- Open Access
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.
- coupled viscoelastic wave equations
- relaxation functions
- damping terms
- source terms
- global nonexistence
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 .
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).
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