On the Energy Decay of a Linear Thermoelastic Bar and Plate

Kim, Jong Uhn

doi:10.1137/0523047

Cited by 129 publications

(83 citation statements)

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“…In this direction, we quote some works dealing with thermoelasic rods and plates. 1,10,11,[15][16][17][19][20][21]23,25 The plan of the paper is as follows. In Sec.…”

Section: Abstract Linear Systems With Memory 629mentioning

confidence: 99%

Stability of Abstract Linear Thermoelastic Systems With Memory

Giorgi

Pata

2001

Math. Models Methods Appl. Sci.

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“…In this direction, we quote some works dealing with thermoelasic rods and plates. 1,10,11,[15][16][17][19][20][21]23,25 The plan of the paper is as follows. In Sec.…”

Section: Abstract Linear Systems With Memory 629mentioning

confidence: 99%

Stability of Abstract Linear Thermoelastic Systems With Memory

Giorgi

Pata

2001

Math. Models Methods Appl. Sci.

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“…And in fact this is indeed the case in linear models, where exponential decay rates for the linear energy have been established [4,5,27] for linear thermoelastic plates and more recently in [8,13] for semilinear plates. The situation in the quasilinear case is much more complex, due to the unboundedness of the nonlinear term with respect to the topology induced by the energy.…”

Section: Introductionmentioning

confidence: 99%

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

Lasiecka

Maad

Sasane

2008

Nonlinear differ. equ. appl.

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Abstract. We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in R n , n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after "recovery" inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1,16,44]. Mathematics Subject Classification (2000). Primary 74F05; Secondary 35B30, 35B40, 74H40. Keywords. Quasilinear thermoelastic plates, existence of weak solutions, uniform decays of finite energy solutions.

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“…Analyticity. In order to prove our main result, note that to show the condition (2.4) from Theorem 2.2 is equivalent to stating that there exists a constant C > 0 such that 1) or state that for any ε > 0, there exists C ε > 0 such that …”

Section: Positive Constant Independent Of λ and U ∈ D(a)mentioning

confidence: 99%

“…Several authors studied this system with different types of boundary conditions (Dirichlet, Neumann, clamped, etc.) and proved the exponential stability of the solutions; see for example [1,7,8,11] and references therein. In the particular case μ = 0, it was proved in [3,4,5] that the semigroup associated to the above system is analytic.…”

Section: Introductionmentioning

confidence: 99%