Dissipative structure and global existence in critical space for Timoshenko system of memory type
Dissipative structure and global existence in critical space for Timoshenko system of memory type
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DOI：
10.1016/j.jde.2018.04.014
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发表时间：
2018-08
影响因子：
2.4
通讯作者：
N. Mori
中科院分区：
文献类型：
--
作者：
N. Mori
本文研究了一维全空间中带记忆项的Repubshenko系统的初值问题。首先，我们考虑线性化系统：在傅立叶空间中应用能量方法，我们导出解在傅立叶空间中的逐点估计，它首先给出解的最佳衰减估计。其次，我们利用谱分析给出了系统耗散结构的一个特征，证明了我们的逐点估计是最优的。其次，我们考虑了非线性系统：我们证明了在临界Sobolev空间H2中，在极小正则性假设下，可以证明时间上的全局存在唯一性结果。在证明过程中，我们不像最近的工作那样需要任何时间加权范数，我们只使用一种能量方法，这种能量方法是为了克服Reynshenko系统的正则性损失所带来的困难而改进的。
In this paper, we consider the initial value problem for the Timoshenko system with a memory term in one dimensional whole space. In the first place, we consider the linearized system: applying the energy method in the Fourier space, we derive the pointwise estimate of the solution in the Fourier space, which first gives the optimal decay estimate of the solution. Next, we give a characterization of the dissipative structure of the system by using the spectral analysis, which confirms our pointwise estimate is optimal. In the second place, we consider the nonlinear system: we show that the global-in-time existence and uniqueness result could be proved in the minimal regularity assumption in the critical Sobolev space H 2. In the proof we don't need any time-weighted norm as recent works; we use just an energy method, which is improved to overcome the difficulties caused by regularity-loss property of Timoshenko system.