Solvability of a Mixed Nonlocal Problem for a Nonlinear Singular Viscoelastic Equation

Mesloub, Said; Mesloub, Fatiha

doi:10.1007/s10440-008-9388-y

Cited by 16 publications

(14 citation statements)

References 14 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%

“…Later, mixed problems with integral conditions for both parabolic and hyperbolic equations were studied by Pulkina [9,10], Yurchuk [11], Kartynnik [12], Mesloub and Bouziani [13], Mesloub and Messaoudi [14,15], Mesloub [16], and Kamynin [17]. For instance, Said Mesloub and Fatiha Mesloub [18] obtained existence and uniqueness of the solution to the following problem:…”

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%

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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“…In Messaoudi, under suitable conditions for the relaxation function, the author showed that solutions with initial negative energy blow‐up in a limited time if p > m and remain if m ≥ p , for the following: problem

\{\begin{array}{l} u_{t t} - normalΔ u + \int_{0}^{t} g false(t - s false) normalΔ u + u_{t} false| u_{t} {false|}^{m - 2} = false| u {false|}^{p - 2} u, in normalΩ \times false(0, \infty false), \\ u = 0 x \in \partial normalΩ, t \in false(0, \infty false), \\ u false(x, 0 false) = u_{0} false(x false), u_{t} false(x, 0 false) = u_{1} false(x false), x \in normalΩ . \end{array}

Then, in Mesloub and Mesloub, a model describing the movement of the viscoelastic two‐dimensional body on the unit disc was investigated in the case of radial solutions. Using some prior estimates and some of the many arguments, the authors have demonstrated the existence and uniqueness of the generalized solution to the nonlocal problem

\{\begin{array}{l} u_{t t} - \frac{1}{x} {false(x u_{x} false)}_{x} + \int_{0}^{t} g false(t - s false) \frac{1}{x} x u_{x} {false(x, s false) false)}_{x} d s = f false(x, t, u, u_{x} false), in Q, \\ u_{x} false(1, t false) = 0, \end{array}

…”

Section: Introductionmentioning

confidence: 99%

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized damping term

Boulaaras

Mezouar

2020

Math Methods in App Sciences

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“…See the works of Cahlon and Shi and Mesloub and Lekrine, 2,3 Ewing and Lin, 4 Shi, 5 Choi and Chan, 6 Cannon, 7 Capasso-Kunisch, 8 Yurchuk, 9 Shi and Shilor, 10 Ionkin and Moiseev, 11 Kamynin, 12 Mesloub, 13,14 Ionkin, 15 Mesloub and Messaoudi, 16,17 Kartynnik, 18 Pulkina, 19,20 and Mesloub and Bouziani. 21,22 The motivation of our work is due to some results regarding the following research papers: In Mesloub, 23 the solvability of the following problem is studied…”

Section: Introductionmentioning

confidence: 99%

“…In Mesloub, 23 the solvability of the following problem is studied

\{\begin{array}{l} u_{t t} ((), t) - \frac{1}{x} {((), x u_{x} ((), t))}_{x} + \int_{0}^{t} g ((), t - s) \frac{1}{x} {((), x u_{x} ((), x, s))}_{x} d s + a u_{t} ((), t) \\ = f ((), x, t, u_{x}, u), in Q, \\ u false(x, 0 false) = u_{0} false(x false), u_{t} false(x, 0 false) = u_{1} ((), x), x \in ((), 0, 1), \\ u_{x} ((), 1, t) = 0, \int_{0}^{1} x u ((), x, t) d x = 0 t \in ([], 0, T), \end{array}

where

Q : = ((), 0, 1) \times ((), 0, T), a > 0

and the bounded relaxation function

g ((), t),

satisfies

g : ℝ_{+} \to ℝ_{+}

of class C 2 such that

g (s) \geq 0, g^{'} (s)

…”

Section: Introductionmentioning

confidence: 99%

Decay rates for the energy of a singular nonlocal viscoelastic system

Alaeddine

2020

Math Methods in App Sciences

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This work deals with decay rates for the energy of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. We prove the decay rates for the energy of a singular one-dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition of relaxation kernels described by the inequality g ′ i (t) ≤ −H (g i (t)) , (i = 1, 2) for all t ≥ 0, with H convex.

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Solvability of a Mixed Nonlocal Problem for a Nonlinear Singular Viscoelastic Equation

Cited by 16 publications

References 14 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

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized damping term

Decay rates for the energy of a singular nonlocal viscoelastic system

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