Evolution Equations in Thermoelasticity

Jiang, Song; Racke, E; Shitikova, Marina V.

doi:10.1115/1.1445336

Cited by 49 publications

(53 citation statements)

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“…Finally, Assumption A.2.6 implies compatibility conditions in sense of [8,Equations (A.8) and (A.9)]. Hence, applying [8,Theorem A.9] at the energy level s−1, we obtain additional regularity for the classical solution satisfying…”

Section: Compatibility Conditionsmentioning

confidence: 99%

“…Hence, applying [8,Theorem A.9] at the energy level s−1, we obtain additional regularity for the classical solution satisfying…”

Section: Compatibility Conditionsmentioning

confidence: 99%

“…A different technique is employed here to obtain a higher solution regularity needed for the fixed-point iteration in Theorem 4.5. Besides, the topologies used for the data (θ 0 ,ḡ) and the solution θ differ from those in [8,Chapter A.3].…”

Section: A1 Linear Wave Equationmentioning

confidence: 99%

“…To derive an estimate for ∂ s t z and ∇∂ s−1 t z, we need to employ a molifier technique similar to [8,Section A.2]. First, we select 0 < δ < ε < T .…”

Section: A1 Linear Wave Equationmentioning

confidence: 99%

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Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

Lasiecka

Pokojovy

Wan

2017

Nonlinear Analysis: Real World Applications

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We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.

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Section: Compatibility Conditionsmentioning

confidence: 99%

“…Hence, applying [8,Theorem A.9] at the energy level s−1, we obtain additional regularity for the classical solution satisfying…”

Section: Compatibility Conditionsmentioning

confidence: 99%

Section: A1 Linear Wave Equationmentioning

confidence: 99%

“…To derive an estimate for ∂ s t z and ∇∂ s−1 t z, we need to employ a molifier technique similar to [8,Section A.2]. First, we select 0 < δ < ε < T .…”

Section: A1 Linear Wave Equationmentioning

confidence: 99%

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Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

Lasiecka

Pokojovy

Wan

2017

Nonlinear Analysis: Real World Applications

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“…The mathematical analysis of thermoelastic systems has attracted a lot of attention over the years. An array of new and fundamental results in the area of wellposedness and stability of solutions to both linear and nonlinear thermoelasticity have been contributed to the field (see [14,15,17,18,26,48,49] and references therein).…”

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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Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

Ebenfeld

2001

Math Methods in App Sciences

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SUMMARYIn this article (which is divided in three parts) we investigate the non-linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type.Our goal is to characterize systematically all admissible couplings between systems of higher order and di erent type.By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem.In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so-called energy method. In the above sense, the regularity assumptions about the coe cients and right-hand sides deÿne the admissible couplings. In part 3, we extend the results of part 2 to the non-linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence deÿne the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems.

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Evolution Equations in Thermoelasticity

Cited by 49 publications

References 0 publications

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

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

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

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