Nonclassical diffusion with memory lacking instantaneous damping

Conti, Monica; Dell’Oro, Filippo; Pata, Vittorino

doi:10.3934/cpaa.2020090

Cited by 22 publications

(20 citation statements)

References 25 publications

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“…

In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [7,14] as the perturbed parameters ν = 0, but the nonlinearity f satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.

…”

supporting

confidence: 62%

“…In [7,14], the authors considered the following nonclassical diffusion equation of autonomous and non-autonomous system with memory lacking instantaneous damping −ν∆u respectively,…”

mentioning

confidence: 99%

“…To achieve this goal, we have to overcome the following difficulties. First of all, we can't obtain the higher regularity of the solutions to the Eq(1) by using the method of [7,14], and then obtain the existence of the global attractors. In addition, due to the perturbed parameter ν ≥ 0, we can't directly construct the contractive function to prove the asymptotic compactness of the solution semigroup {S ν (t)} t≥0 of the Eq(1)(see, e.g., [23,24]).…”

mentioning

confidence: 99%

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Attractors for a class of perturbed nonclassical diffusion equations with memory

Yuan

Zhang

Xie

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [<xref ref-type="bibr" rid="b7">7</xref>,<xref ref-type="bibr" rid="b14">14</xref>] as the perturbed parameters <inline-formula><tex-math id="M1">\begin{document}$ \nu = 0 $\end{document}</tex-math></inline-formula>, but the nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.</p>

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“…

…”

supporting

confidence: 62%

“…In [7,14], the authors considered the following nonclassical diffusion equation of autonomous and non-autonomous system with memory lacking instantaneous damping −ν∆u respectively,…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Attractors for a class of perturbed nonclassical diffusion equations with memory

Yuan

Zhang

Xie

2022

DCDS-B

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“…(1.1) was considered, and the nonlinearity satisfies critical exponential growth conditions. In [25], with memory lacking instantaneous damping for Eq. (1.1) was considered, and the nonlinearity satisfies critical exponential growth conditions.…”

Section: Introductionmentioning

confidence: 99%

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

Xie

Zhu

2021

Adv Differ Equ

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In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order $p-1$ p − 1 ($p \geq 2$ p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

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“…We limit ourselves here to studies concerning the long‐term dynamics 5‐10 . In particular, it has been proved the optimal regularity of attractors for the related semigroup by Conti et al 7 for

α = β = 1

, by Conti et al 8 for

α = 0

, and by Chekroun et al 5 for

β = 0

. For

α = β = 1

, Wang et al 9 showed the regularity of attractors in weak and strong topological spaces.…”

Section: Introductionmentioning

confidence: 99%

Quasi‐stability and upper semicontinuity for coupled parabolic equations with memory

Aouadi

2020

Stud Appl Math

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This current study deals with the long-time dynamics of a nonlinear system of coupled parabolic equations with memory. The system describes the thermodiffusion phenomenon where the fluxes of mass diffusion and heat depend on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin-Pipkin's law. Inspired by the works of Chueshov and Lasiecka on the property of quasistability of dynamic systems, we prove this property for the problem considered in this study. This property allows us to analyze certain properties of global and exponential attractors in a more efficient and practical way. This approach is applied for the first time for coupled parabolic equations. We analyze the continuity of global attractors with respect to a pair of parameters in a residual dense set and their upper semicontinuity in a complete metric space. Finally, we analyze the upper semicontinuity of global attractors with respect to small perturbations of the damping terms.

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Nonclassical diffusion with memory lacking instantaneous damping

Cited by 22 publications

References 25 publications

Attractors for a class of perturbed nonclassical diffusion equations with memory

Attractors for a class of perturbed nonclassical diffusion equations with memory

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

Quasi‐stability and upper semicontinuity for coupled parabolic equations with memory

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