Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms

Boulaaras, Salah; Guefaifia, Rafik; Mezouar, Nadia

doi:10.1080/00036811.2020.1760250

Cited by 11 publications

(16 citation statements)

References 16 publications

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“…In this subsection, we state, without proof, the local and global existence results for system (1), which can be proved similarly to the ones in [14,18] and [21].…”

Section: Local and Global Existencementioning

confidence: 91%

“…They proved the global existence and established a decay rate of solution and also a finite time blow up. Recently, Boulaaras et al [21] treated problem (1) and proved the existence of a global solution to the problem using the potential-well theory. Moreover, they established a general decay result in which the relaxation functions k 1 and k 2 satisfy…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the above work, we prove a general stability result of system (1) replacing the condition (6) used in [21] by a more general assumption of the form:…”

Section: Introductionmentioning

confidence: 99%

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New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

2020

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This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function $k_{i}$ k i , namely, $$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$ k i ′ ( t ) ≤ − ξ i ( t ) Ψ i ( k i ( t ) ) , i = 1 , 2 . We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when $k_{i}(s) = s^{p}$ k i ( s ) = s p and p covers the full admissible range $[1, 2)$ [ 1 , 2 ) .

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“…In this subsection, we state, without proof, the local and global existence results for system (1), which can be proved similarly to the ones in [14,18] and [21].…”

Section: Local and Global Existencementioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

2020

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“…and Q = (0, L) × (0, T), L < ∞, T < ∞, μ ∈ C 1 ((0, L)), g 1 (2). The problems related with localized frictional damping have been extensively studied by many teams [5], where the authors obtained an exponential rate of decay for the solution © The Author(s) 2020.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, in [2] we have studied the following singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory:…”

Section: Introductionmentioning

confidence: 99%

Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms

et al. 2020

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Global Existence and Decay for a System of Two Singular Nonlinear Viscoelastic Equations with General Source and Localized Frictional Damping Terms

Boulaaras

Guefaifia

Mezouar

et al. 2020

Journal of Function Spaces

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The current paper deals with the proof of a global solution of a viscoelasticity singular one-dimensional system with localized frictional damping and general source terms, taking into consideration nonlocal boundary condition. Moreover, similar to that in Boulaaras’ recent studies by constructing a Lyapunov functional and use it together with the perturbed energy method in order to prove a general decay result.

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Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms

Cited by 11 publications

References 16 publications

New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms

Global Existence and Decay for a System of Two Singular Nonlinear Viscoelastic Equations with General Source and Localized Frictional Damping Terms

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