Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping

We study von Karman evolution equations with non-linear dissipation and with partially clamped and partially free boundary conditions. Two distinctive mechanisms of dissipation are considered: (i) internal dissipation generated by non-linear operator, and (ii) boundary dissipation generated by shear forces friction acting on a free part of the boundary. The main emphasis is given to the effects of boundary dissipation. Under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension. We also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which, in turn, depends on the behaviour at the origin of the functions describing the dissipation.     

A nonlinear transmission problem for a compound plate with thermoelastic part

In this paper, we study a nonlinear transmission problem for a plate that consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. The main result is the proof of the asymptotic smoothness of this dynamical system. We also prove the existence of a compact global attractor in special cases when the nonlinearity is of Berger type or scalar. Copyright © 2012 John Wiley & Sons, Ltd.     

Null controllability of von Kármán thermoelastic plates under the clamped or free mechanical boundary conditions

In this paper, we provide results of local and global null controllability for 2-D thermoelastic systems, in the absence of rotational inertia, and under the influence of the (nonLipschitz) von Kármán nonlinearity. The plate component may be taken to satisfy either the clamped or higher order (and physically relevant) free boundary conditions. In the accompanying analysis, critical use is made of sharp observability estimates which obtain for the linearization of the thermoelastic plate (these being derived in [G. Avalos, I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl. 294 (2004) 34-61] and [G. Avalos, I. Lasiecka, Asymptotic rates of blowup for the minimal energy function for the null controllability of thermoelastic plates: The free case, in: Proc. of the Conference for the Control of Partial Differential Equations, Georgetown University, Dekker, in press]). Moreover, another key ingredient in our work to steer the given nonlinear dynamics is the recent result in [A. Favini, M.A. Horn, I. Lasiecka, D. Tataru, Addendum to the paper: Global existence, uniqueness and regularity of solution to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equations 10 (1997) 197-200] concerning the sharp regularity of the von Kármán nonlinearity.     

Plate models with state-dependent damping coefficient and their quasi-static limits

We study long-time dynamics of a class of plate models with a state-dependent damping coefficient and their quasi-static limits. We first present the problem in abstract form and then prove the existence of finite-dimensional global attractors and their upper semicontinuity in the quasi-static limit, i.e., in the case when the mass density of plate tends to zero. Our proofs involve a recently developed method based on “compensated” compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, von Karman and Berger plate models with different types of boundary conditions and damping coefficients. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension.     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a 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. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).     

Long-Time Dynamics for a Nonlinear Viscoelastic Kirchhoff Plate Equation

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.     

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.     

Asymptotic behavior of solutions to plate equations with nonlinear dissipation occurring through shear forces and bending moments

We consider the question of strong stability of solutions to plate equations with nonlinear dissipation in the boundary conditions. Two cases are discussed: (1) dissipation occurring through the nonlinear forces applied on the boundary and (2) dissipation acting through the nonlinear moments. Asymptotic stability results are presented for both cases. In the first case the results are established under the natural geometric conditions imposed on the domain, while in the second case certain restrictions on the curvature on the active portion of the boundary are required.Research partially supported by NSF Grant DMS-8301668 and by AFOSR Grant AFOSR-84-0365.     

Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity

Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$.     

ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT

Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.     

Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition

In this paper, we consider a semilinear elliptic boundary value problem in a smooth bounded domain, having the so-called logistic nonlinearity that originates from population dynamics, with a nonlinear boundary condition. Although the logistic nonlinearity has an absorption effect in the problem, the nonlinear boundary condition is induced by the homogeneous incoming flux on the boundary. The objective of our study is to analyze the existence of a bifurcation component of positive solutions from trivial solutions and its asymptotic behavior and stability. We perform this analysis using the method developed by Lyapunov and Schmidt, based on a scaling argument.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

Solvability of nonlinear elliptic boundary value problem via its associated linear problem

We investigate the existence of solutions of a nonlinear elliptic boundary value problem at resonance. Under the condition that the associated linear boundary value problem has no sign-changing solution or nontrivial solution and some other additional conditions, we prove the existence of solutions or nontrivial (even multiple) solutions to the nonlinear problem. Thus the existence of solutions can be obtained when the nonlinearity may cross any finite number of eigenvalues of the linear problem.     

On the Spectrum of the Dirac Operator and the Existence of Discrete Eigenvalues for the Defocusing Nonlinear Schrödinger Equation

We revisit the scattering problem for the defocusing nonlinear Schrödinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.     

Multiple positive solutions for discrete nonlocal boundary value problems

In this paper, we investigate a second-order nonlinear difference equation with sign-changing nonlinearity subject to two different sets of nonlocal boundary conditions. The explicit expressions of the associated Green's functions are presented. By using a recently developed fixed point theorem, we establish sufficient conditions for the existence of multiple positive solutions of the boundary value problem.     

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ ₀—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ ₀. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ <sub>0</sub>, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ <sub>0</sub>; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ω are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin. \par Accepted 12 May 2000. Online publication 6 October 2000.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks

We consider a dynamic linear shallow shell model, subject to nonlinear dissipation active on a portion of its boundary in physical boundary conditions. Our main result is a uniform stabilization theorem which states a uniform decay rate of the resulting solutions. Mathematically, the motion of a shell is described by a system of two coupled partial differential equations, both of hyperbolic type: (i) an elastic wave in the 2-d in-plane displacement, and (ii) a Kirchhoff plate in the scalar normal displacement. These PDEs are defined on a 2-d Riemann manifold. Solution of the uniform stabilization problem for the shell model combines a Riemann geometric approach with microlocal analysis techniques. The former provides an intrinsic, coordinate-free model, as well as a preliminary observability-type inequality. The latter yield sharp trace estimates for the elastic wave—critical for the very solution of the stabilization problem—as well as sharp trace estimates for the Kirchhoff plate—which permit the elimination of geometrical conditions on the controlled portion of the boundary.