| Title:
|
Asymptotic behavior of solutions for linear evolutionary boundary value problem of viscoelastic damped wave equation (English) |
| Author:
|
Berbiche, Mohamed |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
205-223 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the existence of global in time and uniform decay of weak solutions to the initial-boundary value problem related to the dynamic behavior of evolution equation accounting for rotational inertial forces along with a linear nonlocal frictional damping arises in viscoelastic materials. By constructing appropriate Lyapunov functional, we show the solution converges to the equilibrium state polynomially in the energy space. (English) |
| Keyword:
|
global existence |
| Keyword:
|
uniqueness |
| Keyword:
|
uniform stabilization |
| MSC:
|
35B33 |
| MSC:
|
47J35 |
| idZBL:
|
07217190 |
| idMR:
|
MR4221830 |
| DOI:
|
10.21136/MB.2019.0054-18 |
| . |
| Date available:
|
2020-06-10T13:17:34Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148155 |
| . |
| Reference:
|
[1] Aassila, M., Cavalcanti, M. M., Soriano, J. A.: 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. Zbl 0985.35008, MR 1766431, 10.1137/S0363012998344981 |
| Reference:
|
[2] Achouri, Z., Amroun, N. E., Benaissa, A.: The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type.Math. Methods Appl. Sci. 40 (2017), 3837-3854. Zbl 1366.93484, MR 3668815, 10.1002/mma.4267 |
| Reference:
|
[3] Aizicovici, S., Feireisl, E.: Long-time stabilization of solutions to a phase-field model with memory.J. Evol. Equ. 1 (2001), 69-84. Zbl 0973.35037, MR 1838321, 10.1007/PL00001365 |
| Reference:
|
[4] Cavalcanti, M. M., Oquendo, H. P.: Frictional versus viscoelastic damping in a semilinear wave equation.SIAM J. Control Optimization 42 (2003), 1310-1324. Zbl 1053.35101, MR 2044797, 10.1137/S0363012902408010 |
| Reference:
|
[5] Chill, R., Fašangová, E.: Convergence to steady states of solutions of semilinear evolutionary integral equations.Calc. Var. Partial Differ. Equ. 22 (2005), 321-342. Zbl 1087.45005, MR 2118902, 10.1007/s00526-004-0278-5 |
| Reference:
|
[6] Coleman, B. D., Dill, E. H.: On the thermodynamics of electromagnetic fields in materials with memory.Arch. Rational Mech. Anal. 41 (1971), 132-162. MR 0347245, 10.1007/BF00281371 |
| Reference:
|
[7] Coleman, B. D., Dill, E. H.: Thermodynamic restriction on the constitutive equations of electromagnetic theory.Zeit. Angew. Math. Phys. 22 (1971), 691-702. Zbl 0218.35072, 10.1007/BF01587765 |
| Reference:
|
[8] Dafermos, C. M.: Asymptotic stability in viscoelasticity.Arch. Ration. Mech. Anal. 37 (1970), 297-308. Zbl 0214.24503, MR 0281400, 10.1007/BF00251609 |
| Reference:
|
[9] Desch, W., Fašangová, E., Milota, J., Propst, G.: Stabilization through viscoelastic boundary damping: a semigroup approach.Semigroup Forum 80 (2010), 405-415. Zbl 1193.35149, MR 2647481, 10.1007/s00233-009-9197-2 |
| Reference:
|
[10] Eringen, A. C., Maugin, G. A.: Electrodynamics of Continua. I.Foundations and Solid Media. Springer, New York (1990). MR 1031714, 10.1007/978-1-4612-3226-1 |
| Reference:
|
[11] Fabrizio, M., Morro, A.: Thermodynamics of electromagnetic isothermal system with memory.J. Non-Equilibrium Thermodyn. 22 (1997), 110-128. Zbl 0889.73004, 10.1515/jnet.1997.22.2.110 |
| Reference:
|
[12] Giorgi, C., Naso, M. G., Pata, V.: Energy decay of electromagnetic systems with memory.Math. Models Methods Appl. Sci. 15 (2005), 1489-1502. Zbl 1078.35018, MR 2168942, 10.1142/S0218202505000844 |
| Reference:
|
[13] Han, X., Wang, M.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70 (2009), 3090-3098. Zbl 1173.35579, MR 2503053, 10.1016/j.na.2008.04.011 |
| Reference:
|
[14] Han, X., Wang, M.: General decay of energy for a viscoelastic equation with nonlinear damping.Math. Methods Appl. Sci. 33 (2010), 346-358. Zbl 1161.35319, MR 2484178, 10.1002/mma.1041 |
| Reference:
|
[15] Komornik, V., Zuazua, E.: A direct method for the boundary stabilization of the wave equation.J. Math. Pures Appl. (9) 69 (1990), 33-54. Zbl 0636.93064, MR 1054123 |
| Reference:
|
[16] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Etudes mathematiques. Dunod; Gauthier-Villars, Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[17] Matignon, M., Audounet, J., Montseny, G.: Energy decay rate for wave equations with damping of fractional order.Fourth Int. Conf. Mathematical and Numerical Aspects of Wave Propagation Phenomena (1998), 638-640. |
| Reference:
|
[18] Matos, L. P. V., Dmitriev, V.: On the stability of energy and harmonic waves in conductors with memory.SBMO/IEEE MTT-S Int. Microwave and Optoelectronics Conf. (IMOC) IEEE, Belem (2009), 528-532. 10.1109/IMOC.2009.5427530 |
| Reference:
|
[19] Maugin, G. A.: Continuum Mechanics of Electromagnetic Solids.North-Holland Series in Applied Mathematics and Mechanics 33. North-Holland, Amsterdam (1988). Zbl 0652.73002, MR 0954611, 10.1002/zamm.19890691106 |
| Reference:
|
[20] Mbodje, B.: Wave energy decay under fractional derivative controls.IMA J. Math. Control Inf. 23 (2006), 237-257. Zbl 1095.93015, MR 2211512, 10.1093/imamci/dni056 |
| Reference:
|
[21] Messaoudi, S. A.: General decay of the solution energy in a viscoelastic equation with a nonlinear source.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69 (2008), 2589-2598. Zbl 1154.35066, MR 2446355, 10.1016/j.na.2007.08.035 |
| Reference:
|
[22] Rivera, J. E. Muñoz, Naso, M. G., Vuk, E.: Asymptotic behaviour of the energy for electromagnetic systems with memory.Math. Methods Appl. Sci. 27 (2004), 819-841. Zbl 1054.35103, MR 2055321, 10.1002/mma.473 |
| Reference:
|
[23] Nicaise, S., Pignotti, C.: Stabilization of the wave equation with variable coefficients and boundary condition of memory type.Asymptotic Anal. 50 (2006), 31-67. Zbl 1139.35373, MR 2286936 |
| Reference:
|
[24] Park, J. Y., Park, S. H.: Decay rate estimates for wave equations of memory type with acoustic boundary conditions.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 993-998. Zbl 1202.35032, MR 2738648, 10.1016/j.na.2010.09.057 |
| Reference:
|
[25] Pata, V., Zucchi, A.: Attractors for a damped hyperbolic equation with linear memory.Adv. Math. Sci. Appl. 11 (2001), 505-529. Zbl 0999.35014, MR 1907454 |
| Reference:
|
[26] Wu, S.-T.: General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions.Z. Angew. Math. Phys. 63 (2012), 65-106. Zbl 1242.35053, MR 2878734, 10.1007/s00033-011-0151-2 |
| Reference:
|
[27] Yassine, H.: Stability of global bounded solutions to a nonautonomous nonlinear second order integro-differential equation.Z. Anal. Anwend. 37 (2018), 83-99. Zbl 06852543, MR 3746499, 10.4171/ZAA/1604 |
| Reference:
|
[28] Zacher, R.: Convergence to equilibrium for second order differential equations with weak damping of memory type.Adv. Differ. Equ. 14 (2009), 749-770. Zbl 1190.45007, MR 2527692 |
| . |
