Beam Equation with Viscoelastic Damping and Delay

Abstract

This paper studies the long-time dynamical behavior of a linear delayed beam equation with viscoelastic damping and affine linear boundary conditions. The well-posedness of the corresponding autonomous system is obtained through classical semigroup methods. It is proven that the dynamical system \((\mathcal{H}, S_t)\) generated by mild solutions has a compact global attractor \(\mathfrak{A}\) in the topology of phase space \(\mathcal{H}\), and the attractor has finite fractal dimension and Hausdorff dimension (when the delay \(\tau\) is sufficiently small).

\( u_{tt} + \Delta^2 u - \int_{-\infty}^{t} g(t-s)\Delta^2 u(s)ds + \mu u_t(t-\tau) = 0 \)

Mathematical Model and Transformation

The original equation contains delay terms and memory terms, which are transformed into an autonomous system by introducing historical variables and delay variables:

Historical Variable Definition

\( \eta^t(x,s) = u(x,t) - u(x,t-s), \quad s \ge 0 \)

Delay Variable Definition

\( z(t,p) = u_t(x, t - \tau p), \quad p \in [0,1] \)

Autonomous System

\[ \begin{cases} u_{tt} + \kappa\Delta^2 u + \int_0^\infty g(s)\Delta^2 \eta^t(s)ds + \mu z(1) = 0 & \text{in } \Omega \times \mathbb{R}^+ \\ \eta_t^t = -\eta_s^t + u_t & \text{in } \Omega \times \mathbb{R}^+ \times \mathbb{R}^+ \\ \tau z_t + z_p = 0 & \text{in } \Omega \times \mathbb{R}^+ \times \mathbb{R}^+ \end{cases} \]

Delay Parameter τ: 0.5

Damping Parameter μ: 1.0

Real-time Parameter Impact Observation

Delay Effect Comparison

Adjust delay parameter τ to observe changes in system response

Energy Decay Rate

Impact of damping parameter μ on system energy decay

Phase Space Construction

The extended phase space \(\mathcal{H}\) "instantizes" all historical information of the system as part of the current state:

Symbol	Space Definition	Physical Meaning	Inner Product/Norm Features
W	\(W = \{ v \in H^2(\Omega) \mid v\vert_{\Gamma_0} = \frac{\partial v}{\partial \nu} \vert_{\Gamma_0} = 0 \}\)	Displacement field space satisfying boundary conditions, corresponding to elastic potential energy	\((u, v)_W = (\Delta u, \Delta v)\)
\(L^2(\Omega)\)	Standard square integrable function space	Velocity field space, whose norm squared represents the system's kinetic energy	\((u, v) = \int_\Omega u(x)v(x)dx\)
\(L_g^2(\mathbb{R}^+; W)\)	Function space with range in \(W\), square integrable on \(\mathbb{R}^+\) with respect to weight function \(g(s)\)	Viscoelastic memory history space, describing historical differences in displacement	\((\eta, \xi)_{g,W} = \int_0^\infty g(s)(\Delta \eta(s), \Delta \xi(s)) ds\)
\(L^2((0,1); L^2(\Omega))\)	Square integrable function space defined on interval \((0,1)\) with range in \(L^2(\Omega)\)	Delay variable space, describing the state of delay effects in the transport channel	\((z_1, z_2) = \tau \vert \mu \vert \int_0^1 (z_1(p), z_2(p))_{L^2(\Omega)} dp\)

\[ \mathcal{H} = W \times L^2(\Omega) \times L_g^2(\mathbb{R}^+; W) \times L^2((0,1); L^2(\Omega)) \]

Key Concepts Visualization

Viscoelastic Memory Effect

Material stress depends not only on current strain but also on strain history. The memory kernel \(g(s)\) describes the memory decay rate.

Delay Effect

System response depends not only on current state but also on past states. Delays can cause system instability or oscillations.

Energy Decay

System energy decays exponentially over time, proving the system is dissipative, which is a prerequisite for the existence of global attractors.

Main Results and Numerical Simulation

Global Attractor Existence

When the delay is sufficiently small, the system has a compact global attractor \(\mathfrak{A}\).

Current parameters, attractor existence: established

Energy Functional Decay

System energy \(E(t)\) decays exponentially over time, with decay rate dependent on delay parameter \(\tau\) and damping parameter \(\mu\).

Initial energy: 10.0, Final energy: 0.5

Main Theorems

Well-posedness: For any initial value \(V_0 \in \mathcal{H}\), the system has a unique mild solution.
Dissipativity: The system has a bounded absorbing set and is a dissipative dynamical system.
Global Attractor: The system \((\mathcal{H}, S_t)\) has a compact global attractor \(\mathfrak{A}\).
Finite Dimensionality: The attractor \(\mathfrak{A}\) has finite fractal dimension and Hausdorff dimension.
Regularity: Complete trajectories in the attractor have higher regularity.

Phase Space Trajectory Visualization

Evolution trajectory of the system state in phase space \(\mathcal{H}\), showing the convergence process to the global attractor.