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

In the paper, we study a Balakrishnan–Taylor quasilinear wave equation|zt|α⁢zt⁢t-Δ⁢zt⁢t-(ξ1+ξ2⁢∥∇⁡z∥2+σ⁢(∇⁡z,∇⁡zt))⁢Δ⁢z-Δ⁢zt+β⁢(x)⁢f⁢(zt)+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)=0in a bounded domain ofℝn{\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.

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.

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

Long-time dynamics of Kirchhoff wave models with strong nonlinear damping

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.

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.

Global existence and stabilization of the quasilinear Petrovsky equation with localized nonlinear damping

Acknowledgement.- Overview of the algorithm and the proof of the main theorem.- Reduction of main theorem to three propositions.- Proof of proposition 1.- Proof of proposition 2.- Proof of proposition 3.

Unilateral 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.

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.

The long-term behaviour of solutions to a model for acoustic–structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension—established in a previous work by Bucci et al (2007 Commun. Pure Appl. Anal. 6 113–40) only in the presence of full-interior acoustic damping—holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping.

In this paper, we study the existence of global attractors for the extensible beam equation with localized nonlinear damping and linear memory. Copyright © 2011 John Wiley & Sons, Ltd.

* Introduction* Part 1. Weakly degenerate operators with Dirichlet boundary conditions* Controllability and inverse source problems* Notation and main results* Global Carleman estimates for weakly degenerate operators* Some Hardy-type inequalities (proof of Lemma 3.18)* Asymptotic properties of elements of $H^2 (\Omega) \cap H^1 _{A,0}(\Omega)$* Proof of the topological lemma 3.21* Outlines of the proof of Theorems 3.23 and 3.26* Step 1: computation of the scalar product on subdomains (proof of Lemmas 7.1 and 7.16)* Step 2: a first estimate of the scalar product: proof of Lemmas 7.2, 7.4, 7.18 and 7.19* Step 3: the limits as $\Omega^\delta \to \Omega$ (proof of Lemmas 7.5 and 7.20)* Step 4: partial Carleman estimate (proof of Lemmas 7.6 and 7.21)* Step 5: from the partial to the global Carleman estimate (proof of Lemmas 7.9-7.11)* Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14 and 7.15)* Proof of observability and controllability results* Application to some inverse source problems: proof of Theorems 2.9 and 2.11* Part 2. Strongly degenerate operators with Neumann boundary conditions* Controllability and inverse source problems: notation and main results* Global Carleman estimates for strongly degenerate operators* Hardy-type inequalities: proof of Lemma 17.10 and applications* Global Carleman estimates in the strongly degenerate case: proof of Theorem 17.7* Proof of Theorem 17.6 (observability inequality)* Lack of null controllability when $\alpha\geq 2$: proof of Proposition 16.5* Explosion of the controllability cost as $\alpha\to 2^-$ in space dimension $1$: proof of Proposition 16.7* Part 3. Some open problems* Some open problems* Bibliography* Index

In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.

Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity