Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation

We consider a semilinear wave equation, defined on a two-dimensional bounded domain Ω, with a nonlinear dissipation. Our main result is that the flow generated by the model is attracted by a finite dimensional global attractor. In addition, this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation. To our knowledge this is a first result of this type in the context of higher dimensional wave equations.     

Upper semicontinuity of global attractors for a viscoelastic equations with nonlinear density and memory effects

This paper is devoted to showing the upper semicontinuity of global attractors associated with the family of nonlinear viscoelastic equations in a three‐dimensional space, for f growing up to the critical exponent and dependent on ρ ∈ [0,4), as ρ→0⁺. This equation models extensional vibrations of thin rods with nonlinear material density ?(?_tu) = |?_tu|^ρ and presence of memory effects. This type of problems has been extensively studied by several authors; the existence of a global attractor with optimal regularity for each ρ ∈ [0,4) were established only recently. The proof involves the optimal regularity of the attractors combined with Hausdorff's measure.     

Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous 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 nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993).We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.     

Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain

In this paper, we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent. We prove the existence, regularity and finite dimensionality of a global attractor in .     

Stability for the time-dependent Hartree equation with positive energy

In this paper, we show that the H¹ solutions to the time-dependent Hartree equation

     

Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors Aη, η∈[0,1], corresponding to a singularly perturbed damped wave equation

     

Long-time dynamics of an extensible plate equation with thermal memory

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.     

Generalized nonlinear Schrödinger equation with time-dependent dissipation

The generalized nonlinear Schrödinger equation with time-dependent dissipation [1] is considered using decomposition [2].     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

A general stability result for a nonlinear wave equation with infinite memory

In this work we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, among which those of exponential decay type, usually considered in the literature, are only special cases.     

Nonclassical diffusion with memory

We consider the nonclassical diffusion equation with hereditary memory on a bounded three‐dimensional domain. Setting the problem in the past history framework, and with very general assumptions on the memory kernel κ, we prove that the related solution semigroup possesses a global attractor of optimal regularity. Copyright © 2014 John Wiley & Sons, Ltd.     

Long-time behavior for a nonlinear plate equation with thermal memory

We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic. Moreover, we provide an estimate on the convergence rate.     

Global attractors for wave equations with nonlinear interior damping and critical exponents

In this paper we study the global attractors for wave equations with nonlinear interior damping. We prove the existence, regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter, when the growth of the nonlinear terms is critical.     

一类非线性非自治可伸缩板方程的一致吸引子

讨论了一类带有初值条件和固定边界条件的非线性非自治可伸缩板方程u_(tt)+△~2u+αu-△u_(tt)-M(||▽u||~2)△u-γ△u_t+f(u)=σ(x,t)证明了该系统的整体适定性.进一步研究了该系统的长时间动力学,利用建立整体弱解所生成的半过程一致渐近紧性,得到了有界区域Ω■R~n(n≥1)或无界开集中一致吸引子的存在性.     

Study of a finite element method for the time-dependent generalized Stokes system associated with viscoelastic flow

A three-field finite element scheme designed for solving systems of partial differential equations governing time-dependent viscoelastic flows is studied. Once a classical backward Euler time discretization is performed, the resulting three-field system of equations allows for a stable approximation of velocity, pressure and extra stress tensor, by means of continuous piecewise linear finite elements, in both two- and three- dimensional space. This is proved to hold for the linearized form of the system. An advantage of the new formulation is the fact that it provides an algorithm for the explicit iterative resolution of system nonlinearities. Convergence in an appropriate sense applying to these three flow fields is demonstrated.     

Controllability and observability of a heat equation with hyperbolic memory kernel

The exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered. Due to the appearance of such a kind of memory, the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations, and is significantly different from that of the usual parabolic equations. By means of Carleman estimate, we establish a positive controllability and observability result under some geometric condition. On the other hand, by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory, we present a negative controllability and observability result when the geometric condition is not satisfied.