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The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the vertical displacement
Applicable Analysis ( IF 1.1 ) Pub Date : 2020-09-02 , DOI: 10.1080/00036811.2020.1811974
Aissa Guesmia 1
Affiliation  

In this paper, we study the energy decay for two one-dimensional thermoelastic Bresse-type systems in a bounded open interval under mixed homogeneous Dirichlet–Neumann boundary conditions and with two different kinds of dissipation working only on the vertical displacement and given by heat conduction of types I and III. The two systems are consisting of three wave equations (Bresse-type system) coupled, in a certain manner, with one heat equation (type I) or with one wave equation (type III). We prove that, independently of the values of the coefficients, these systems are not exponentially stable. Moreover, we show the polynomial stability for each system with a decay rate depending on the smoothness of the initial data. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach. Our results complete our study [Guesmia A. Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement. Nonauton Dyn Syst. 2017;4:78–97] for the case of a dissipation generated by an infinite memory.



中文翻译:

I型和III型热传导通过垂直位移对布雷斯系统衰减率的影响

在本文中,我们研究了两个一维热弹性 Bresse 型系统在混合齐次 Dirichlet-Neumann 边界条件下的有界开区间的能量衰减,其中两种不同的耗散仅作用于垂直位移并由热传导给出Ⅰ型和Ⅲ型。这两个系统由三个波动方程(Bresse 型系统)组成,它们以某种方式与一个热方程(I 型)或一个波动方程(III 型)耦合。我们证明,独立于系数的值,这些系统不是指数稳定的。此外,我们展示了每个系统的多项式稳定性,其衰减率取决于初始数据的平滑度。证明是基于半群理论和能量方法和频域方法的结合。我们的结果完成了我们的研究 [Guesmia A. Bresse 系统的非指数和多项式稳定性结果,在垂直位移中有一个无限记忆。Nonauton Dyn Syst。2017;4:78–97] 对于无限内存产生的耗散情况。

更新日期:2020-09-02
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