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We consider the Wentzell Laplacian and its square in general domains. Applications to the Cauchy problems associated with Wentzell heat, wave and plate equations are presented.

We consider an evolution plate equation aiming to model the motion of the deck of a periodically forced strongly prestressed suspension bridge. Using the prestress assumption, we show the appearance of multiple time-periodic uni-modal longitudinal solutions and we discuss their stability. Then, we investigate how these solutions exchange energy with a torsional mode. Although the problem is forced, we find a portrait where stability and instability regions alternate. The techniques used rely on ODE analysis of stability and are complemented with numerical simulations.

We consider the initial value problem for the plate equation with (t, x) −depending complex valued lower order terms. Under suitable decay conditions as |x|→∞ on the imaginary part of the subprincipal term we prove energy estimates in weighted Sobolev spaces. This provides also well posedness of the Cauchy problem in the Schwartz space \(\mathcal {S}(\mathbb R^n)\) and in \(\mathcal {S}^\prime (\mathbb R^n)\).

In this note, we study a damped plate equation. On the one hand, the action of the damping creates a smoothing effect in Gevrey classes, on the other hand, it dissipates the energy of the solution.

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babu\v{s}ka paradox, highlighting the geometrically non-conforming aspect of the HHJ method., Comment: 27 pages, 4 figures, 4 tables. To appear in SIAM Journal on Numerical Analysis

International audience; We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas we add a damping in the plate equation. We use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if we assume the presence of frictions in the boundary conditions.

This brief paper mainly investigates the problem of approximate controllability of a bilinear vibrating plate equation for a class of desired states using distributed controls and a relation between the controllability and stabilization. Besides, this approach is successfully illustrated by simulations.

Two classes of plate equations with past history and strong time-dependent delay in the internal feedback are considered. Our results contain the global well-posedness and exponential stability of the two systems. We prove the global well-posedness of a system with rotational inertia without any restrictions on $\mu_1$, $\mu_2$, and the system without rotational inertia under the assumption $|\mu_2| \leq \mu_1$. For the system with rotational inertia, we establish exponential stability to the plate equation with the memory term only to control the delay term if the amplitude of the time delay term is small, and the stability result also holds for the plate equation with strong anti-damping. For the system without rotational inertia, we obtain the exponential stability under the assumption $|\mu_2| \lt \sqrt{1-d} \mu_1$.

In this paper we consider a nonlinear class viscoelastic plate equation with a lower order by perturbation of p ( x , t ) -Laplace operator of the form u t t + Δ 2 u − Δ p ( x , t ) u + ∫ 0 t g ( t − s ) Δ u ( s ) d s − ϵ Δ u t + f ( u ) = 0 , ( x , t ) ∈ Q T = Ω × ( 0 , T ) , associated with initial and Dirichlet–Neumann boundary conditions. Under suitable conditions on g , f and the variable exponent of the p ( x , t ) -Laplace operator, we prove a blow up in finite time with negative initial energy in the presence of a strong damping ϵ Δ u t ( ϵ > 0 ) acting in the domain. This equation corresponds to a viscoelastic version arising in dynamics of elastoplastic flows and plate vibrations.

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in \cite{AB01, ACK01, AB02, AL12} for the linear damping case, and in \cite{AB07} for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of \cite{AB05, AB10}., 32 pages, no figures

In this paper we consider the initial value problem of an inertial model for a generalized semilinear plate equation with memory in Rn (n ≥ 1). We study the decay and the regularity-loss property for this type of equations in the spirit of [1, 2]. The novelty of this paper is that we extend the order of derivatives from integer to fraction and refine the results in the related literature [1, 3].

In this paper we consider a viscoelastic plate equation with a nonlinear weakly dissipative feedback localized on a part of the boundary. Without imposing restrictive assumptions on the boundary frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

In this paper, we study the long-time dynamics of solutions to a nonlinear nonautonomous extensible plate equation with a strong damping. Under some suitable assumptions on the initial data, the nonlinear term and external force, we establish the existence of global solutions that generate a family of processes for the problem and obtain uniform attractors corresponding to strong and weak symbol spaces in a bounded domain Ω⊆Rn(n⩾1). Copyright © 2016 John Wiley & Sons, Ltd.

The dynamics of functionally graded micropolar plates is considered. The derivation process is based on power series expansions in the thickness coordinate. Using the three-dimensional equations of motion for micropolar continuum, variationally consistent equations of motion and end boundary conditions are derived in a systematic fashion up to arbitrary order. Numerical results are presented for simply supported plates using different material distributions for both low and high order truncation orders. These results illustrate that the present approach renders benchmark solutions provided higher order truncations are used, and act as engineering plate equations using low order truncation.

We prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped plate equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent.

In this paper we are concerned with local existence, regularity and continuous dependence upon the initial data of $$\epsilon $$ -regular mild solutions for the abstract integrodifferential equation (1, 2). We also present a result on unique continuation and a blow-up alternative for an $$\epsilon $$ -regular mild solution of (1, 2). Finally, we apply our results to three interesting models: Navier–Stokes equations with memory, diffusion equations with memory and a strongly damped plate equation with memory.

Recently, Ferrero and Gazzola, [Disc. Cont. Dyn. Syst. 35: 5879–5908 (2015)], suggested and investigated a rectangular plate model describing the statics and dynamics of a suspension bridge. The plate is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. This reliable model aims to describe more accurately the motion of suspension bridges compared to all previous known models. In the present paper, we consider a plate equation in the presence of memory and subject to the above-mentioned boundary conditions. We give a rigorous well-posedness result and establish the existence of a global attractor.

In this work we study decay rates for the L 2 -norm of solutions to the plate equation with fractional damping and a pseudo fractional rotational inertia term. We also show that the decay rates depending on the fractional power of the damping term are optimal using an asymptotic expansion of the corresponding solution of the related equation in the Fourier space.

We discuss the solvability of the periodic Navier problem for the plate equation with forced vibrationsxtt(t, y)+Δ2x(t, y)+l(t, y, x(t, y)) = 0 in higher dimensions with side lengths being irrational numbers and the nonlinearity being superlinear. We also derive a new dual variational method.

In this paper, we consider the plate equation with both weak frictional damping and viscoelastic damping acting simultaneously and complementarily in the domain. An energy decay rate formula is obtained under nonrestrictive hypotheses on the relaxation function and the frictional damping term. Our results improve and generalize previous results existing in the literature.

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow-up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory.

In this paper we consider an initial-boundary value problem of a semilinear regularity-loss-type plate equation with memory in a bounded domain of \begin{document}$ \mathbb{R}^n $\end{document} (\begin{document}$ n = 1,2,\cdots $\end{document}). By using the Faedo-Galërkin method and some theories of ordinary differential equations, we obtain the local existence and uniqueness of weak solutions. Then, we study the dynamics of the weak solutions, such as global existence and finite time blow-up, by energy estimation and some ordinary differential inequalities. Moreover, the upper bound of blow-up time for the blow-up solutions is also considered.

In this paper, we consider the following plate problem: u t t − σ Δ u t t + Δ 2 u − ∫ 0 + ∞ k ( s ) Δ 2 u ( t − s ) d s = 0 , and we show that the stability of this problem holds for a much larger class of kernels. More precisely, we consider the kernel k : [ 0 , + ∞ ) → ( 0 , + ∞ ) satisfies k ′ ( t ) ≤ − ξ ( t ) Ψ ( k ( t ) ) , t ≥ 0 , where ξ and Ψ are functions satisfying some specific properties. Under this very general assumption on the behavior of k at infinity, we establish a relation between the decay rate of the solution and the growth of k at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumption on the history data, usually made in the literature.

In this paper we explicitly calculate the first correction in the small amplitude homogenization limit of a sequence of periodic tensors describing material properties in the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. We also discuss smooth dependence of H-limit on a parameter and calculate H-limit of a periodic sequence of tensors.

We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity. (C) 2019 Elsevier Inc. All rights reserved.

International audience; In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han and Wang [26].

We consider the linear thermoelastic plate equations with free boundary conditions in uniform \begin{document}$ C^4 $\end{document} -domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in \begin{document}$ L^p $\end{document} -spaces for all \begin{document}$ p\in(1, \infty) $\end{document} and has maximal \begin{document}$ L^q $\end{document} - \begin{document}$ L^p $\end{document} -regularity on finite time intervals. On bounded \begin{document}$ C^4 $\end{document} -domains, we obtain exponential stability.

Due to an error in the typesetting process, reference [1] is incorrectly published in the original publication of the article.

We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.

In this work we study decay rates for a hyperbolic plate equation under effects of an intermediate damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient. We obtain decay rates with very general conditions on the time-dependent coefficient (Theorem 2.1, Section 2), for the power fractional exponent of the Laplace operator $(-\Delta)^\theta$, in the damping term, $\theta \in [0,1]$. For the special time-dependent coefficient $b(t)=\mu (1+t)^{\alpha}$, $\alpha \in (0,1]$, we get optimal decay rates (Theorem 3.1, Section 3).

The initial boundary value problem for a Kirchhoff type plate equation in a bounded domain is considered. We show the blow-up of solutions and the lifespan estimates for three different ranges of initial energy. Global existence of solutions is proved by the potential well theory, and decay estimates of the energy function are established by using Nakao's inequality.

In this paper, we consider a viscoelastic plate equation withp-Laplacianu ′′ + ∆ 2 u − div(|∇u| p−2 ∇u) + σ(t)Z t0 g(t − s)∆u(s)ds − ∆u ′ = 0.By introducing suitable energy and Lyapunov functionals, we establish ageneral decay estimate for the energy, which depends on the behavior ofboth σ and g. 1. IntroductionThis work is concerned with the asymptotic stability of solutions for a vis-coelastic plate equation with p-Laplacian of the form(1.1) u ′′ +∆ 2 u−div(|∇u| p−2 ∇u)+σ(t)Z t0 g(t −s)∆u(s)ds −∆u ′ = 0 in Ω×(0,∞),(1.2) u = ∆u = 0 on Γ×(0,∞),(1.3) u(0) = u 0 , u ′ (0) = u 1 in Ω,where Ω is a bounded domain in R n with smooth boundary Γ, σ and g are realfunctions.Thismodel canbe regardedasafourth orderweakviscoelasticplateequationwith a lower order perturbation of p-Laplacian type, and which is related toone-dimensional nonlinear equation of elastoplastic microstructure flows givenby(1.4) u ′′ +u xxxx −a(u 2x ) x = 0. Received April 21, 2014; Revised August 28, 2014.2010 Mathematics Subject Classification. 35L70, 35B40.Key words and phrases. plate equation, p-laplacian, weak viscoelasticity, general decay.This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea(NRF) funded by the Ministry of Science, ICT and FuturePlanning(Grant No. 2014R1A1A3A04049561).

In this paper we consider a plate equation with viscoelastic damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalize previous results existing in the literature.

This paper is addressed to a stabilization problem of a system coupled by a wave and a Euler-Bernoulli plate equation. Only one equation is supposed to be damped. Under some assumption about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically at infinity without any geometric conditions on the effective damping domain. The proofs of these decay results rely on the interpolation inequalities for the coupled elliptic-parabolic systems and make use of the estimate of the resolvent operator for the coupled system. The main tools to derive the desired interpolation inequalities are global Carleman estimates.

We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.

We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.

A Kirchhoff type plate equation with memory is investigated. Under the suitable assumptions, we establish the existence of a global attractor by using the contraction function method.