Arbitrary decays in linear viscoelasticity

Tatar, Nasser–eddine

doi:10.1063/1.3533766

Cited by 44 publications

(60 citation statements)

References 21 publications

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“…Now we define We use here the following identity due to [1], to give a better estimate for the term ( ) ( ) can be estimated as in [1]:…”

Section: Arbitrary Rate Of Decaymentioning

confidence: 99%

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Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

Ma¹

2017

JAMP

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“…Now we define We use here the following identity due to [1], to give a better estimate for the term ( ) ( ) can be estimated as in [1]:…”

Section: Arbitrary Rate Of Decaymentioning

confidence: 99%

“…+∞ for some nonnegative function ( ) t γ , Tatar [1] generalized these works to an arbitrary decay for wave equation with a viscoelastic damping term. Moreover, we would like to mention some results in [25]- [30].…”

Section: Introductionmentioning

confidence: 99%

Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

Ma¹

2017

JAMP

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“…This last condition was shown to be necessary condition for exponential decay [7]. More recently Tatar [25] investigated the asymptotic behavior to problem (1.1) with r = g = 0 when h(t)g(t) L 1 (0, ∞) for some nonnegative function h(t). He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate.…”

Section: Introductionmentioning

confidence: 99%

“…He generalized earlier works to an arbitrary decay not necessary of exponential or polynomial rate. Motivated by previous works [21,25], in this paper, we consider the initial boundary value problem for the following nonlinear viscoelastic equation: 4) with initial conditions 5) and boundary condition 6) where Ω ⊂ R N , N ≥ 1, is a bounded domain with a smooth boundary ∂Ω. Here r, p > 0 and g represents the kernel of the memory term, with conditions to be stated later [see assumption (A1)-(A3)].…”

Section: Introductionmentioning

confidence: 99%

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Arbitrary decays for a viscoelastic equation

2011

Bound Value Probl

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In this paper, we consider the nonlinear viscoelastic equation | u t | ρ u tt − u − u tt + t 0 g(t − s) u(s)ds + | u| p u = 0 , in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential (polynomial) type, which is a necessary condition for the exponential (polynomial) decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata (Q Appl Math 64:499-513, 2006) IntroductionIt is well known that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are of great importance and interest. From the mathematical point of view, their memory effects are modeled by an integro-differential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. Many authors have focused on this problem for the last two decades and several results concerning existence, decay and blow-up have been obtained, see and the reference therein.In [3], Cavalcanti et al. studied the following problemwhere Ω ⊂ R N , N ≥ 1, is a bounded domain with a smooth boundary ∂Ω, g ≥ 0,, and the function g: R + R + is a nonincreasing function. This type of equations usually arise in the theory of viscoelasticity when the material density varies according to the velocity. In that paper, they proved a global existence result of weak solutions for g ≥ 0 and a uniform decay result for g > 0. Precisely, they showed that the solutions goes to zero in an exponential rate for g > 0 and g is a positive bounded C 1 -function satisfyingandfor all t ≥ 0 and some positive constants ξ 1 and ξ 2 . Later, this result was extended by Messaoudi and Tatar [15] to a situation where a nonlinear source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. Recently Messaoudi and Tatar [14] studied problem (1.1) for the case of g = 0, they improved the result in [3] by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g.The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (r = 0), see [1,2,[4][5][6]13,22,23,[29][30][31]. Lately, these conditions have been weakened by some researchers. For instance, instead of (1.3) Furati and Tatar [8] required the functions e at g(t) and e at g'(t) to have sufficiently small L 1 -norm on (0, ∞) for some a > 0 and they can also have an exponential decay of solutions. In particular, they do not impose a rate of decreasingness for g. Later on Messaoudi and Tatar [21] improved this result further by removing the condition on g'. They established an exponential decay under the conditions g'(t) ≤ 0 and e at g(t) L 1 (0, ∞) ...

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Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

Liu

Chen

2015

Mathematische Nachrichten

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In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi-valued hyperbolic differential inclusionswith memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: h(∇u) = −∇φ · (a∇u) and prove the general decay estimates of equivalent energy.

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Arbitrary decays in linear viscoelasticity

Abstract: A decay result to a viscoelastic problem in R n with an oscillating kernel General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms

Cited by 44 publications

References 21 publications

Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

Arbitrary decays for a viscoelastic equation

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

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