Exponential decay of solutions for the plate equation with localized damping

Simsek, Sema; Khanmamedov, Azer

doi:10.1002/mma.3185

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…This is caused by the lack of unique continuation result for the weak solutions of the plate equation with nonsmooth coefficients. This question has been recently solved by Simsek and Khanmamedov 37 (see also Khanmamedov and Yayla 38 ) when a localized linear weak damping (and a localized linear strong damping), respectively, are considered. To this end, in Simsek and Khanmamedov, 37 and using the sequentially limit transition technique, 29,39 the authors first prove the uniformly asymptotic compactness of the family of semigroups.…”

Section: Introductionmentioning

confidence: 98%

“…This question has been recently solved by Simsek and Khanmamedov 37 (see also Khanmamedov and Yayla 38 ) when a localized linear weak damping (and a localized linear strong damping), respectively, are considered. To this end, in Simsek and Khanmamedov, 37 and using the sequentially limit transition technique, 29,39 the authors first prove the uniformly asymptotic compactness of the family of semigroups. Then, using point dissipativity property for the semilinear plate equation established in Khanmamedov 40 and borrowing the energy inequalities obtained in Zuazua 41 the superlinear case, they show the contraction of the energy for the plate equations, which leads to exponential decay of energy for the problem ().…”

Section: Introductionmentioning

confidence: 98%

“…The main contribution of the present article is to complement the works aforementioned 37,38 in twofold: (i) by considering a nonlinear localized damping term (which seems to be new in the literature) and a beam term

b false‖ \nabla u {false‖}_{L^{2} false(normalΩ false)}^{2} normalΔ u

, (ii) introducing a new approach when dealing with uniform decay rate estimates associated to problem () using the observability inequality for the linear problem as well as the fact that the application

false{z_{0}, z_{1}, f false} \mapsto false{z, \partial_{t} z false} \in L^{\infty} false(0, T; H_{0}^{2} false(normalΩ false) false) \times L^{\infty} false(0, T; L^{2} false(normalΩ false) false)

associating the initial data

false{z_{0}, z_{1}, f false} \in H_{0}^{2} false(normalΩ false) \times L^{2} false(normalΩ false) \times L^{1} false(0, T; L^{2} false(normalΩ false) false)

to the unique solution to the linear problem

\{\begin{array}{ll} \partial_{t}^{2} z + {normalΔ}^{2} z = f & in normalΩ \times false(0, T false) \\ z = \partial_{ν} z = 0 & on \partial normalΩ \times false(0, T false), \\ z false(0 false) = z_{0}, \partial_{t} z false(0 false) = z_{1} & in normalΩ, \end{array}

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

Cavalcanti

Delatorre

Cavalcanti

et al. 2021

Math Methods in App Sciences

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In this paper, we study the semilinear beam equation with a locally distributed nonlinear damping on a smooth bounded domain. We first construct approximate solutions, and we show that the aforementioned approximate solutions decay uniformly in the weak phase space by using an observability inequality associated to the linear problem and a unique continuation property. Then, we prove the global existence as well as the uniform decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively.

show abstract

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

b false‖ \nabla u {false‖}_{L^{2} false(normalΩ false)}^{2} normalΔ u

false{z_{0}, z_{1}, f false} \mapsto false{z, \partial_{t} z false} \in L^{\infty} false(0, T; H_{0}^{2} false(normalΩ false) false) \times L^{\infty} false(0, T; L^{2} false(normalΩ false) false)

associating the initial data

false{z_{0}, z_{1}, f false} \in H_{0}^{2} false(normalΩ false) \times L^{2} false(normalΩ false) \times L^{1} false(0, T; L^{2} false(normalΩ false) false)

to the unique solution to the linear problem

\{\begin{array}{ll} \partial_{t}^{2} z + {normalΔ}^{2} z = f & in normalΩ \times false(0, T false) \\ z = \partial_{ν} z = 0 & on \partial normalΩ \times false(0, T false), \\ z false(0 false) = z_{0}, \partial_{t} z false(0 false) = z_{1} & in normalΩ, \end{array}

…”

Section: Introductionmentioning

confidence: 99%

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Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

Cavalcanti

Delatorre

Cavalcanti

et al. 2021

Math Methods in App Sciences

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show abstract

“…The theory of attractors for second-order evolution equations was studied and summarized by Chueshov and Lasiecka [1]. Due to the wide applications to the material science, optics, and wave motion, the (second-order) plate PDE (1) (or other forms) has received the attention and research of many scholars; see the case of bounded domains [2][3][4][5][6][7][8][9], the case of unbounded domains [10,11], and the stochastic version [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Residually continuous pullback attractors for stochastic discrete plate equations

Han,

Li,

Xia

2024

Math Methods in App Sciences

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For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system.

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Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type

Cavalcanti

Silva

et al. 2021

Journal of Differential Equations

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Exponential decay of solutions for the plate equation with localized damping

Cited by 5 publications

References 23 publications

Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

Uniform decay rate estimates for the beam equation with locally distributed nonlinear damping

Residually continuous pullback attractors for stochastic discrete plate equations

Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type

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