System Of Viscoelastic Wave Equations Research Articles

Decay rate of solutions to the Cauchy problem for a coupled system of viscoelastic wave equations with a strong delay in Rⁿ

"Using weighted spaces, we establish a general decay rate properties of solutions as T → ∞ for a coupled system of viscoelastic wave equations in Rⁿ under some conditions on g1, g2, φ. We exploit a density function to introduce weighted spaces for solutions and using an appropriate Lyapunov function. Keywords: Lyapunov function, relaxation function, density, decay rate, weighted spaces."

Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity

This work deals with a system of Kirchhoff‐type equations with viscoelastic term and logarithmic nonlinearity. First, we proved the global existence by combining the potential well method with Faedo–Galerkin's procedure. Later, we established decay rate estimates by using multiplier method.

Existence and decay of solution to coupled system of viscoelastic wave equations with strong damping in Rn

In this paper, we establish a general decay rate properties of solutions for a coupled system of viscoelastic wave equations in IRn under some assumptions on g1; g2 and linear forcing terms. We exploit a density function to introduce weighted spaces for solutions and using an appropriate perturbed energy method. The questions of global existence in the nonlinear cases is also proved in Sobolev spaces using the well known Galerkin method.

New general decay result for a system of viscoelastic wave equations with past history

<p style='text-indent:20px;'>This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels <inline-formula><tex-math id="M1">\begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document}</tex-math></inline-formula> satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H_i $\end{document}</tex-math></inline-formula> are functions satisfying some specific properties. Under this very general assumption on the behavior of <inline-formula><tex-math id="M4">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity, we establish a relation between the decay rate of the solutions and the growth of <inline-formula><tex-math id="M5">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchho type with a delay term in a bounded domain. At first,the energy decay rate is proved by Nakao's technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow-upof solutions is obtained.

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

General decay and blow up of solutions for a system of viscoelastic wave equations with nonlinear boundary source terms

In this work, an initial boundary value problem for a system of viscoelastic wave equations with nonlinear boundary source term of the form(ui)tt−Δui−Δ(ui)tt+∫0tgi(t−s)Δui(s)ds−Δ(ui)t=0,inΩ×(0,T),ui(x,0)=φi(x),(ui)t(x,0)=ψi(x),inΩ,ui(x,t)=0,onΓ0×(0,T),∂ν(ui)tt+∂νui−∫0tgi(t−s)∂νui(s)ds+∂ν(ui)t+fi(u)=0,onΓ1×(0,T), where i=1,...,l (l≥2) is considered in a bounded domain Ω in RN (N≥1). By the Faedo–Galerkin approximation method we obtain existence and uniqueness of weak solutions. Under appropriate assumptions on initial data and the relaxation functions, we establish general decay and blow up results associated to solution energy. Estimates for lifespan of solutions are also given.

GENERAL DECAY OF SOLUTION FOR COUPLED SYSTEM OF VISCOELASTIC WAVE EQUATIONS OF KIRCHHOFF TYPE WITH DENSITY IN Rn

A system of viscoelastic wave equations of Kirchhoff type is considered. For a wider class of relaxation functions, we use spaces weighted by the density function to establish a very general decay rate of the solution.

Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms

This paper deals with the initial boundary value problem for the viscoelastic wave equations$$\left\{\begin{array}{c}u_{t t}-\Delta u+\int_0^t g_1(t-\tau) \Delta u(\tau) d \tau+u_t=f_1(u, v), \\v_{t t}-\Delta v+\int_0^t g_2(t-\tau) \Delta v(\tau) d \tau+v_t=f_2(u, v)\end{array}\right.$$in a bounded domain. We obtain the global nonexistence of solutions by applying a lemma due to Y. Zhou [Global existence and nonexistence for a nonliear wave equation with damping and source terms, Math. Nacht, 278 (2005) 1341-1358].

Global Existence and Asymptotic Behavior for a Coupled System of Viscoelastic Wave Equations with a Delay Term

Global Existence and Asymptotic Behavior for a Coupled System of Viscoelastic Wave Equations with a Delay Term

Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms

The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ $ {{\mathbb{R}}^n} $ , n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of $ {{\mathbb{R}}^n} $ , n ≥ 1, the initial energy is positive, and the strongly nonlinear functions f1 and f2 satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari.

Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered. We prove that, under suitable assumptions on relaxation functions and certain initial data, the decay rate of the solutions energy is exponential.

General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms

The initial boundary value problem for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms in a bounded domain is considered. Under some assumptions on the relaxation functions, degenerate damping and source terms, we obtain the decay rate of the energy function for certain initial data. This improves earlier results in the literature.

General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.

On decay and blow-up of solutions for a system of nonlinear wave equations

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.

General decay of solutions of a nonlinear system of viscoelastic wave equations

This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247–263, 2008) and Liu (Nonlinear Anal. 71:2257–2267, 2009) in which only the exponential and polynomial decay rates are considered.

Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.