Nonlinear Boundary Dissipation Research Articles

Longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiros

This paper is concerned with the deduction of a mathematical model for the small longitudinal vibrations of a bar where one end is clamped and in the other end is glued a concentrated mass. The existence of global solutions of this model perturbed with a general nonlinear internal dissipation and with a general nonlinear boundary dissipation is derived. For a particular case, the exponential decay rate of the energy is obtained.

Blow-Up of Solution of Lamé Wave Equation with Fractional Damping and Logarithmic Nonlinearity Source Terms

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

Existence and nonexistence of solution of fractional Lamé wave equation with polynomial nonlinearity source terms

In this work, we examine the existence and nonexistence of solution for lamé wave equation with a nonlinear source terms and fractional boundary dissipation. By using the semi group theory approach, we prove the existence of a global solution. Furthermore, we established the blow-up results under the non-positive initial energy condition.

Existence of Global Attractors for a Semilinear Wave Equation with Nonlinear Boundary Dissipation and Nonlinear Interior and Boundary Sources with Critical Exponents

Existence of Global Attractors for a Semilinear Wave Equation with Nonlinear Boundary Dissipation and Nonlinear Interior and Boundary Sources with Critical Exponents

Blow‐up of solutions of wave equation with a nonlinear boundary condition and interior focusing source of variable order of growth

In this paper, being investigated an initial‐boundary value problem for a one‐dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow‐up of weak solutions in a finite time interval.

Energy decay and blow-up of solutions for a viscoelastic equation with nonlocal nonlinear boundary dissipation

In this paper, we consider a nonlinear viscoelastic equation with nonlocal nonlinear boundary damping and two nonlinear source terms (boundary and interior). We obtain the global existence of solution via the potential well method. By introducing suitable Lyapunov functionals, using the multiplier method, and constructing convex differential inequality, we establish an explicit and general decay rate result. This improves previous decay results concerning the problem. We also get a finite time blow-up result of solution with positive initial energy under suitable conditions on the initial data and positive memory function. This is the first result for blow-up of the problem with nonlocal nonlinear boundary damping.

On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation

<p style='text-indent:20px;'>Long time behavior of a semilinear wave equation with variable coefficients with nonlinear boundary dissipation is considered. It is shown that the existence of global and compact attractors depends on the curvature properties of a Riemannian metric given by the variable coefficients.</p>

Asymptotic behavior of the Timoshenko-type system with nonlinear boundary control

AbstractIn this paper, we consider the Timoshenko-type system with nonlinear boundary dissipation. We prove the existence and uniqueness of the solution and we establish an explicit and general decay result for a wide class of the relaxation function, which depends on the length of the beam.

Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary

We consider a (nonlinear) Berger plate in the absence of rotational inertia acted upon by nonlinear boundary dissipation. We take the boundary to have two disjoint components: a clamped (inactive) portion and a controlled portion where the feedback is active via a hinged-type condition. We emphasize the damping acts only in one boundary condition on a portion of the boundary . In [24] this type of boundary damping was considered for a Berger plate on the whole boundary and shown to yield the existence of a compact global attractor. In this work we address the issues arising from damping active only on a portion of the boundary, including deriving a necessary trace estimate for $(\Delta u)\big|_{\Gamma_0}$ and eliminating a geometric condition in [24] which was utilized on the damped portion of the boundary. Additionally, we use recent techniques in the asymptotic behavior of hyperbolic-like dynamical systems [11, 18] involving a ``stabilizability estimate to show that the compact global attractor has finite fractal dimension and exhibits additional regularity beyond that of the state space (for finite energy solutions).

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.

Note on intrinsic decay rates for abstract wave equations with memory

In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g′ + H(g) ⩽ 0, s ⩾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [“Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,” Differential and Integral Equations 6, 507–533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) ∼ sp) and for the full range of admissible parameters p ∈ [1, 2). While such result has been known for certain restrictive ranges of the parameters p ∈ [1, 3/2), the methods introduced previously break down when p ⩾ 3/2. The present paper develops a new and general tool that is applicable to all admissible parameters.

Global nonexistence for a semilinear wave equation with nonlinear boundary dissipation

In this paper, we consider the following problem{ytt−yxx+yt=|y|p−1y,(x,t)∈(0,L)×(0,T),y(0,t)=0,t∈(0,T),yx(L,t)+y(L,t)+|yt(L,t)|m−1yt(L,t)=0,t∈(0,T),y(x,0)=y0(x),yt(x,0)=y1(x),x∈(0,L), where (0,L) is a bounded open interval in R, p>1 and m⩾1. We are interested in the interaction between the boundary damping |yt(L,t)|m−1yt(L,t) and the interior source |y(t)|p−1y(t). Under some appropriate assumptions on the initial data, two blow-up results with positive initial energy are established. Furthermore, we obtain that the solutions blow up if p or |y0(L)| is sufficiently large.

Nonlinear exact controllability and nonlinear stabilization of hyperbolic or hyperbolic-like evolution equations

The present manuscript provides a comprehensive and detailed version of the content of the lectures delivered by the author at the Winter School on Nonlinear Analysis, Instituto de Matematica/UFRJ, August 3-7, 2009, Rio de Janeiro, Brazil. They encompassed two distinct yet related topics: (1) nonlinear global exact controllability, in the space of finite energy, for a second-order hyperbolic equation with Neumannboundary control; and(2) sharp uniform decay rates, at the L2(Ω)-level, of the Schrodinger equation with nonlinear boundary dissipation. ∗Research partially supported by the National Science Foundation under grant NSF-DMS01043-5 and by the Army Research Office under grant ARO-DAAD19-02-1-0179. Mathematics Subject Classification. Primary: 35L05, 35L20. Secondary: 93.

Uniform Decay of Solutions for von Karman Equations of Dynamic Viscoelasticity with Memory

In this paper, we consider the dynamical von Karman equations for viscoelastic plates with nonlinear boundary dissipation. We show the existence of solutions using the Galerkin method and then prove the asymptotic behaviour of the corresponding solutions by choosing suitable Lyapunov functional.

Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation

Long time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. It is known [CEL1] that the attractor consists of all full trajectories emanating from the set of stationary points. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary points at an exponential rate.

Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions

This paper studies a wave equation on a bounded domain in $\bbR^d$ with nonlinear dissipation which is localized on a subset of the boundary. The damping is modeled by a continuous monotone function without the usual growth restrictions imposed at the origin and infinity. Under the assumption that the observability inequality is satisfied by the solution of the associated <em>linear</em> problem, the asymptotic decay rates of the energy functional are obtained by reducing the <em>nonlinear PDE</em> problem to a <em> linear PDE</em> and a <em>nonlinear ODE</em>. This approach offers a generalized framework which incorporates the results on energy decay that appeared previously in the literature; the method accommodates systems with variable coefficients in the principal elliptic part, and allows to dispense with linear restrictions on the growth of the dissipative feedback map.

Well-posedness and sharp uniform decay rates at the L2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schrodinger equation defined on a bounded open domain of \( \mathbb{R}^{n} \) and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L2(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L2(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L2(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L2(Ω)-level for a fully general Schrodinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H1(Ω)-level energy estimate to the L2(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.

Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations

We first consider the wave equation in an exterior domain Ω in R N with two separated boundary parts Γ 0 , Γ 1 . On Γ 0 , the Dirichlet condition u | Γ 0 = 0 is imposed, while on Γ 1 , Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = − g ( u t ) is assumed. Further, a ‘half-linear’ localized dissipation is attached on Ω . For such a situation we derive a precise rate of decay of the energy E ( t ) for solutions of the initial boundary value problem. We impose no geometrical condition on the shape of the boundary ∂ Ω = Γ 0 ∪ Γ 1 . Secondly, when a T periodic forcing term works we prove the existence of a T periodic solution on R under an additional growth assumption on ρ ( x , v ) and g ( v ) .

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.

Finite Dimensionality of the Attractor for a Semilinear Wave Equation with Nonlinear Boundary Dissipation

Long-time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary point at an exponential rate. This result makes it possible to prove that the global attractor is exponential, i.e., it attracts every bounded set with exponential speed.