Volume 1 • Issue 1 • PP: 47-55 • 2023
The Decay of The Solutions of a Nonlinear Viscoelastic Hyperbolic Equation
Open Access & Copyright
© 2023 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
View PDF Abstract
We study under some conditions on p, m and suitable conditions on g, the decay of solutions of the nonlinear viscoelastic hyperbolic equation in problem (P) as t→+∞, with Ω is a bounded domain in R^N (N>1), with smooth boundary Γ, and a, b, w are positive constants, m≥2, P≥2, and the function g(t) satisfying some conditions. We show that the energy of solutions decays exponentially if m =2 and polynomial if m >2, provided that the initial data are small enough.
Keywords
References
[1] Ball, J. (1977)."Remarks on blow up and nonexistence theorems for nonlinear evolutions equations". Quart. J. Math. Oxford, (2) 28. 473-486.
[2] Berrimi, S. & Messaoudi, S. (2004). "Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping". Electronic journal of differential equations. 88. 1-10.
[3] Berrimi, S. & Messaoudi, S. (2006). "Existence and decay of solutions of a viscoelastic equation with a nonlinear source". Nonlinear analysis. 64. 2314-2331.
[4] Cavalcanti, M. M. Calvalcanti, V. N. D. & Soriano, J. A. (2002). "Exponential decay for the solutions of semilinear viscoelastic wave equations with localized damping". Electronic journal of differential equations. 44. 1-44.
[5] Cavalcanti, M. M. & Oquendo, H. P. (2000). "Frictional versus viscoelastic damping in a semilinear wave equation". SIAM journal on control and optimization. 42(4). 1310-1324.
[6] Cavalcanti, M. M. Cavalcanti D. V, N. & Ferreira, J. (2001). "Existence and uniform decay for nonlinear viscoelastic equation with strong damping". Math. Meth. Appl. Sci, 24. 1043-1053.
[7] Cavalcanti, M. M. Cavalcanti, D. V. N, P. J. Filho, S. & Soriano, J. A. (2001). "Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping". Differential and integral equations, 14(1). 85-116.
[8] Dafermos, C. M. (1970). "Asymptotic stability in viscoelasticity". Arch. Rational Mech. Anal. 37. 297-308.
[9] Gazzola, F. & Sequassina, M. (2006). "Global solution and finite time blow up for damped semilinear wave equation". Ann. I. H. Pointcaré-An 23185-207.
[10] Georgiev, V. & Todorova, G. (1994). "Existence of solution of the wave equation with nonlinear damping and source terms". Journal of differential equations 109. 295-308.
[11] Gerbi, S. & Said-Houari, B. (2010). "Exponential decay for solutions to semilinear damped wave equation". Discrete and Continuous Dynamical Systems.
[12] Haraux, A. & Zuazua, E. (1988). "Decay estimates for some semilinear damped hyperbolic problems". Arch. Rational Mech. Anal. 150. 191-206.
[13] Hrusa, W. J. & Renardy, M. March (1988). "A model equation for viscoelasticity with a strongly singular kernel" SIAMJ. Math. Anal. 19(2).
[14] Ikehata, R. (1996). "Some remarks on the wave equations with nonlinear damping and source terms". Nonlinear Analysis. 27(10). 1165-1175.
[15] Kalantarov, V. K. & Ladyzhenskaya, O. A. (1978). "The occurrence of collapse for quasilinear equation of parabolic and hyperbolic type". J. Soviet Math. 10. 53-70.
[16] Kopackova, M. (1989). "Remarks on bounded solutions of a semilinear dissipative hyperbolic equation". Comment Math. Univ. Carolin, 30(4). 713-719.
[17] Levine, H. A. (1974). "Instability and nonexistence of global solutions of nonlinear wave equation of the form Pu_tt=Au+F(u"). Trans. Amer. Math. Sci, 192. 1-21.
[18] Levine, H. A. (1974). "Some additional remarks on the nonexistence of global solutions of nonlinear wave equation" SIAMJ. Math. Anal. 5. 138-146.
[19] Levine, H. A. & Serrin, J. (1997). "A global nonexistence theorem for quasilinear evolution equation with dissipative". Arch. Rational Mech Anal. 137. 341-361.
[20] Messaoudi, S. (2003). "Blow up and global existence in a nonlinear viscoelastic wave equation". Maths Nachr. 260. 58-66.
[21] Messaoudi, S. & Said-Houari, B. (2004). "Global nonexistence of solutions of a class of wave equations with nonlinear damping and source terms". Math. Meth. Appl. Sc. 27. 1687-1696.
[22] Messaoudi, S. & Tatar, N-E. (2003). "Global existence and asymptotic behavior for a nonlinear viscoelastic problem". Mathematical. Sciences research journal.7(4). 136-149.
[23] Messaoudi, S. & Tatar, N-E. (2007). "Global existence and uniform stability of a solutions for quasilinear viscoelastic problem". Math. Meth. Appl. Sci. 30. 665-680.
[24] Messaoudi, S. (2001). "Blow up in a nonlinearly damped wave equation" Math. Nachr. 231. 1-7.
[25] Munoz Rivera, J. E. & Naso, M. G. (2006). "On the decay of the energy for systems with memory and indefinite dissipation". Asymptote. anal. 49(34). 189204.
[26] Pata, Vi. September (2006). "Exponential stability in viscoelasticity". Quarterly of applied mathematics volume LXIV. number3. 499-513.
[27] Rivera, J. E. M. & Lapa, E. C. & Barreto, R. K. (1996). "Decay rates for viscoelastic plates with memory" Journal of elasticity 44. 61-87.
[28] Vittilaro, E. (1999). "Global nonexistence theorems for a class of evolution equations with dissipation". Arch. Rational Mech. Anal. 149. 155-182.
Cite This Article
Choose your preferred format
Publisher's Note
The statements, opinions, and data presented in this article are solely those of the author(s) and do not necessarily represent those of ASPG, the journal, or its editors. ASPG and the editors disclaim responsibility for any harm arising from the use of any ideas, methods, instructions, or products described in this article, to the fullest extent permitted by applicable law.