Numerical and Theoretical Stability Study of a Viscoelastic Plate Equation with Nonlinear Frictional Damping Term and a Logarithmic Source Term

Abstract

This paper is designed to explore the asymptotic behaviour of a two dimensional visco-elastic plate equation with a logarithmic nonlinearity under the influence of nonlinear frictional damping. Assuming that relaxation function g satisfies g′(t)≤−ξ(t)G(g(t)), we establish an explicit general decay rates without imposing a restrictive growth assumption on the damping term. This general condition allows us to recover the exponential and polynomial rates. Our results improve and extend some existing results in the literature. We preform some numerical experiments to illustrate our theoretical results.

Long time behavior of weak solutions for the nonlinear damped beam equation

“…Then, by using Lemma 4.3 developed by Martinez [30], we find an accurate exponential decay rate of the energy for the weak solutions. So, the results obtained in this paper can be regarded as a supplementary extension of [25, 27–29].…”

“…For a special case of () with $$$ k=0 $$$ and $$$ h\left({u}_t\right)=0 $$$, Alabau‐Boussouira et al [25] and Li and Zhu [27] obtained an optimized uniform decay estimate of the weak solutions. Recently, Al‐Gharabli et al also considered the model () with $$$ k>0 $$$ and $$$ h\left({u}_t\right)=0 $$$ in [28] and the same model () with $$$ k>0 $$$ and $$$ h\left({u}_t\right) $$$ being nontrivial but asymptotically linear for large $$$ \mid {u}_t\mid $$$ in [29]. In both papers, they proved the global existence of the weak solutions and derived the precise decay rates of energy for the global weak solutions.…”

Dynamic analysis of viscoelastic foundation plate with fractional Kelvin–Voigt model using shifted Bernstein polynomials

“…Dehghan [14] has proposed various numerical schemes for solving partial integral‐differential equations with weak singular nuclei concerning viscoelastic phenomena. Al‐Gharabli et al [15] studied the free vibration analysis of a three‐layer viscoelastic sandwich plate modeled by fractional theory, derived the motion governing equation using Lagrange equations, and obtained the frequency response of sandwich plates by Rayleigh–Ritz method. Krusser and Shitikova [16] used FKV constitutive relation to depict nonlinear vibration of partially viscoelastic polyethylene terephthalate (PET) films under external excitation forces.…”

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

“…Moreover, they also proved that the exponential growth of the solution in $$$ {L}^2 $$$ norm under the assumption $$$ h\left({u}_t\right) $$$ is linear. For more related works, we refer to previous studies 25,26 and references therein.…”