A note on the regularity of a family of thermoelastic plates

• DOI: https://doi.org/10.22533/at.ed.13176626040614

• Palavras-chave: Euler-Bernoulli thermoelastic plate, Kirchhoff thermoelastic plate, sharp Gevrey class, fractional rotational force, analiticity, lack differentiable.

• Keywords: Euler-Bernoulli thermoelastic plate, Kirchhoff thermoelastic plate, sharp Gevrey class, fractional rotational force, analiticity, lack differentiable.

• Abstract: In these notes, we present an alternative analysis providing direct proofs of the regularity of solutions for a family of thermoelastic plate models that incorporate a fractional rotational inertia term of the form (−∆)τutt, where the parameter τ ranges in [0,1]. It is worth noting that the limiting cases τ = 0 and τ = 1 correspond to the classical Euler-Bernoulli and Kirchhoff plate models, respectively. The first results on the regularity of the Euler-Bernoulli thermoelastic plate date back to 1995, in the work of Liu and Renardy [14], where the authors established the analyticity of the semigroup S(t) under clamped mechanical boundary conditions for u and Dirichlet thermal boundary condition for θ. Regularity results for Kirchhoff thermoelastic plates under various boundary conditions were subsequently obtained in 1998 in [3].In the same year, and later in 2000, Lasiecka and Triggiani [8, 12] showed that the associated semigroup S(t) is not differentiable, which in particular implies the lack of analyticity. A decade later, in 2020, Keyantuo et al. [6], motivated by the results of Taylor’s thesis [23] on Gevrey classes, investigated the regularity properties of thermoelastic plate models. More recently, in 2021, Kuang et al. [7] investigated the regularity of a broader class of thermoelastic plate models encompassing the model considered in the present work. In that study, the authors identified the corresponding Gevrey classes and proved their sharpness for τ ∈ (0,1). Their analysis relies on spectral methods, frequency-domain techniques, and multiplier arguments combined with contradiction principles. We also carried out a spectral analysis of the system based on the properties of the operator −∆. In addition, we employed frequency-domain techniques, multiplier methods, and Lions’ interpolation theorem for τ ∈ [0,1]. Through direct estimates, we identified the sharp Gevrey classes  for the semigroup S(t) when τ ∈ (0,1). Moreover, we showed that S(t) is analytic for τ = 0, and loses analyticity for τ ∈ (0,1]. Finally, we proved that, in the case τ = 1, the semigroup associated with the model is not differentiable.