Extensible Beam Equation Research Articles

Stability of strong attractors for the extensible beam equation with gentle dissipation

The paper investigates the stability of the strong global and exponential attractors for the extensible beam equation with gentle dissipation: utt−βM(‖∇u‖2)Δu+Δ2u+(−Δ)2αut+f(u)=g(x), where the perturbed parameter α∈(0,1/2] is a dissipative index and β∈[0,1] is an extensibility parameter. We show that when the nonlinearity f(u) is of the optimally subcritical growth depending on α: 1≤p<pα:=N+8α(N−4)+, (i) the strong (H,H2)-global attractors obtained in [15] for this model are upper semicontinuous with respect to α in H2-topology and with respect to β in stronger V4×V2+α1-topology, with α1∈(0,α), respectively; (ii) for any fixed β∈[0,1] and α0∈(0,1/2], the related dynamical system (Sα(t),H) has a family of strong (H,H2)-exponential attractors Eexpα which are continuous with respect to α at α0 in H2-topology; (iii) for any fixed α∈(0,1/2], the related dynamical system (Sβ(t),H) has a family of strong (H,V4×V2+2α1)-exponential attractors Eexpβ which are Hölder continuous with respect to β at any point β0∈[0,1] in V4×V2+2α1-topology. All these strong global and exponential attractors are also the standard global and exponential attractors of optimal regularity. The method used here allows to obtain the stability of the strong attractors with respect to the perturbed parameters in strong topology instead of only standard energy topology.

Attractor for the Extensible Beam Equation with Nonlocal Weak Damping on Time–Dependent Space

Attractor for the Extensible Beam Equation with Nonlocal Weak Damping on Time–Dependent Space

Blow-Up Phenomena for a Class of Extensible Beam Equations

Blow-Up Phenomena for a Class of Extensible Beam Equations

Minimax optimal control problems for an extensible beam equation with uncertain initial velocity

This paper is devoted to the problem of minimax optimal control problems of an extensible beam equation with distributed controls and initial velocity disturbances (or noises). The existence of optimal solutions for distributed control with fixed disturbance, namely the (P_{v}) problem, and the existence of minimax optimal solutions to (P) problem without restricting the initial disturbance are proved. We derive the necessary optimality conditions for optimal solutions of the (P_{v}) and (P) problems in the form of Pontryagin’s maximum principle.

Bilinear Minimax Optimal Control Problems of the Velocity Term in an Extensible Beam Equation with Rotational Inertia

The author studies the minimax optimal control problem for an extensible beam equation that takes into account rotational inertia effects. A velocity term multiplied by the bilinear control and disturbance functions to construct a minimax optimal control strategy is added. The existence of an optimal pair, meaning optimal control and perturbation, is proved by assuming some conditions on the considered quadratic cost function. The optimal conditions for optimal pairs are provided by the adjoint systems that correspond to some physically meaningful observation cases.

The well-posedness and attractor on an extensible beam equation with nonlocal weak damping

&lt;p style='text-indent:20px;'&gt;In this paper, we consider some new results on the well-posedness and the asymptotic behavior of the solutions for a class of extensible beams equation with the nonlocal weak damping and nonlinear source terms. Our contribution is threefold. First, we establish the well-posedness by means of the monotone operator theory with locally Lipschitz perturbation. Then we show that the related solution semigroup &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \{S_{t}\}_{t\geq0} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in phase space &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal{H} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; has a finite-dimensional global attractor &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathcal{A} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; which has some regularity when the growth exponent of the nonlinearity &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ f(u) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is up to the subcritical and critical case, respectively. Finally, we obtain the exponential attractor &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathcal{A}_{exp} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the dynamical system &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ (\mathcal{H}, S_{t}) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. These results deepen and extend our previous works([&lt;xref ref-type="bibr" rid="b31"&gt;31&lt;/xref&gt;], [&lt;xref ref-type="bibr" rid="b30"&gt;30&lt;/xref&gt;]), where we only considered the existence of the global attractors in the case of degenerate damping.&lt;/p&gt;

On the dynamics of the damped extensible beam 1D-equation

We show that the solutions of the linearized damped extensible beam equation exhibit a chaotic or stable behavior that depends on the distribution of the physical parameters of the equation. Such dynamical behavior is achieved in Herzog-like spaces. Our results provide new insights into the damped extensible beam equation by finding a critical parameter whose sign determines such qualitative properties.

Optimal control problems for an extensible beam equation with pointwise state constraints

This paper is devoted to deal with optimal control problems of an extensible beam equation with pointwise state constraints. By making use of Ekeland's variational principle combined with penalization and spike variation techniques, we deduce necessary optimality condition in the form of a Pontryagin's maximum principle for an observation case.

Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models

Abstract For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.

Ground state solutions for the generalized extensible beam equations

In this paper, we study the generalized extensible beam equations with steep potential Δ2u−M‖∇u‖L22Δu+λV(x)u=f(x,u)inRN,u∈H2(RN),where N≥4, M(t)=at+b with a>0,λ>0 and b∈R are parameters, V∈C(RN,R) and f∈C(RN×R,R). Unlike most other papers on this problem, we allow the parameter b to be any negative number, which has the physical significance. Under some suitable assumptions on V(x) and f(x,u), when λ is large enough, we prove the existence of ground state solutions.

Attractors and their continuity for an extensible beam equation with rotational inertia and nonlocal energy damping

This paper investigates the well-posedness of global solutions, the existence of global and exponential attractors and their continuity for an extensible beam equation with rotational inertia and nonlocal energy damping in Ω⊂RN: (1−αΔ)utt+Δ2u−ϕ(‖∇u‖2)Δu−M(‖Δu‖2+‖ut‖2)Δut+f(u)=h, where α∈[0,1] is a rotational coefficient. We find a better subcritical index p⁎:=N+2N−4, with N≥5, and show that when the growth exponent p of the nonlinearity f(u) is up to the range: 1≤p≤p⁎, the equation is well-posed. In particular, when 1≤p<p⁎: (i) the related solution semigroup Sα(t) has a global attractor Aα for each α∈[0,1] and the family of {Aα}α∈[0,1] is continuous on α in a residual subset of [0,1], and upper semi-continuous on [0,1]; (ii) Sα(t) has an exponential attractor Aexpα for each α and the family of {Aexpα}α∈[0,1] is Hölder continuous on α. The main contributions of this paper are that: (a) it breaks through the existing growth restriction: 1≤p≤NN−4 on this issue in literature; (b) it gives a new criterion on the continuity (rather than only upper semi-continuity) of the global attractors in a residual subset of the parameter space and its application; (c) it provides a new method to establish the continuity of the exponential attractors for pure hyperbolic model.

Attractors for a class of extensible beams with strong nonlinear damping

&lt;p style='text-indent:20px;'&gt;We concern to stablish the existence and qualitative properties of the compact global attractor associate to solutions of a class of extensible beam equations with strong nonlinear damping arising from the wave model proposed by Prestel [&lt;xref ref-type="bibr" rid="b18"&gt;18&lt;/xref&gt;].&lt;/p&gt;

Long-time dynamics of a class of nonlocal extensible beams with time delay

This work is devoted to the following nonlocal extensible beam equation with time delay: u t t + Δ 2 u − M ( ∫ Ω | ∇ u | 2 d x ) Δ u + a 0 u t ( x , t ) + a 1 u t ( x , t − τ ) + φ ( u ) = f , on a bounded smooth domain . The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

Global Existence and Uniform Decay of Solutions for a Kirchhoff Beam Equation with Nonlinear Damping and Source Term

An extensible beam equation of Kirchhoff type with internal damping and source term is investigated. We apply the potential well and establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created from the Nehari manifold. Moreover, by using Nakao’s method, we prove the exponential stability of the solution for $$p=1$$ and the polynomial stability for $$p>1$$ .

Attractor for a model of extensible beam with damping on time-dependent space

In this paper, we study the asymptotic behavior of the following extensible beam equations: $$ \varepsilon(t) u_{tt}+\Delta^2 u-M\bigg(\int_\Omega |\nabla u|^2dx\bigg) \Delta u +\alpha u_t+\varphi (u)=f, \quad t&gt; \tau, $$ where $\varepsilon(t)$ is a decreasing function of time vanishing at infinity. After generalizing the abstract results on time dependent space, we establish an invariant time-dependent global attractor for the equation by proving the well-posedness (thereby, the existence of process), dissapativity and the compactness of the process. Our work supplements the theoretical results on time-dependent space and the results on the longtime behavior of the model.

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping

The paper investigates the well-posedness, regularity and longtime behavior of solutions for an extensible beam equation with fractional rotational inertia and structural nonlinear damping: (1+(−Δ)θ)utt+Δ2u−M(‖∇u‖2)Δu+N(‖∇u‖2)(−Δ)αut+f(u)=g, with dissipative index α∈(0,1] and rotational index θ∈[0,α). We find a new critical exponent pθα≡N+2(2α−θ)(N−4)+(>p⁎=N(N−4)+) depending on both θ and α and show that when the growth exponent p of the nonlinearity f(u) is up to the range: 1≤p<pθα, the well-posedness, regularity and longtime behavior of the solutions are like parabolic, in the concrete, (i) the IBVP of the equation admits a unique strong solution while its initial data is in weaker energy space; (ii) the related solution semigroup has a strong global attractor and a strong exponential attractor for each θ∈[0,α), where the “strong” means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the strong solution space; (iii) the family of strong global attractors {Aθ}θ∈[0,α) is upper-semicontinuous on the perturbed index θ at any point θ0∈[0,α). These results improve and extend those in recent literatures [17,24].

Solvability of Impulsive Nonlinear Partial Differential Equations with Nonlocal Conditions

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

Multiplicity of solutions to the generalized extensible beam equations with critical growth

In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in RN: Δ2u−M∫RN|∇u|2dxΔu+V(x)u=k(x)|u|q−2u+λ|u|2∗∗−2u,x∈RN, where Δ2u=Δ(Δu) is the biharmonic operator, M:R+→R+ is a continuous function, λ>0 is a parameter, V:RN→R+ is a potential function, and k(x) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions’ second Concentration-compactness principle and Concentration-compactness principle at infinity.

Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity

This paper is devoted to establishing the long-time behavior of extensible beam equation with the nonlocal weak damping on a bounded smooth domain of Rn with hinged (clamped) boundary condition. It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity f(u) is up to the critical case in natural energy space.