具有Coleman-Gurtin热定律的波热系统的最优衰减
Optimal decay for a wave-heat system with Coleman-Gurtin thermal law
摘要：
本文研究了一类具有Coleman-Gurtin热定律的一维波热耦合系统解的长期行为。我们的方法是基于$C_0$ -半群的渐近理论和最近的结果耦合控制系统。作为我们的主要结果，我们表示系统的波部分和Coleman-Gurtin部分之间的反馈互连，我们表明，在Dafermos的历史框架的相关半群是多项式稳定的最佳衰减率$t^{-2}$为$t\to\infty$。特别是，我们得到了一个尖锐的估计能量衰减率的经典解的问题。
英文摘要：
We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman-Gurtin thermal law. Our approach is based on the asymptotic theory of $C_0$-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman-Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate $t^{-2}$ as $t\to\infty$. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.
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