Han, Xiaosen; Wang, Mingxin Global existence and uniform decay for a nonlinear viscoelastic equation with damping. (English) Zbl 1173.35579 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, 3090-3098 (2009). Summary: We investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established. Cited in 1 ReviewCited in 58 Documents MSC: 35L35 Initial-boundary value problems for higher-order hyperbolic equations 45K05 Integro-partial differential equations 35L90 Abstract hyperbolic equations 74Dxx Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) 35B40 Asymptotic behavior of solutions to PDEs Keywords:global existence; exponential decay; nonlinear viscoelastic equation; damping PDFBibTeX XMLCite \textit{X. Han} and \textit{M. Wang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, 3090--3098 (2009; Zbl 1173.35579) Full Text: DOI References: [1] Andrade, D.; Fatori, L. H.; Rivera, J. M., Nonlinear transmission problem with a dissipative boundary condition of memory type, Electronic Journal of Differential Equations, 53, 1-16 (2006) [2] Berrimi, S.; Messaoudi, S. A., Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 64, 2314-2331 (2006) · Zbl 1094.35070 [3] Berrimi, S.; Messaoudi, S. A., Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electronic Journal of Differential Equations, 88, 1-10 (2004) · Zbl 1055.35020 [4] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ferreira, J., Existence and uniform decay for a nonlinear viscoelastic equation with strong damping, Mathematical Methods in the Applied Sciences, 24, 1043-1053 (2001) · Zbl 0988.35031 [5] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ma, T. F.; Soriano, J. A., Global existence and asymptotic stability for viscoelastic problems, Differential and Integral Equation, 15, 731-748 (2002) · Zbl 1015.35071 [6] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ma, T. F., Exponential decay of the viscoelastic Euler-Bernouli equation with a nonlocal dissipation in general domains, Differential and Integral Equation, 17, 495-510 (2004) · Zbl 1174.74320 [7] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Soriano, J. A., Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations, 44, 1-14 (2002) · Zbl 0997.35037 [8] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Martinez, P., General decay rate estimates for viscoelastic dissipative systems, Nonlinear Analysis, 68, 177-193 (2008) · Zbl 1124.74009 [9] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Prates Filho, J. S.; Soriano, J. A., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations, 14, 1, 85-116 (2001) · Zbl 1161.35437 [10] Cavalcanti, M. M.; Oquendo, H. P., Frictional versus viscoelastic damping in a semilinear wave equation, SIAM Journal on Control and Optimization, 42, 1310-1324 (2003) · Zbl 1053.35101 [11] Messaoudi, S. A.; Tatar, N., Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Analysis, 68, 785-793 (2008) · Zbl 1136.35013 [12] Messaoudi, S. A.; Tatar, N., Global existence and uniform decay of solutions for a quasilinear viscoelastic problem, Mathematical Methods in the Applied Sciences, 30, 665-680 (2007) · Zbl 1121.35015 [13] Messaoudi, S. A., Blow-up and global existence in a nonlinear viscoelastic wave equation, Mathematische Nachrichten, 260, 58-66 (2003) · Zbl 1035.35082 [14] Messaoudi, S. A., Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, Journal of Mathematical Analysis and Applications, 320, 902-915 (2006) · Zbl 1098.35031 [15] Zheng, S. M., Nonlinear Evolution Equation (2004), CRC Press: CRC Press Boca Raton This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.