Global nonexistence of solutions for nonlinear coupled viscoelastic wave equations with damping and source terms
© Hao et al.; licensee Springer. 2014
Received: 29 October 2014
Accepted: 18 November 2014
Published: 28 November 2014
In this paper, we are concerned with a nonlinear coupled viscoelastic wave equations with initial-boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on relaxation functions, damping terms, and source terms, by using the energy method we proved a global nonexistence result for certain solutions with negative initial energy.
He established blow-up result for solutions with negative initial energy and , and gave a global existence result for arbitrary initial (in the appropriate space) if . This work was later improved by Messaoudi .
under some conditions on the relaxation function g and . They proved an exponential stability result when the relaxation function g is decaying exponentially and the function h is linear and a polynomial stability when g is decaying polynomially and h is nonlinear.
on a compact Riemannian manifold subject to a combination of locally distributed viscoelastic and frictional dissipations. It is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping.
Muñoz Rivera and Naso  studied a viscoelastic systems with nondissipative kernels and showed that if the kernel function decays exponentially to zero, then the solution decays exponentially to zero. On the other hand, if the kernel function decays polynomially as , then the corresponding solution also decays polynomially to zero with the same rate of decay.
Wang and Wang  studied a one-dimensional wave equation with viscoelastic damping under the Dirichlet boundary condition, where the kernel was taken for the finite sum of exponential polynomials. Using the asymptotic analysis technique, the authors proved an exponential stability result. Zhao and Wang  considered a coupled system of an ODE and a wave equation with Kelvin-Voigt damping, where the velocity of the wave equation entered the ODE and the output was fed into the boundary of the wave equation. They presented the asymptotic expressions of eigenvalues and eigenfunctions and established the exponential stability result.
with Dirichlet boundary condition, where , , , . An exponential decay result for and has been obtained. For and , Messaoudi and Tatar ,  showed that there exists an appropriate set, called stable set, such that if the initial data are in stable set, the solution continues to live there forever, and the solution goes to zero with an exponential or polynomial rate depending on the decay rate of relaxation function.
for , , g is a positive summable kernel, H be a real Hilbert space and the operator be a self-adjoint linear positive definite operator with domain such that the embedding is dense and compact. He introduced some new concepts such as the flatness of a kernel and gave the asymptotic result.
Lasiecka et al. discussed (1.6) with , with memory kernel satisfying the inequality , where is a given continuous positive increasing and convex function such that . They developed an intrinsic method for determining decay rates of the energy given in terms of the function .
Local existence, global existence, uniqueness, and blow-up in finite time were obtained when , , , , and the initial values satisfy some conditions.
Messaoudi and Said-Houari  dealt with the problem (1.7) and proved a global nonexistence of solutions for a large class of initial data for which the initial energy takes positive values. Also, Said-Houari et al. discussed (1.7) and proved a general decay result.
The author used perturbed energy method to show that dissipations given by the viscoelastic terms are strong enough to ensure the decay of the corresponding energy function.
Our purpose in this paper is to give the global nonexistence of solutions for coupled viscoelastic equations with damping and source terms by using the energy method.
The present work is organized as follows. In Section 2, we give some notations and material needed for this work. Section 3 contains the main result and the proof of the global nonexistence result.
In this section, we give some notations and material needed for the proof of our result.
We shall write to denote the usual norm and to denote the usual norm .
3 Global nonexistence result
In this section, we give the global nonexistence result and its proof.
Then any solution of (1.1) cannot exist for all time.
Throughout, C and represent generic positive constants.
Next we estimate .
This completes the proof of Theorem 3.1. □
The authors would like to express their gratitude to the anonymous referees for helpful and fruitful comments, and very careful reading on this paper. This research was partially supported by National Natural Science Foundation of China (61374089), Natural Science Foundation of Shanxi Province (2014011005-2), Shanxi Scholarship Council of China (2013-013), and Shanxi International Science and Technology Cooperation Projects (2014081026).
- Dafermos CM: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 1970, 37: 297-308.MathSciNetView ArticleGoogle Scholar
- Hrusa WJ: Global existence and asymptotic stability for a semilinear Volterra equation with large initial data. SIAM J. Math. Anal. 1985, 16(1):110-134. 10.1137/0516007MathSciNetView ArticleGoogle Scholar
- Muñoz Rivera JE: Asymptotic behavior in linear viscoelasticity. Q. Appl. Math. 1994, 52(4):628-648.Google Scholar
- Messaoudi SA: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 2003, 260: 58-66. 10.1002/mana.200310104MathSciNetView ArticleGoogle Scholar
- Messaoudi SA: Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 2006, 320: 902-915. 10.1016/j.jmaa.2005.07.022MathSciNetView ArticleGoogle Scholar
- Cavalcanti MM, Oquendo HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 2003, 42(4):1310-1324. 10.1137/S0363012902408010MathSciNetView ArticleGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I, Nascimento FAF: Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete Contin. Dyn. Syst., Ser. B 2014, 19(7):1987-2012. 10.3934/dcdsb.2014.19.1987MathSciNetView ArticleGoogle Scholar
- Muñoz Rivera JE, Naso MG: On the decay of the energy for systems with memory and indefinite dissipation. Asymptot. Anal. 2006, 49(3-4):189-204.MathSciNetGoogle Scholar
- Wang J, Wang JM: Spectral analysis and exponential stability of one-dimensional wave equation with viscoelastic damping. J. Math. Anal. Appl. 2014, 410(1):499-512. 10.1016/j.jmaa.2013.08.034MathSciNetView ArticleGoogle Scholar
- Zhao DX, Wang JM: Spectral analysis and stabilization of a coupled wave-ODE system. J. Syst. Sci. Complex. 2014, 27(3):463-475. 10.1007/s11424-014-2219-5MathSciNetView ArticleGoogle Scholar
- Cavalcanti MM, Domingos Cavalcanti VN, Ferreira J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 2001, 24: 1043-1053. 10.1002/mma.250MathSciNetView ArticleGoogle Scholar
- Messaoudi SA, Tatar NE: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 2007, 30: 665-680. 10.1002/mma.804MathSciNetView ArticleGoogle Scholar
- Messaoudi SA, Tatar NE: Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal. TMA 2007, 68: 785-793. 10.1016/j.na.2006.11.036MathSciNetView ArticleGoogle Scholar
- Pata V: Exponential stability in linear viscoelasticity. Q. Appl. Math. 2006, LXIV(3):499-513.MathSciNetView ArticleGoogle Scholar
- Lasiecka I, Messaoudi SA, Mustafa M: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 2013., 54(3): 10.1063/1.4793988Google Scholar
- Han X, Wang M: Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source. Nonlinear Anal. TMA 2009, 71: 5427-5450. 10.1016/j.na.2009.04.031MathSciNetView ArticleGoogle Scholar
- Messaoudi SA, Said-Houari B: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 2010, 365: 277-287. 10.1016/j.jmaa.2009.10.050MathSciNetView ArticleGoogle Scholar
- Said-Houari B, Messaoudi SA, Guesmia A: General decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear Differ. Equ. Appl. 2011, 18: 659-684. 10.1007/s00030-011-0112-7MathSciNetView ArticleGoogle Scholar
- Liu W: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. TMA 2009, 71: 2257-2267. 10.1016/j.na.2009.01.060View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.