Thermoelastic Plate Equations Research Articles

Optimal large‐time estimates and singular limits for thermoelastic plate equations with the Fourier law

In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space , where we introduce a reduction methodology basing on third‐order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when , bounded estimates when , and decay estimates when for the vertical displacement in the norm. Particularly, the new critical dimension for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates but also rigorously demonstrate a new second‐order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity.

On the existence theory for the nonlinear thermoelastic plate equation

In this paper, we analyze the nonlinear thermoelastic plates, with Fourier heat conduction, and consider a polynomial-type nonlinearity. We first develop a theoretical analysis of the corresponding linear system to derive time decay estimates in L ∞ ( R n ) and H s ( R n ) . Then, using that set of decay estimates and controlling the nonlinearity, we prove the existence and uniqueness of local solutions with initial data ( u ( 0 ) , u t ( 0 ) , θ ( 0 ) ) = ( u 0 , Δ u 1 , Δ θ 1 ) , with u 0 ∈ H s , and u 1 , θ 1 ∈ H s + 1 , for s > n 2 + 1 .

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C∞ Semigroups

We consider Fourier multiplier systems on Rn with components belonging to the standard Hörmander class S1,0mRn, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector Λ⊂C (introduced by Denk, Saal, and Seiler) we show the generation of both C∞ semigroups and analytic semigroups (in a particular case) on the Sobolev spaces WpkRn,Cq with k∈N0, 1≤p<∞ and q∈N. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces.

Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

Stabilization of coupled thermoelastic Kirchhoff plate and wave equations

We consider a coupled system consisting of a Kirchhoff thermoelastic plate and an undamped wave equation. It is known that the Kirchhoff thermoelastic plate is exponentially stable. The coupling is weak. First, we show that the coupled system is not exponentially stable. Afterwards, we prove that the coupled system is polynomially stable, and provide an explicit polynomial decay rate of the associated semigroup. Our proof relies on a combination of the frequency domain method and the multipliers technique.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/121/abstr.html

Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications

In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted $L^1$ spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on decay properties and asymptotic profiles of solutions. We find that the regularity-loss structure is destroyed by structural damping. Finally, we give some applications of our results on thermoelastic plate equations and damped Moore-Gibson-Thompson equation.

Cauchy problem for thermoelastic plate equations with different damping mechanisms

In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^p-L^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates for lower bound and upper bound of solutions in the $\dot{H}^s$ norm ($s\geq0$) are shown.

Dispersive mixed-order systems in $L^p$-Sobolev spaces and application to the thermoelastic plate equation

We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation with and without inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

Exact controllability of thermoelastic plate equation with memory

In this paper, we consider exact control problem for a coupled system of plate with Gurtin‐Pipkin equation. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the dual system. Firstly, we obtain the observability inequality of the dual system by means of multiplier method. Then, we prove that the system is exactly controllable based on the Hilbert Unique Method.

On some nonlinear problem for the thermoplate equations

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal \begin{document}$ L_p $\end{document} - \begin{document}$ L_q $\end{document} regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of \begin{document}$ C_0 $\end{document} analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

Generation of semigroups for the thermoelastic plate equation with free boundary conditions

We consider the linear thermoelastic plate equations with free boundary conditions in uniform \begin{document}$ C^4 $\end{document} -domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in \begin{document}$ L^p $\end{document} -spaces for all \begin{document}$ p\in(1, \infty) $\end{document} and has maximal \begin{document}$ L^q $\end{document} - \begin{document}$ L^p $\end{document} -regularity on finite time intervals. On bounded \begin{document}$ C^4 $\end{document} -domains, we obtain exponential stability.

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by urn:x-wiley:mma:media:mma4687:mma4687-math-0004 with the initial condition u|(t=0)=u0, ut|(t=0)=u1, and θ|(t=0)=θ0 in Ω and the boundary condition u=∂νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x1,⋯,xn)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C0 analytic semigroup and the maximal Lp‐Lq‐regularity of time‐dependent problem are derived.

Nonlinear thermoelastic plate equations – Global existence and decay rates for the Cauchy problem

We consider the Cauchy problem in Rn for quasilinear thermoelastic Kirchhoff-type plate equations where the heat conduction is modeled by either the Cattaneo law or by the Fourier law. Additionally, we take into account possible inertial effects. Considering nonlinearities which are of fourth-order in the space variable, we deal with a quasilinear system which triggers difficulties typical for nonlinear Schrödinger equations. The different models considered are systems of mixed type comparable to Schrödinger–parabolic or Schrödinger–hyperbolic systems. The main task consists in proving sophisticated a priori estimates leading to obtaining the global existence of solutions for small data, neither known nor expected for the Cauchy problem in pure plate theory nor available before for the coupled system under investigation, where only special cases (bounded domains within with analytic semigroup setting, or the Cauchy problem with semilinear nonlinearities) had been treated before.

Maximal regularity for the thermoelastic plate equations with free boundary conditions

We consider the linear thermoelastic plate equations with free boundary conditions in the $$L_p$$ in time and $$L_q$$ in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform $$C^4$$ -domain, which includes the cases of a bounded domain and of an exterior domain with $$C^4$$ -boundary. Moreover, we prove uniform a priori estimates for the solution. The proof is based on the existence of $${\mathcal R}$$ -bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

Dissipative structures for thermoelastic plate equations in $\mathbb R^n$

We consider the Cauchy problem in $\mathbb R^n$ for linear thermoelastic plate equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law -- described by the relaxation parameter $\tau$, where $\tau>0$ corresponds to Cattaneo's law and $ \tau=0 $ corresponds to Fourier's law. Additionally, we take into account possible inertial effects characterized by a parameter $\mu\geq 0$, where $\mu=0$ corresponds to the situation without inertial terms. For the Catteneo system without inertial term, being a coupling of a Schrödinger type equation (the elastic plate equation) with a hyperbolic system for the temperature and the heat flux, we shall show that a regularity-loss phenomenon appears in the asymptotic behavior as time tends to infinity, while this is not given in the standard model where the Cattaneo law is replaced by the standard Fourier law. This kind of effect of changing the qualitative behavior when moving from Fourier to Cattaneo reflects the effect known for bounded domains, where the system with Fourier law is exponentially stable while this property is lost when going to the Cattaneo law. In particular, we shall describe in detail the singular limit as $\tau\to 0$. For the system with inertial term we demonstrate that it is of standard type, not of regularity loss type. The corresponding limit of a vanishing inertial term is also described. All constants appearing in the main results are given explicitly, allowing for quantitative estimates. The optimality of the estimates is also proved.

Structural stability for a linear system of thermoelastic plate equations

The continuous dependence on parameters of strong solutions of initial boundary value problems for thermoelastic plate equation is established.

Local strong solutions for the non-linear thermoelastic plate equation on rectangular domains in L p -spaces

We consider the non-linear thermoelastic plate equation in rectangular domains $\Omega$. More precisely, $\Omega$ is considered to be given as the Cartesian product of whole or half spaces and a cube. First the linearized equation is treated as an abstract Cauchy problem in $L^p$-spaces. We take advantage of the structure of $\Omega$ and apply operator-valued Fourier multiplier results to infer an $\mathcal R$-bounded $\mathcal H^\infty$-calculus. With the help of maximal $L^p$-regularity existence and uniqueness of local real-analytic strong solutions together with analytic dependency on the data is shown.

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lionsʼ interpolation inequalities.

Instability of coupled systems with delay

We consider linear initial-boundary value problems that are a couplinglike second-order thermoelasticity, or the thermoelastic plate equation or its generalization (the $\alpha$-$\beta$-system introduced in [1, 26]). Now, there is a delay term given in part of the coupled system, and we demonstrate that the expected inherent damping will not prevent the system from not being stable; indeed, thesystems will shown to be ill-posed: a sequence of bounded initial data may lead to exploding solutions (at any fixed time).

Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U 0 ∈ H s (ℝ) ∩ L 1(ℝ). In addition, by restricting the initial data to U 0 ∈ H s (ℝ) ∩ L 1,γ(ℝ) and γ ∈ [0, 1], we can derive faster decay estimates with the decay rate improvement by a factor of t −γ/2.