] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. [26] Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. [27] S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048. [28] S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis T. M. A., 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. [29] S. A. Messaoudi and N. E. Tatar, Global existence and asymptotic behavior for a nonlinear viscoelastic problem, Math. Meth. Sci. Res. J., 4 (2003), 136-149. [30] S. A. Messaoudi and N. E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Meth. Appl. Sci., 30 (2007), 665-680. doi: 10.1002/mma.804. [31] J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707. doi: 10.1016/j.jmaa.2006.03.022. [32] M. I. Mustafa, Exponential decay in thermoelastic systems with boundary delay, J. Abst. Diff. Equa. Appl., 2 (2011), 1-13. [33] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 5 (2006), 1561-1585. doi: 10.1137/060648891. [34] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Diff. Integral Equa., 9-10 (2008), 935-958. [35] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Elect. J. Diff. Equa., 41 (2011), 1-20. [36] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Disc. Cont. Dyna. Syst. Series S, 3 (2011), 693-722. doi: 10.3934/dcdss.2011.4.693. [37] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim., 2 (2010), 420-456. doi: 10.1051/cocv/2009007. [38] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Disc. Cont. Dyna. Syst. Series S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559. [39] V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 3 (2006), 499-513. doi: 10.1007/s00032-009-0098-3. [40] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure. Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721. [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [42] B. Said-Houari, A stability result for a Timoshenko system with past history and a delay term in the internal feedback, Dynamic Systems and Applications, 20 (2011), 327-354. [43] B. Said-Houari and F. Falcão Nascimento, Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction, Comm. Pure Appl. Anal., 12 (2013), 375-403. doi: 10.3934/cpaa.2013.12.375. [44] N. E. Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Z. angew. Math. Phys., 60 (2009), 640-650. doi: 10.1007/s00033-008-8030-1. [45] N. E. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp., 215 (2009), 2298-2306. doi: 10.1016/j.amc.2009.08.034. [46] N. E. Tatar, How far can relaxation functions be increasing in viscoelastic problems? Appl. Math. Letters, 22 (2009), 336-340. doi: 10.1016/j.aml.2008.04.005. [47] N. E. Tatar, A new class of kernels leading to an arbitrary decay in viscoelasticity, Mediterr. J. Math., 6 (2010), 139-150. doi: 10.1007/s00009-012-0177-5. [48] N. E. Tatar, On a perturbed kernel in viscoelasticity, Appl. Math. Letters, 24 (2011), 766-770. doi: 10.1016/j.aml.2010.12.035. [49] N. E. Tatar, Arbitrary decays in linear viscoelasticity, J. Math. Phys., 52 (2011), 1-12. doi: 10.1063/1.3533766. [50] N. E. Tatar, Uniform decay in viscoelasticity for kernels with small non-decreasingness zones, Appl. Math. Comp., 218 (2012), 7939-7946. doi: 10.1016/j.amc.2012.02.012. [51] N. E. Tatar, Oscillating kernels and arbitrary decays in viscoelasticity, Math. Nachr., 285 (2012), 1130-1143. doi: 10.1002/mana.201000053. [52] A. Vicente, Wave equation with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat., 27 (2009), 29-39. doi: 10.5269/bspm.v27i1.9066.
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