A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models
Article
Keyantuo, V, Tebou, L, Warma, M. (2020). A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models
. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 40(5), 10.3934/dcds.2020152
Keyantuo, V, Tebou, L, Warma, M. (2020). A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models
. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 40(5), 10.3934/dcds.2020152
In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter 0 ≤ θ ≤ 1. The model includes both the Euler-Bernoulli (θ = 0) and Kirchhoff (θ = 1) models for thermoelastic plate as special cases. First, we show that the underlying semigroup is of Gevrey class δ for every δ > (2 − θ)/(2 − 4θ) for both the clamped and hinged boundary conditions when the parameter θ lies in the interval (0, 1/2). Then, we show that the semigroup is exponentially stable for hinged boundary conditions, for all values of θ in [0, 1]. Finally, we prove, by constructing a counterexample, that, under hinged boundary conditions, the semigroup is not analytic, for all θ in the interval (0, 1]. The main features of our Gevrey class proof are: the frequency domain method, appropriate decompositions of the components of the system and the use of Lions’ interpolation inequalities.
