Asymptotic Stability of Energy for a Weak Viscoelastic Plate Equation with Complementary Frictional Damping

A plate equation with a memory term and a time delay term in the internal feedback is investigated. Under suitable assumptions, we establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, by using energy perturbation method, we prove a general decay result of the energy provided that the weight of the delay is less than the weight of the damping.

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Decay of an Extensible Viscoelastic Plate Equation with a Nonlinear Time Delay

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-018-0602-4 ◽

2018 ◽

Vol 42 (5) ◽

pp. 2265-2285 ◽

Cited By ~ 1

Author(s):

Baowei Feng ◽

Khaled Zennir ◽

Lakhdar Kassah Laouar

Keyword(s):

Time Delay ◽

Viscoelastic Plate ◽

Plate Equation

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New general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

Boundary Value Problems ◽

10.1186/s13661-019-01308-0 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 3

Author(s):

Mohammad M. Al-Gharabli

Keyword(s):

Acta Phys ◽

Relaxation Function ◽

Viscoelastic Plate ◽

General Assumption ◽

General Decay ◽

Minimal Condition ◽

Plate Equation ◽

General Energy ◽

The Stability ◽

Appl Math

AbstractIn this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal condition $$ g^{\prime }(t)\le -\xi (t) G\bigl(g(t)\bigr), $$g′(t)≤−ξ(t)G(g(t)), where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when $G(s) = s^{p}$G(s)=sp and p covers the full admissible range $[1, 2)$[1,2). Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019).

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