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Keywords:
exponential decay; polynomial decay; porous-elastic system; neutral delay; multipliers method; Faedo-Galerkin approximations
exponential decay; polynomial decay; porous-elastic system; neutral delay; multipliers method; Faedo-Galerkin approximations
Summary:
We consider a one-dimensional porous-elastic system with porous-viscosity and a distributed delay of neutral type. First, we prove the global existence and uniqueness of the solution by using the Faedo-Galerkin approximations along with some energy estimates. Then, based on the energy method with some appropriate assumptions on the kernel of neutral delay term, we construct a suitable Lyapunov functional and we prove that, despite of the destructive nature of delays in general, the damping mechanism considered provokes an exponential decay of the solution for the case of equal speed of wave propagation. In the case of lack of exponential stability, we show that the solution decays polynomially.
References:
[1] Apalara, T. A.: Exponential decay in one-dimensional porous dissipation elasticity. Q. J. Mech. Appl. Math. 70 (2017), 363-372. DOI 10.1093/qjmam/hbx012 | MR 1423.35033 | Zbl 1423.35033
[2] Apalara, T. A.: Corrigendum to: "Exponential decay in one-dimensional porous dissipation elasticity". Q. J. Mech. Appl. Math. 70 (2017), 553-555. DOI 10.1093/qjmam/hbx027 | MR 3737355 | Zbl 1465.35053
[3] Apalara, T. A.: A general decay for a weakly nonlinearly damped porous system. J. Dyn. Control Syst. 25 (2019), 311-322. DOI 10.1007/s10883-018-9407-x | MR 3953144 | Zbl 1415.35257
[4] Apalara, T. A.: General decay of solutions in one-dimensional porous-elastic system with memory. J. Math. Anal. Appl. 469 (2019), 457-471. DOI 10.1016/j.jmaa.2017.08.007 | MR 3860432 | Zbl 1402.35042
[5] Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences 183. Springer, New York (2013). DOI 10.1007/978-1-4614-5975-0 | MR 2986590 | Zbl 1286.76005
[6] Casas, P. S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32 (2005), 652-658. DOI 10.1016/j.mechrescom.2005.02.015 | MR 2158183 | Zbl 1192.74156
[7] Casas, P. S., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43 (2005), 33-47. DOI 10.1016/j.ijengsci.2004.09.004 | MR 2112810 | Zbl 1211.74060
[8] Choucha, A., Boulaaras, S. M., Ouchenane, D., Cherif, B. B., Abdalla, M.: Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term. J. Funct. Spaces 2021 (2021), Article ID 5581634, 8 pages. DOI 10.1155/2021/5581634 | MR 4225540 | Zbl 1462.35073
[9] Choucha, A., Ouchenane, D., Zennir, K.: General decay of solutions in one-dimensional porous-elastic with memory and distributed delay term. Tamkang J. Math. 52 (2021), 479-495. DOI 10.5556/j.tkjm.52.2021.3519 | MR 4334409 | Zbl 1483.35027
[10] Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26 (1988), 697-713. DOI 10.1137/0326040 | MR 0937679 | Zbl 0643.93050
[11] Datko, R., Lagnese, J., Polis, M. P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986), 152-156. DOI 10.1137/032400 | MR 0818942 | Zbl 0592.93047
[12] Driver, R. D.: A mixed neutral system. Nonlinear Anal., Theory Methods Appl. 8 (1984), 155-158. DOI 10.1016/0362-546X(84)90066-X | MR 0734448 | Zbl 0553.34042
[13] Driver, R. D.: A neutral system with state-dependent delays. Trends in Theory and Practice of Nonlinear Differential Equations Lecture Notes in Pure and Applied Mathematics 90. Marcel Dekker, Basel (1984), 157-161. MR 0741499 | Zbl 0543.34053
[14] Fareh, A., Messaoudi, S. A.: Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Appl. Math. Comput. 293 (2017), 493-507. DOI 10.1016/j.amc.2016.08.040 | MR 3549686 | Zbl 1411.74023
[15] Feng, B.: Global well-posedness and stability for a viscoelastic plate equation with a time delay. Math. Probl. Eng. 2015 (2015), Article ID 585021, 10 pages. DOI 10.1155/2015/585021 | MR 3335470 | Zbl 1394.35535
[16] Foughali, F., Zitouni, S., Bouzettouta, L., Khochemane, H. E.: Well-posedness and general decay for a porous-elastic system with microtemperatures effects and time-varying delay term. Z. Angew. Math. Phys. 73 (2022), Artile ID 183, 31 pages. DOI 10.1007/s00033-022-01801-0 | MR 4462689 | Zbl 1495.35031
[17] Goodman, M. A., Cowin, S. C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44 (1972), 249-266. DOI 10.1007/BF00284326 | MR 1553563 | Zbl 0243.76005
[18] Guesmia, A.: Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inf. 30 (2013), 507-526. DOI 10.1093/imamci/dns039 | MR 3144921 | Zbl 1279.93090
[19] Morales, M. E. Hernández, Henríquez, H. R.: Existence results for second order partial neutral functional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 15 (2008), 645-670. MR 2446031 | Zbl 1180.34089
[20] Morales, M. E. Hernández, Henríquez, H. R., McKibben, M. A.: Existence of solutions for second order partial neutral functional differential equations. Integral Equations Oper. Theory 62 (2008), 191-217. DOI 10.1007/s00020-008-1618-1 | MR 2447914 | Zbl 1187.35268
[21] Khochemane, H. E., Bouzettouta, L., Guerouah, A.: Exponential decay and well-posedness for a one-dimensional porous-elastic system with distributed delay. Appl. Anal. 100 (2021), 2950-2964. DOI 10.1080/00036811.2019.1703958 | MR 4309873 | Zbl 1476.35047
[22] Khochemane, H. E., Zitouni, S., Bouzettouta, L.: Stability result for a nonlinear damping porous-elastic system with delay term. Nonlinear Stud. 27 (2020), 487-503. MR 4103707 | Zbl 1450.35252
[23] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Études mathématiques. Dunod, Paris (1969), French. MR 0259693 | Zbl 0189.40603
[24] Liu, G., Yan, J.: Global asymptotic stability of nonlinear neutral differential equation. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1035-1041. DOI 10.1016/j.cnsns.2013.08.035 | MR 3119279 | Zbl 1457.34110
[25] Loucif, S., Guefaifia, R., Zitouni, S., Khochemane, H. E.: Global well-posedness and exponential decay of fully dynamic and electrostatic or quasi-static piezoelectric beams subject to a neutral delay. Z. Angew. Math. Phys. 74 (2023), Article ID 83, 22 pages. DOI 10.1007/s00033-023-01972-4 | MR 4572110 | Zbl 1524.74134
[26] Magaña, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43 (2006), 3414-3427. DOI 10.1016/j.ijsolstr.2005.06.077 | MR 2221521 | Zbl 1121.74361
[27] Rivera, J. E. Muñoz, Lapa, E. C., Barreto, R.: Decay rates for viscoelastic plates with memory. J. Elasticity 44 (1996), 61-87. DOI 10.1007/BF00042192 | MR 1417809 | Zbl 0876.73037
[28] Nicaise, S., Pignotti, C.: 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 10.1137/060648891 | MR 2272156 | Zbl 1180.35095
[29] Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21 (2008), 935-958. DOI 10.57262/die/1356038593 | MR 2483342 | Zbl 1224.35247
[30] Nunziato, J. W., Cowin, S. C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72 (1979), 175-201. DOI 10.1007/BF00249363 | MR 0545517 | Zbl 0444.73018
[31] Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16 (2003), 487-491. DOI 10.1016/S0893-9659(03)00025-9 | MR 1983718 | Zbl 1040.74023
[32] Racke, R.: Instability of coupled systems with delay. Commun. Pure. Appl. Anal. 11 (2012), 1753-1773. DOI 10.3934/cpaa.2012.11.1753 | MR 2911109 | Zbl 1267.35246
[33] Seghour, L., Tatar, N.-e., Berkani, A.: Stability of a thermoelastic laminated system subject to a neutral delay. Math. Methods Appl. Sci. 43 (2020), 281-304. DOI 10.1002/mma.5878 | MR 4044239 | Zbl 1445.35074
[34] Wang, J.: Existence and stability of solutions for neutral differential equations with delay. International Conference on Multimedia Technology IEEE, Piscataway (2011), 2462-2465. DOI 10.1109/ICMT.2011.6002527
[35] Xu, G. Q., Yung, S. P., Li, L. K.: Stabilization of wave systems with input delay in the boundary control. ESAIM, Control Optim. Calc. Var. 12 (2006), 770-785 \99999DOI99999 10.1051/cocv:2006021 \goodbreak. DOI 10.1051/cocv:2006021 | MR 2266817 | Zbl 1105.35016
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