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This work focusses on the well–posedness and the exponential stability of a simplified porous elastic system. By omitting the second time derivative term of the function φ in the porous elastic system, we make the system free of the damaging consequences of the second spectrum. We consider an elastic dissipation and prove the well–posedness by the use of Hille–Yosida therorem and some arguments of elliptic equations. Next, we use the multiplier method and establish an exponential decay result without any restriction on coefficients of the system. [ABSTRACT FROM AUTHOR]

This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We also gave some examples to illustrate our theoretical finding. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following viscoelastic swelling porous-elastic system 1 ρ z z tt - a 1 z xx - a 2 u xx + ∫ 0 t g (t - s) z xx (x , s) d s = 0 , in (0 , 1) × (0 , ∞) , ρ u u tt - a 3 u xx - a 2 z xx = 0 , in (0 , 1) × (0 , ∞) , and establish explicit energy decay result with a very general assumption on the behavior of g at infinity; that is 2 g ′ (t) ≤ - k (t) H g (t) , t ≥ 0 , where k and H are two functions satisfying some conditions mentioned in the paper. Our result substantially improves earlier related result in the literature. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, g ′ (t) ≤ − ξ (t) G (g (t)) . We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a viscoelastic Moore–Gibson–Thompson equation with a type-II memory term and a relaxation function satisfying g ′ (t) ≤ - η (t) g (t) . By constructing appropriate Lyapunov functionals in the Fourier space, we establish a general decay estimate of the solution under the condition β - γ α - ϱ 2 > 0. We then give the decay rate of the L 2 -norm of the solution. We also give two examples to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We study the following wave equation u tt - Δ u + α (t) u t m (·) - 2 u t = 0 with a nonlinear damping having a variable exponent m(x) and a time-dependent coefficient α (t) . We use the multiplier method to establish energy decay results depending on both m and α . We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme. [ABSTRACT FROM AUTHOR]

In this paper, we establish general decay estimates of the solution for the Cauchy problem of a Moore–Gibson–Thompson equation with a viscoelastic term using the energy method in the Fourier space. We first discuss the well-posedness. Then we establish our main result and present two illustrative examples by the end. [ABSTRACT FROM AUTHOR]