General Decay Estimates Research Articles

General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

In this work we investigate a viscoelastic wave equation involving a logarithmic nonlinear source and dynamic Wentzell boundary condition. Making some assumptions on the memory kernel function and using convex function theory and Lyapunov method, we establish the general decay estimate of the solutions. Finally we give two examples to illustrate our results.

General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions

In this paper, we consider a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions in domains with non‐locally reacting boundary. By potential well method, we get the global existence of solution and through several important lemmas, we prove the general energy decay estimate of the system.

Existence and general decay estimates of the Dirichlet problem for a nonlinear Kirchhoff wave equation in an annular with viscoelastic term

This paper is devoted to the study of a Kirchhoff wave equation with a viscoelastic term in an annular associated with homogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we establish a sufficient condition such that any global weak solution is general decay as t → +∞.

General energy decay estimate for a viscoelastic damped swelling porous elastic soils with time delay

This paper is concerned with the viscoelastic damped swelling porous elastic soils with time delay. Under more general assumption on the relaxation function and some specific conditions on the weight of the delay, we establish general decay results by using multiplier method and some properties of convex functions. These results generalize and improve some earlier related results in the literature.

A new general decay for a transmission problem of viscoelastic Timoshenko systems

AbstractIn this paper, we investigate the long‐time behavior for a transmission problem of viscoelastic Timoshenko systems with different speeds of wave propagation. By constructing a new Lyapunov functional and combining the technique of perturbation energy with some precise estimates for multipliers, we establish a general uniform decay estimates for the energy.

Global solvability and decay estimates for a type III thermo‐viscoelastic coupled system with infinite memory and boundary interaction feedback

AbstractA type III thermo‐viscoelastic coupled system with infinite memory and distributed delay is considered. The interaction feedback between the nonlinear damping and the acoustic conditions are reacted on portion of the boundary. We obtain the well posedness and regularity of the system by using semigroup theory which is combined with Schauder's fixed point theorem. Moreover, the general decay estimates are established under a much larger class of relaxation functions. Our results are obtained without the boundedness condition of initial data assumed in many earlier papers in the literature. This work generalizes the composite stability between infinite memory and nonlinear damping.

General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary

<abstract><p>In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value $ w_0 \equiv 0 $ on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels $ k_i(i = 1, 2) $, namely $ k_i''(t) \geq - \xi_i(t) G_i(-k_i'(t)) $, where $ G_i $ are convex and increasing functions near the origin and $ \xi_i $ are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions $ \xi_i $ and $ G_i. $ These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature.</p></abstract>

General decay estimate for a viscoelastic wave equation with Balakrishnan-Taylor damping and strong dissipation

General decay estimate for a viscoelastic wave equation with Balakrishnan-Taylor damping and strong dissipation

General decay and blow-up results for a class of nonlinear pseudo-parabolic equations with viscoelastic term

In this paper, we consider initial boundary value problem of the generalized pseudo-parabolic equation contain viscoelastic term. We establish firstly the local existence of solutions by Banach fixed point theorem. Then we prove blow-up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow-up rate for the weak solution via finding the upper bound and the lower bound for the blow-up times and the upper bound and the lower bound for the blow-up rate. For negative energy, we introduce a new method to prove blow-up results with sharper estimate for upper bound for the blow-up times. Finally, we prove the global existence of the solution and a general decay estimate under some suitable assumptions.

General decay estimate for coupled plate problem with memory

This article considers a coupled system of non-linear plate equations with partially hinged boundary conditions. By assuming general conditions on the relaxation functions, we show that the associated energy functional satisfy a general decay estimate. The importance of this work is the extension and generalization of a considerable number of results in the case of wave equations to that of plate equations.

General Decay of a Nonlinear Viscoelastic Wave Equation with Balakrishnân-Taylor Damping and a Delay Involving Variable Exponents

This paper was aimed at investigating the stability of the following viscoelastic problem with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term u t t − M ∇ u 2 2 Δ u + α t ∫ 0 t g t − s Δ u s d s + μ 1 u t p . − 2 u t + μ 2 u t t − τ p . − 2 u t t − τ = 0 in Ω × ℝ + , where Ω is a bounded domain of ℝ n , p . : Ω ¯ ⟶ ℝ is a measurable function, g > 0 is a memory kernel that decays exponentially, α ≥ 0 is the potential, and M ∇ u 2 2 = a + b ∇ u t 2 2 + σ ∫ Ω ∇ u ∇ u t d x for some constants a > 0 , b ≥ 0 , and σ > 0 . Under some assumptions on the relaxation function, we use some suitable Lyapunov functionals to derive the general decay estimate for the energy. The problem considered is novel and meaningful because of the presence of the flutter panel equation and the spillover problem including memory and variable-exponent time delay control. Our result generalizes and improves previous conclusion in the literature.

Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study

<abstract><p>In this work, we consider a viscoelastic wave equation with boundary damping and variable exponents source term. The damping terms and variable exponents are localized on a portion of the boundary. We first, prove the existence of global solutions and then we establish optimal and general decay estimates depending on the relaxation function and the nature of the variable exponent nonlinearity. Finally, we run two numerical tests to demonstrate our theoretical decay results. This study generalizes and enhances existing literature results, and the acquired results are thus of significant importance when compared to previous literature results with constant or variable exponents in the domain.</p></abstract>

Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function.

Existence and general decay estimates for a Petrovsky-Petrovsky coupled system with nonlinear strong damping

In this paper, we consider a coupled system of Petrovsky-Petrovsky equations with nonlinear dissipative terms. We proved the existence and stability of solution to the coupled system (\ref{P}) under some assumptions (\ref{Ep1})-(\ref{2g}) based on the work \cite{benaissaM}. The approach adopted is the Faedo-Galerkin method. Furthermore, by applying the multiplier method and some weighted integral inequalities, we strictly proved the decay properties (\ref{26}).

General decay estimate for a two-dimensional plate equation with time-varying feedback and time-varying coefficient

In this paper, we consider a plate equation as a model for a suspension bridge with a general nonlinear internal feedback and time-varying weight. Under some conditions on the feedback and the coefficient functions, we establish a general decay estimate for the associated energy functional, from which the exponential and the polynomial, are particular cases.

A General Stability Result for a Viscoelastic Moore–Gibson–Thompson Equation in the Whole Space

In this paper, we are interested in a viscoelastic Moore–Gibson–Thompson equation with a type-II memory term and a relaxation function satisfying $$g^{\prime }(t)\le -\eta (t)g(t)$$ . By constructing appropriate Lyapunov functionals in the Fourier space, we establish a general decay estimate of the solution under the condition $$\left( \beta -\frac{\gamma }{\alpha }-\frac{\varrho }{2}\right) >0.$$ We then give the decay rate of the L $$^{2}$$ -norm of the solution. We also give two examples to illustrate our theoretical results.

A new optimal and general stability result for a thermoelastic Bresse system with Maxwell–Cattaneo heat conduction

The present paper investigates the stability of a one-dimensional thermoelastic-Bresse system, where viscoelastic damping is acting on the vertical angle displacement and thermal dissipation governed by Maxwell–Cattaneo’s law is effective on the shear angle displacement. Assuming very general conditions on the memory term and physical parameters, we show that the solution energy has a general and optimal decay estimate from which the exponential and polynomial decay estimates are only particular cases.

Asymptotic stability of a viscoelastic problem with time-varying delay in boundary feedback

In this paper, we investigate a nonlinear viscoelastic equation. By assuming time-varying delay feedback acting on the boundary, under certain assumptions on the given data, the general decay estimates for the energy are established by introducing suitable Lyapunov functionals. This model improves earlier ones in the literature in which only the dissipative term in the feedback condition is considered.

Stability Result for a New Viscoelastic–Thermoelastic Timoshenko System

In this work, we prove a general and optimal decay estimates for the solution energy of a new thermoelastic Timoshenko system with viscoelastic law acting on the transverse displacement. Therefore, exponential and polynomial decay rates are obtained as particular cases. The result is obtained under the assumption of equal speed of wave propagation.

Optimal decay for a wave transmission problem with fading memory

The transmission of waves over a body composed of two different materials are considered in this paper. One component is a simple elastic part while the other is a viscoelastic component with time-varying delay and damping effect. Besides, one part of the boundary is controlled by a nonlinear memory source term while the other part of the boundary is clamped. Under the influence of time-varying delay and damping effect, we derive a general decay estimate of the energy when the internal viscoelastic memory and the boundary memory decay simultaneously. Moreover, we present some examples to illustrate that the decay rate of the energy is optimal in some range of decay exponent of the memory.