Nonlinear Localized Damping Research Articles

Dynamics for wave equations connected in parallel with nonlinear localized damping

Abstract This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

Stability for coupled thermoelastic systems with nonlinear localized damping and Wentzell boundary conditions

Stability for coupled thermoelastic systems with nonlinear localized damping and Wentzell boundary conditions

Exponential stability of extensible beams equation with Balakrishnan–Taylor, strong and localized nonlinear damping

Exponential stability of extensible beams equation with Balakrishnan–Taylor, strong and localized nonlinear damping

Global attractor of the Euler-Bernoulli equations with a localized nonlinear damping

Global attractor of the Euler-Bernoulli equations with a localized nonlinear damping

Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping

Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{\alpha}z_{tt}-\Delta z_{tt}-\bigl{(}\xi_{1}+\xi_{2}\|\nabla z\|^{2}+% \sigma(\nabla z,\nabla z_{t})\bigr{)}\Delta z-\Delta z_{t}+\beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {\mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.

Double-wall carbon nanotubes model with nonlinear localized damping: Asymptotic stability

We consider the one-dimensional equations for the double-wall carbon nanotubes modeled by coupled Timoshenko elastic beam system with nonlinear arbitrary localized damping. We prove that the damping placed on an arbitrary small support, unquantized at the origin, leads to uniform decay rates (asymptotic in time) for the energy function. This study generalizes and improves previous literature outcomes.

Well-posedness and stability for a viscoelastic Petrovsky equation with a localized nonlinear damping

Well-posedness and stability for a viscoelastic Petrovsky equation with a localized nonlinear damping

Long-time dynamics of ternary mixtures with localized dissipation

In this paper, we are considering a system modeling a mixture of three interacting continua with localized nonlinear damping acting in an arbitrary small region of the interval under consideration and external forces. The main goal is to construct a smooth global attractor with a finite fractal dimension using the recent quasi-stability theory. We also study the convergence of these attractors with respect to a parameter ϵ that multiplies the external forces. This study generalizes and improves the previous paper by Freitas et al. [Discrete Contin. Dyn. Syst. B 27, 3563 (2021)].

Dynamics of locally damped Timoshenko systems

In this paper, we study the long-time dynamics of a Timoshenko system modeling vibrations of beams with non-linear localized damping mechanisms acting on both displacement and angular rotation and subjected to non-linear source terms. Using recent quasi-stability methods, we prove the existence of smooth finite dimensional global attractor, which is characterized as unstable manifold of the set of stationary solutions. Moreover, the existence of exponential attractors is shown. These aspects were not previously considered for the Timoshenko system with localized damping.

Stability results for laminated beam with thermo-visco-elastic effects and localized nonlinear damping

In this article, we investigate a one-dimensional thermoelastic laminated beam system with nonlinear damping and viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under minimal conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the first two equations, we show that the solution energy has an explicit and optimal decay rate from which the exponential and polynomial stability are just particular cases. Moreover, we establish a weaker decay result in the case of non-equal wave of speed propagation and give some examples illustrate our results. This work extends and improves the earlier results in the literature, particularly the result of Mukiawa et al. (2021).

Energy decay for a porous-elastic system with nonlinear localized damping

Energy decay for a porous-elastic system with nonlinear localized damping

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

Energy uniform decay rates for the semilinear wave equation with nonlinear localized damping and source terms of critical variable exponent

Energy uniform decay rates for the semilinear wave equation with nonlinear localized damping and source terms of critical variable exponent

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Unilateral problems for the wave equation with degenerate and localized nonlinear damping: well‐posedness and non‐stability results

AbstractUnilateral problems related to the wave model subject to degenerate and localized nonlinear damping on a compact Riemannian manifold are considered. Our results are new and concern two main issues: (a) to prove the global well‐posedness of the variational problem; (b) to establish that the corresponding energy functional is not (uniformly) stable to equilibrium in general, namely, the energy does not converge to zero on the trajectory of every solution, even if a full linear damping is taken in place.

Stability for the Mixed Problem Involving the Wave Equation, with Localized Damping, in Unbounded Domains with Finite Measure

This paper is concerned with the study of the uniform decay rates of the energy associated with mixed problems involving the wave equation with nonlinear localized damping. The domain is an unbound...

Energy decay rates for solutions of the wave equations with nonlinear damping in exterior domain

In this paper we study the behaviors of the energy of solutions of the wave equations with localized nonlinear damping in exterior domains.

Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping

We consider the Timoshenko model for vibrating beams under effect of two nonlinear and localized frictional damping mechanisms acting on the transverse displacement and on the rotational angle. We prove that the damping placed on an arbitrarily small support, unquantitized at the origin and without assuming equal speeds of propagation of waves, leads to uniform decay rates (asymptotic in time) for the energy function. This result removes the necessity (as long as both transverse displacements and rotational angles are minimally damped) of the assumption on equal speeds which has been imposed in the prior literature. The proof of this result relies on the method introduced in Daloutli et al. (Discret Contin Dyn Syst 2(1):67-94, 2009), which reduces the nonlinear stabilization to the observability inequality established for the associated linear problem. The latter is important on its own rights within the context of internal and localized controllability/observability of free linear systems. Mathematics Subject Classification. 35B40 · 35L53. ϕ(x, 0) = ϕ0(x) ,ϕ t(x, 0) = ϕ1(x) ,ψ (x, 0) = ψ0(x) ,ψ t(x, 0) = ψ1(x), (1.4)

Asymptotic stability for Bresse systems

In this paper consider a vibrating system of Bresse type in a one-dimensional bounded domain with a nonlinear damping acting in the equation of the shear angle displacement, and nonlinear localized damping in other equations. We show the asymptotic stability without imposing conditions about the equal-speed wave propagation, using a method developed by Lasiecka and Tataru [9].

Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.