Dynamics of a Cauchy problem related to extensible beams under nonlocal and localized damping effects

A Finite Difference Method for Solving the Wave Equation with Fractional Damping

“…Recently, the study of damping effects has been a hot research topic because it appears in a variety of the dynamic processes of complex systems, including electromagnetic shunt [1], extensible beams [2], swelling porous elastic [3], vibration [4], and so on [5][6][7][8][9][10][11][12][13].…”

“…where Ω = (a, b) ⊂ R 1 or Ω = (a, b) × (c, d) ⊂ R 2 , ∂Ω is the boundary of Ω, ∆ is the Laplacian operator, D(> 0) is the diffusion coefficient, f (x, t), ψ(x, t), ϕ(x) and φ(x) are given functions, and the operator C 0 D α t denotes the Caputo fractional derivative defined by…”

“…Wu et al [34] used the reduced-order method to develop a Crank-Nicolson compact difference scheme of the time-fractional damped plate vibration equation, in which the equation includes the second-order derivative and the Caputo fractional derivative of order α ∈ (1, 2) in the temporal direction. Based on an efficient sum-of-exponentials (SOE) approximation for the kernel function t −β−1 with β ∈ (0, 1), Lyu et al [35] presented a fast linearized finite difference scheme for the 1D nonlinear multi-term time-fractional wave equation, where all fractional orders are in (1,2).…”

Global attractor of the Euler-Bernoulli equations with a localized nonlinear damping

“…There are also some achievements in the global attractor for the plate/beam equations with fully interior/boundary dissipation (see [6,20,22,26,28,15]) and the long-time behavior of solutions for the beam/plate equations with a localized damping (see [2,4,11,12,13,14,16,18,24,27,30]). In particular, as the locally distributed and unbounded nature of the damping, the authors [18,24] established the exponential decay of solutions by using a frequency domain method and a contradiction argument.…”

“…Moreover, the existence, regularity and finite dimensionality of compact global attractors as well as the upper semicontinuity of attractors with respect to the rotational inertia terms have been proved in [11] for the von Karman evolution equations with rotational inertial forces, critial nonlinearity and a nonlinear localized damping by the flux multiplier methods. Recently, the authors [30] have proved the existence, characterization and regularity of a compact global attractor for a Cauchy problem related to extensible beams with dissipative effects from the nonlocal Balakrishnan-Taylor and the localized weak damping terms in R n by establishing a new unique continuation property for models associated with extensible beam. However, to the best of our knowledge, there is no results concerning the existence of a global attractor for the Euler-Bernoulli equations with a localized nonlinear damping.…”