基于位移的Gurtin-Pipkin热弹性Bresse型和Breschenko型系统的稳定性研究
Study on the stability of thermoelastic Bresse and Timoshenko type systems with Gurtin-Pipkin's law via the vertical displacements
摘要：
本文研究了有界区域中一维线性热弹性Bresse系统的稳定性，其中耦合通过Bresse模型的第一分量给出，热传导为Gurtin-Pipkin型。考虑了两种耦合，第一种耦合是关于空间变量的一阶耦合，第二种耦合是零阶耦合。我们的状态的适定性和多项式稳定的系统，其中的衰减率依赖于初始数据的光滑性。此外，在一阶耦合的情况下，我们证明了系统的指数稳定性与一些新的参数条件之间的等价性。然而，当耦合是零阶时，我们证明了非指数稳定性独立于系统的参数。应用到相应的特殊的Eschlishenko模型，我们证明了这两个耦合导致指数稳定的当且仅当某些条件的参数的系统被满足，和这两个耦合保证多项式稳定的参数无关。证明是基于半群理论和结合的能量方法和频域方法。
英文摘要：
The objective of this paper is to study the stability of a linear one-dimensional thermoelastic Bresse system in a bounded domain, where the coupling is given through the first component of the Bresse model with the heat conduction of Gurtin-Pipkin type. Two kinds of coupling are considered; the first coupling is of order one with respect to space variable, and the second one is of order zero. We state the well-posedness and show the polynomial stability of the systems, where the decay rates depend on the smoothness of initial data. Moreover, in case of coupling of order one, we prove the equivalence between the exponential stability and some new conditions on the parameters of the system. However, when the coupling is of order zero, we prove the non-exponential stability independently of the parameters of the system. Applications to the corresponding particular Timoshenko models are also given, where we prove that both couplings lead to the exponential stability if and only if some conditions on the parameters of the systems are satisfied, and both couplings guarantee the polynomial stability independently of the parameters of the systems. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.
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