General Decay Result Research Articles

A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay

Abstract In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

On a class of a coupled nonlinear viscoelastic Kirchhoff equations variable-exponents: global existence, blow up, growth and decay of solutions

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

Stabilization of a Rao–Nakra Sandwich Beam System by Coleman–Gurtin’s Thermal Law and Nonlinear Damping of Variable-Exponent Type

In this paper, we explore the asymptotic behavior of solutions in a thermoplastic Rao–Nakra (sandwich beam) beam equation featuring nonlinear damping with a variable exponent. The heat conduction in this context adheres to Coleman–Gurtin’s thermal law, encompassing linear damping, Fourier, and Gurtin–Pipkin’s laws as specific instances. By employing the multiplier approach, we establish general energy decay results, with exponential decay as a particular manifestation. These findings extend and generalize previous decay results concerning the Rao–Nakra sandwich beam equations.

Well-posedness and general energy decay for a nonlinear piezoelectric beams system with magnetic and thermal effects in the presence of distributed delay

In this paper, we consider one-dimensional nonlinear piezoelectric beams with thermal and magnetic effects in the presence of a distributed delay term acting on the heat equation. First, we show that the system is well-posed in the sense of a semigroup. Through the construction of an appropriate Lyapunov functional, we establish a general decay result for the solutions of the system, for which the exponential and polynomial decays are only special cases, under a suitable assumption on the weight of the delay that the damping effect through heat conduction is strong enough to stabilize the system even in the presence of a time delay. Furthermore, our result does not depend on any relationship between system parameters.

The influence of damping on the asymptotic behavior of solution for laminated beam

<p>This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.</p>

Well-posedness and general energy decay of solutions for a nonlinear damping piezoelectric beams system with thermal and magnetic effects

In this article, we study the piezoelectric beams with thermal and magnetic effects in the presence of a nonlinear damping term acting on the mechanical equation. First, we prove that the system is well-posed in the sense of semigroup theory. And by constructing a suitable Liapunov functional, we show a general decay result of the solution for the system from which the polynomial and exponential decay are only special cases. Furthermore, our result does not depend on any relationship between system parameters.

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

Theoretical and numerical decay results of a viscoelastic suspension bridge with variable exponents nonlinearity

AbstractStrong vibrations can cause considerable damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exists evidence that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes and winds, features and modifications such as dampers are used to stabilize these bridges. In this work, we study a nonlinear viscoelastic plate equation with variable exponents, which models the deformation of a suspension bridge. First, we establish explicit and general decay results of the system depending on the decay rate of the relaxation function and the nature of the variable‐exponents nonlinearity. Then, we perform several numerical tests to illustrate our theoretical decay results. Our results extend and generalize many earlier works in the literature.

Blow-up and general decay of solutions for a Kirchhoff-type equation with distributed delay and variable-exponents

A nonlinear Kirchhoff-type equation with distributed delay and variableexponents is studied. Under suitable hypothesis the blow-up of solutions is proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case b = 0.

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.

General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term

General Decay and Blow-up Results of a Robin-Dirichlet Problem for a Pseudoparabolic Nonlinear Equation of Kirchhoff-Carrier Type with Viscoelastic Term

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

General Decay Results for a Viscoelastic Euler–Bernoulli Equation with Logarithmic Nonlinearity Source and a Nonlinear Boundary Feedback

Stability Results for a Weakly Dissipative Viscoelastic Equation with Variable-Exponent Nonlinearity: Theory and Numerics

In this paper, we study the long-time behavior of a weakly dissipative viscoelastic equation with variable exponent nonlinearity of the form utt+Δ2u−∫0tg(t−s)Δu(s)ds+a|ut|n(·)−2ut−Δut=0, where n(.) is a continuous function satisfying some assumptions and g is a general relaxation function such that g′(t)≤−ξ(t)G(g(t)), where ξ and G are functions satisfying some specific properties that will be mentioned in the paper. Depending on the nature of the decay rate of g and the variable exponent n(.), we establish explicit and general decay results of the energy functional. We give some numerical illustrations to support our theoretical results. Our results improve some earlier works in the literature.

New general decay results for a multi-dimensional Bresse system with viscoelastic boundary conditions

<p style='text-indent:20px;'>In this paper we consider a multi-dimensional Bresse with memory-type boundary conditions. By assuming minimal conditions on the resolvent kernel, we establish an optimal explicit and general energy decay result. This result is new and substantially improves earlier results in the literature.</p>

On the existence and decay of a viscoelastic system with variable-exponent nonlinearity

In the present paper, we discuss the existence and the long-time behavior of a system of two viscoelastic equations with variable-exponent nonlinearities acting in both equations. First, we establish in details the existence of solutions, then prove explicit and general decay results, under a very general assumption on the relaxation function and for suitable given data, to solutions with enough regularities. Our results extend and generalize many earlier results in the literature.

On the time decay for a thermoelastic laminated beam with microtemperature effects, nonlinear weight, and nonlinear time-varying delay

<abstract><p>This article examines the joint impacts of microtemperature, nonlinear structural damping, along with nonlinear time-varying delay term, and time-varying coefficient on a thermoelastic laminated beam, where, the equation representing the dynamics of slip is affected by the last three mentioned terms. A general decay result was established regarding the system concerned given equal wave speeds and particular assumptions related to nonlinear terms.</p></abstract>

A general decay result for the Cauchy problem of plate equations with memory

<p style='text-indent:20px;'>In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} u_{tt}+\Delta^2 u+ u+ \int_0^t g(t-s)A u(s)ds = 0, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with <inline-formula><tex-math id="M3">\begin{document}$ A = \Delta $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ A = -Id $\end{document}</tex-math></inline-formula>. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end.</p>

Global existence and general decay of Moore–Gibson–Thompson equation with not necessarily decreasing kernel

In this paper, we consider the Moore.Gibson.Thompson equations. By using the potential well theory we obtain the existence of a global solution. Then, we prove the general decay result of solutions under weaker assumptions than the ones frequently used in the literature. In particular, the kernels we are considering are not necessarily exponentially decaying to zero as was assumed before. The present results improve also a previous work of the authors.

A general decay result for the Cauchy problem of a fractional Laplace viscoelastic equation

In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplace viscoelastic equations in the whole space , given by with and satisfying . We prove the existence of a solution formula then use the energy method in the Fourier space to establish a general decay result which depends on and . We also present three illustrative examples by the end.