Bhargav Kumar Kakumani, Suman Prabha Yadav
In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Using the the Faedo-Galerkin method we establish the global existence of the solution of the problem and we also prove few general decay rate results.
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In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalised Young's inequality are mainly used to prove the decay estimate.
We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von Kármán type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and Müller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and iden…
Energy decay and blow-up of viscoelastic wave equations with polynomial nonlinearity and damping
本文研究带多项式非线性和弱阻尼的粘弹性波动方程，建立了小初始能量下解的显式衰减结果，并推导了初始能量低于势阱深度时爆破时间的上下界。
Energy decay of nonlocal viscoelastic equations with nonlinear damping and polynomial nonlinearity
本文通过Lyapunov方法研究非局部粘弹性方程的能量衰减，利用核函数的凸性假设改进能量衰减率，并通过实例验证了更快的多项式或指数衰减。
In this paper, we consider a weakly coupled system consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions.
Energy decay of a viscoelastic wave equation with variable exponent logarithmic nonlinearity and weak damping
本文研究了具有变量指数对数非线性和弱阻尼的粘弹性波方程的能量衰减问题，推导出一般衰减估计并改进了现有结果，获得了指数和多项式衰减率。
We consider the initial value problem for the semilinear plate equation with nonlocal nonlinearity. We prove the existence of global attractor and then establish the regularity and finite dimensionality of this attractor.
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.
We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates. Computational experiments are provided, too.
Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory
本文研究非自治粘弹性Kirchhoff板方程的记忆效应下均匀吸引子的上半连续性，通过能量估计和反证法证明其存在性及连续性。
In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. Our aim is to obtain the threshold, to classify the global existence of solution for small data or the finite time blow-up pf the solution, with respect to the growth order of the nonlinearity.
We consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin's-Voigt's rheology derived from a three-dimensional model at a finite-strain setting. As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty and complement the $Γ$-convergence analysis of elastic von Kármán ribbons in [Fr…
On the Control of Solutions of a Viscoelastic Plate Problem with a Frictional Damping Term
本文研究具有边界摩擦阻尼和对数源的粘弹性板问题的解的稳定性，通过松弛函数满足的微分不等式条件，扩展了先前工作并改进了早期结果。
Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation
本文研究了黏弹性 Klein-Gordon 方程中小解的长期行为，通过时空共振方法分析了解的全局存在性和扩散衰减特性。
Global existence for wave and beam equations with double damping and a new power nonlinearity
研究双重阻尼和新幂非线性对波方程和梁方程解的全局存在性影响，通过定义新的量Q[u](t)证明线性模型中摩擦和粘弹性阻尼的相互作用导致Q[u](t)指数衰减。
We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.
Inviscid fluid interacting with a nonlinear two-dimensional plate
该研究针对无粘流体与二维非线性Koiter板相互作用的移动边界问题，采用ALE坐标固定定义域，推导了无粘弹性平滑条件下的多类先验估计以证明解的局部时间存在性，并拓展至分数阶Sobolev空间与归一化第二基本形式的情形。
One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams
本文基于三维 Kelvin-Voigt 模型，通过令薄壁梁的高度与宽度同时趋于零，利用静态与演化 Γ-收敛及梯度流理论，严格导出包含拉伸、弯曲和扭转的一维粘弹性 von Kármán 极限模型。
The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogenous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly non- linear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in th…
Stabilization and control of the nonlinear plate equation
本文基于非线性板方程的线性可观察性，提出了一种新的唯一延续性质，实现了半全局稳定性和精确控制，通过分析非线性能量衰减和控制属性。
Derivation of a von Kármán plate theory for thermoviscoelastic solids
本文从三维拟静态非线性模型推导出热粘弹性固体的 von Kármán 板理论，证明在降维极限下方程组弱解收敛于耦合线性热传导方程的二维有效系统，并克服了薄区域先验估计的挑战。
Stability and Finite-Time Blow-Up for a Fractionally Damped Nonlinear Plate Equation: Numerical and Analytical Insights
本文研究分数阻尼非线性板方程的稳定性与有限时间爆破问题，通过分析与数值模拟揭示了分数阻尼与延迟反馈对系统行为的竞争效应。
Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation
本文研究了具有过去历史和非局部边界阻尼的粘弹性梁方程的动力学，证明了其动力系统具有紧致全局吸引子和分形指数吸引子。
We consider the family of singularly nonautonomous plate equation with structural damping \[ u_{tt} + a(t,x)u_{t} + (- Δ) u_{t} + (-Δ)^{2} u + λu = f(u), \] in a bounded domain $Ω\subset \R^n$, with Navier boundary conditions. When the nonlinearity $f$ is dissipative we show that this problem is globally well posed in $H^2_0(Ω) \times L^2(Ω)$ and has a family of pullback attractors which is upper-semicontinuous under small perturbations of the damping $a$.
We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} …
This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild solutions for small initial data in $L^{1}(\mathbb{R}^{n})\cap H^s(\mathbb{R}^{n})$-spaces. We also prove the existence and uniqueness of local and global solutions in the framework of Bessel-potential spaces $H^s_p(\mathbb{R}^n)=(I-Δ)^{s/2}L^p(\mathbb{R}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the…
In this paper we study a nonlinear transmission problem for a plate which consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. Main result is the proof of the asymptotic smoothness of this dynamical system. Also we prove the existence of a compact global attractor in particular cases when the nonlinearity is of Berger type or scalar.
The critical exponent for semilinear wave equations with damped oscillations
该研究确定了一类带幂次非线性项且具特殊耗散的演化方程的全局小初值解的存在性指数，针对n=3的带黏弹性耗散的波动方程，相关爆破结果已被证明，且该指数最优。
Convergent numerical schemes for the viscoelastic Giesekus model in two dimensions
本文提出了一种用于二维粘弹性Giesekus模型的稳定收敛数值方案，并证明其对大数据全局弱解的收敛性，同时通过实验验证了方法的有效性。
The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von K{á}rm{á}n \cite{Karman}. {This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM).} Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation q…
System of partial differential equations with a convolution terms and non-local nonlinearity describing oscillations of plate due to Berger approach and with accounting for thermal regime in terms of Coleman-Gurtin and Gurtin-Pipkin law and fading memory of material is considered. The equation is transformed into a dynamical system in a suitable Hilbert space which asymptotic behaviour is analysed. Existence of the compact global attractor in this dynamical system and some of its properties are proved in this article. Main tool in analysis of asymptotic behaviour is stabilizability inequality.
In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well-known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of "hyperbolic type" with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained and nonlinear damping. In this paper we first show the existence of a comp…
We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von Kármán and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitti…
We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system,…
We consider a nonlinear (Berger or Von Karman) clamped plate model with a {\em piston-theoretic} right hand side---which include non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of st…
Inverse problems for nonlinear Kirchhoff plate equations with multiple unknown parameters
本文研究非线性基尔霍夫板方程的逆边值问题，建立正问题整体适定性，针对被动、主动测量情形分别实现初始数据、初始数据与系数的恢复，提出Runge逼近等方法，可推广至多类板模型。
In this article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ \begin{cases} u_{tt} +ηΔ^2 u+a(t)Δθ=f(t,u), & t>τ,\ x\inΩ,\\ θ_t-κΔθ-a(t)Δu_t=0, & t>τ,\ x\inΩ, \end{cases} $$ subject to boundary conditions $$ u=Δu=θ=0,\ t>τ,\ x\in\partialΩ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $η>0$ and $κ>0$ are constants, $a$ is an Hölder continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.
Quasi-periodic response solutions of nonlinear plate models with nonlocal energy damping
本文研究具有非局部能量阻尼的d维非线性板模型，通过约化结合Nash–Moser迭代克服导数损失，证明了小拟周期强迫下响应解的持久性。
Weakly nonlinear models for hydroelastic water waves
本文推导了耦合非线性粘弹性板的水弹性水波界面简化模型，在弱非线性小坡度 regime 下获得双向非局部演化方程，捕捉界面动力学至二次项，并推导了描述单向传播的两种模型，保留板诱导的主导色散和耗散效应。
Time-periodic solutions for viscous fluids interacting with nonlinear Koiter plates
本文研究了粘性流体与非线性Koiter板相互作用系统中时周期弱解的存在性，采用单阶段Leray-Schauder固定点方法直接应用于完全耦合的Galerkin系统，克服了非线性Koiter能量破坏解映射凸性的问题。
A full theory for hinged beams and degenerate plates with multiple intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem in one dimension. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is underlined. A complete spectral theorem is then proved, explicitly determining the eigenvalues on varying of the position of the piers and exhibiting the fundamental modes of oscillation. The obtained eigenfunctions are used to tackle the study of the nonlinear evolution prob…
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