General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions

Boukhatem, Yamna; Benabderrahmane, Benyattou

doi:10.1007/s10440-017-0137-y

Cited by 9 publications

(10 citation statements)

References 28 publications

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“…(2.9) (H2) (Boukhatem et al 22 ) For i = 1, 2, g i (s) ∶ R → R are continuous and monotone increasing functions, there exist positive constants m i , M i and the increasing functions G i ∶ R + → R + of the class C 1 (R + ) ∩ C 2 (R + ) satisfying G i (0) = 0, and G i are linear or nonlinear with G ′ i (0) = 0 and…”

Section: Assumptionsmentioning

confidence: 99%

“…Therefore, we should give some additional conditions or control terms to eliminating the effect of time delay, so as to obtain the stability of energy; see, for example, previous studies. [18][19][20][21][22] Benaissa et al 21 investigated a nonlinear wave equation with time varying delay term in weakly internal feedbacks and obtained the global solvability and energy decay estimate under some conditions on damping and time varying delay using the energy method, multiplier method, and some general weighted integral inequalities.…”

Section: Introductionmentioning

confidence: 99%

“…Particularly in Boukhatem and Benabderrahmane, 22 the authors considered the following viscoelastic equation with acoustic boundary conditions, nonlinear damping, and nonlinear delay term in the boundary feedback…”

Section: Introductionmentioning

confidence: 99%

\{\begin{array}{ll} u_{t t} - d i v false(A false(x false) \nabla u false(t false) false) + \int_{0}^{\infty} g false(s false) d i v false(A false(x false) \nabla u false(t - s false) false) d s = 0 & in normalΩ \times false(0, \infty false), \\ u = 0 & on {normalΓ}_{0} \times false(0, \infty false), \\ \frac{\partial u}{\partial ν_{scriptA}} false(t false) - \int_{0}^{\infty} g false(s false) \frac{\partial u}{\partial ν_{scriptA}} false(t - s false) normald s + μ_{1} k_{1} false(u_{t} false(x, t false) false) + μ_{2} k_{2} false(u_{t} false(x, t - τ false) false) = h false(x false) z_{t} & on {normalΓ}_{1} \times false(0, \infty false), \\ u_{t} + f false(x false) z_{t} + m false(x false) z = 0 & on {normalΓ}_{1} \times false(0, \infty false), \\ u false(x, - t false) = u_{0} false(x, t false) & in normalΩ \times false(0, \infty false), \\ u_{t<...>} \end{array}

…”

Section: Introductionmentioning

confidence: 99%

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Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

Hao

2020

Math Methods in App Sciences

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In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.

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Section: Assumptionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\{\begin{array}{ll} u_{t t} - d i v false(A false(x false) \nabla u false(t false) false) + \int_{0}^{\infty} g false(s false) d i v false(A false(x false) \nabla u false(t - s false) false) d s = 0 & in normalΩ \times false(0, \infty false), \\ u = 0 & on {normalΓ}_{0} \times false(0, \infty false), \\ \frac{\partial u}{\partial ν_{scriptA}} false(t false) - \int_{0}^{\infty} g false(s false) \frac{\partial u}{\partial ν_{scriptA}} false(t - s false) normald s + μ_{1} k_{1} false(u_{t} false(x, t false) false) + μ_{2} k_{2} false(u_{t} false(x, t - τ false) false) = h false(x false) z_{t} & on {normalΓ}_{1} \times false(0, \infty false), \\ u_{t} + f false(x false) z_{t} + m false(x false) z = 0 & on {normalΓ}_{1} \times false(0, \infty false), \\ u false(x, - t false) = u_{0} false(x, t false) & in normalΩ \times false(0, \infty false), \\ u_{t<...>} \end{array}

…”

Section: Introductionmentioning

confidence: 99%

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Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

Hao

2020

Math Methods in App Sciences

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“…where G is an increasing positive strictly convex function satisfying some additional properties. For the general decay results in the case of constant or varying time delay, see previous works [28][29][30][31][32][33][34] and references therein. In this work, we are interested in giving an optimal explicit and a general decay rates of the solution of the problem (1) under some assumptions on the function h, 1 and the weight of delay 2 .…”

Section: Introductionmentioning

confidence: 99%

Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Chellaoua

Boukhatem

2020

Math Methods in App Sciences

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In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with infinite memory, time delay, and a kernel function h ∶ R + → R + satisfying, for all t ≥ 0, h ′ (t) ≤ − (t)G(h(t)) where and G are functions satisfying some specific properties. For this much larger class of kernel functions and under a suitable conditions, we prove well-posedness of solution by using semi-group theory. Then, we establish an explicit and general decay results of the energy solution by introducing a suitable Lyapunov functional and some properties of the convex functions. Finally, some applications are given. This work improves the previous results with finite memory to infinite memory and without time delay term to those with delay.

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Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay

Chellaoua

Boukhatem

2021

Z. Angew. Math. Phys.

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General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions

Cited by 9 publications

References 28 publications

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay

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