Asymptotic behavior of second sound thermoelasticity with internal time-varying delay

Mustafa, Muhammad I.

doi:10.1007/s00033-012-0268-y

Cited by 21 publications

(15 citation statements)

References 28 publications

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“…Define

g \circ v = \int_{0}^{1} \int_{0}^{t} g (t - s) {(v (x, t) - v (x, s))}^{2} dsdx, v \in L^{2} ((0, \infty); (0, 1)) .

Let us introduce as in the auxiliary variables

z (ρ, t) = ϕ_{t} (1, t - τ (t) ρ), ρ \in (0, 1), t > 0,

and

u (x, t) = \int_{0}^{t} θ (x, s) ds + χ (x),

where χ ( x ) is the solution of

\{\begin{matrix} - k χ^{′′} = δ θ_{0}^{′′} - ρ_{3} θ_{1} - β ϕ_{0}^{′′} - β ψ_{1}, & x \in (0, 1), \\ χ (0) = χ (1) = 0, \end{matrix}

then we can transform the system as follows:

({, \begin{matrix} ρ_{1} ϕ_{tt} - k {(ϕ_{x} + ψ)}_{x} + u_{xt} = 0, \end{matrix})

…”

Section: Transformations and Preliminariesmentioning

confidence: 99%

“…Mustafa considered the following thermoelastic system with initial time‐varying delay,

\{\begin{matrix} a u_{tt} (x, t) - d u_{xx} (x, t) + β θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ b θ_{t} (x, t) + q_{x} (x, t) + β u_{xt} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ τ_{0} q_{t} (x, t) + k_{1} q (x, t) + k_{2} q (x, t - τ (t)) + θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \end{matrix}

and showed that the damping effect through heat conduction is strong enough to uniformly stabilize the system even in the presence of time delay by the energy method.…”

Section: Introductionmentioning

confidence: 99%

“…Later, Kirane et al [13] extended this result to the case of time-varying delay. Mustafa [14] considered the following thermoelastic system with initial time-varying delay,…”

mentioning

confidence: 99%

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Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

Hao

Wang

2015

Math Methods in App Sciences

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“…Define

g \circ v = \int_{0}^{1} \int_{0}^{t} g (t - s) {(v (x, t) - v (x, s))}^{2} dsdx, v \in L^{2} ((0, \infty); (0, 1)) .

Let us introduce as in the auxiliary variables

z (ρ, t) = ϕ_{t} (1, t - τ (t) ρ), ρ \in (0, 1), t > 0,

and

u (x, t) = \int_{0}^{t} θ (x, s) ds + χ (x),

where χ ( x ) is the solution of

\{\begin{matrix} - k χ^{′′} = δ θ_{0}^{′′} - ρ_{3} θ_{1} - β ϕ_{0}^{′′} - β ψ_{1}, & x \in (0, 1), \\ χ (0) = χ (1) = 0, \end{matrix}

then we can transform the system as follows:

({, \begin{matrix} ρ_{1} ϕ_{tt} - k {(ϕ_{x} + ψ)}_{x} + u_{xt} = 0, \end{matrix})

…”

Section: Transformations and Preliminariesmentioning

confidence: 99%

“…Mustafa considered the following thermoelastic system with initial time‐varying delay,

\{\begin{matrix} a u_{tt} (x, t) - d u_{xx} (x, t) + β θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ b θ_{t} (x, t) + q_{x} (x, t) + β u_{xt} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ τ_{0} q_{t} (x, t) + k_{1} q (x, t) + k_{2} q (x, t - τ (t)) + θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \end{matrix}

and showed that the damping effect through heat conduction is strong enough to uniformly stabilize the system even in the presence of time delay by the energy method.…”

Section: Introductionmentioning

confidence: 99%

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Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

Hao

Wang

2015

Math Methods in App Sciences

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“…In recent years, differential equations with time delay effects have become an active area of research; see for example [20][21][22][23][24][25][26] and the references therein. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary.…”

Section: Introductionmentioning

confidence: 99%

Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions

Lee

Park

2018

Bound Value Probl

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“…Mustafa investigated the thermoelastic system of second sound with internal time‐varying delay

({, \begin{matrix} a u_{tt} (x, t) - d u_{xx} (x, t) + β θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ b θ_{t} (x, t) + q_{x} (x, t) + β u_{xt} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty), \\ τ_{0} q_{t} (x, t) + k_{1} q (x, t) + k_{2} q (x, t - τ (t)) + θ_{x} (x, t) = 0, & (x, t) \in (0, l) \times (0, \infty) \end{matrix})

and showed that the damping effect through heat conduction is strong enough to uniformly stabilize the system even in the presence of time delay by the energy method.…”

Section: Introductionmentioning

confidence: 99%

Asymptotical stability for memory‐type porous thermoelastic system of type III with constant time delay

Hao

Wang

2016

Math Methods in App Sciences

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Asymptotic behavior of second sound thermoelasticity with internal time-varying delay

Cited by 21 publications

References 28 publications

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions

Asymptotical stability for memory‐type porous thermoelastic system of type III with constant time delay

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