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In this paper, the initial value problem for a plate equation with a memory is considered. Under weaker assumptions on the memory function, we obtain the decay estimates of solutions which is of the regularity-loss type by applying the energy method in Fourier space. Moreover, if the memory kernel is an exponential function, we derive the large-time behavior and asymptotic profile of solutions in the one-dimensional space by virtue of the technique of spectral analysis. [ABSTRACT FROM AUTHOR]

We first introduce a new notation so called "asymptotic contractive process" on the time-dependent entire space, and then prove that such asymptotic contractive process is pullback asymptotically compact. Finally, we show that the existence of time-dependent attractor for the plate equation on H 2 (R n) × L 2 (R n) by using this new abstract result. [ABSTRACT FROM AUTHOR]

In this paper, we derive L 1 − L 1 long time estimates for the strongly damped plate equation u t t + Δ 2 u + Δ 2 u t = 0 x ∈ R n , t ∈ R + , u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x). In particular, we prove that ‖ u (t , ⋅) ‖ L 1 ≤ C (1 + t) n 4 (‖ u 0 ‖ L 1 + (1 + t) 1 2 ‖ u 1 ‖ L 1 ) , for any t ≥ 0 , in space dimension n ≥ 5. [ABSTRACT FROM AUTHOR]

We introduce an energy stable, high-order-accurate finite difference approximation of the dynamic, pure bending Kirchhoff plate equations for complex geometries and spatially variable properties. We utilize the summation-by-parts (SBP) framework to discretize the biharmonic operator with variable coefficients, with attention given to free and clamped boundary conditions and corner conditions. Energy conservation is established by combining SBP boundary closures with weak enforcement of the boundary and interface conditions using a penalty (simultaneous approximation term, SAT) technique. Then we couple the plate equations to the shallow water equations to study flexural-gravity wave propagation, and prove that the semi-discrete system of equations is self-adjoint. We demonstrate the stability and accuracy properties of the method on curvilinear multiblock grids using the method of manufactured solutions. The method, which we provide in an open-source code, is then used to model ocean wave interactions with the Thwaites Glacier and Pine Island Ice Shelf in the Amundsen Sea off the coast of West Antarctica. • Finite difference discretization of variable coefficient biharmonic operator with boundary treatment. • Proof of energy stability and semi-discrete self-adjointness. • Application problems in ocean wave-ice shelf interactions off the coast of West Antarctica. [ABSTRACT FROM AUTHOR]

In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of a frictional damping. We prove, for a suitably chosen initial data, a global existence as well as a decay result. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. The problem corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system and guarantees the existence of a finite-dimensional global attractor. In addition, the existence of exponential attractors is also considered. [Copyright &y& Elsevier]

Abstract: We study the exponential stability of the plate equations with potential of second order and indefinite sign damping term. By means of a global Carleman-type estimate and the usual perturbation method, we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small. Both the energy decay rate and the upper bound estimate on the negative damping are given explicitly. [Copyright &y& Elsevier]

Abstract: In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space . [Copyright &y& Elsevier]

Abstract: Let A be a possibly unbounded skew-adjoint operator on the Hilbert space X with compact resolvent. Let C be a bounded operator from to another Hilbert space Y. We consider the system governed by the state equation with the output . We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 (2004) 443–471) and Miller (J. Funct. Anal. 218 (2) (2005) 425–444). We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrödinger equation, the Bernoulli–Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result on shifted squares. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]