Non-uniform stability for bounded semi-groups on Banach spaces
Non-uniform stability for bounded semi-groups on Banach spaces
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DOI：
10.1007/s00028-008-0424-1
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发表时间：
2008-10
影响因子：
1.4
通讯作者：
C. Batty;Thomas Duyckaerts
中科院分区：
文献类型：
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作者：
C. Batty;Thomas Duyckaerts
设S（t）是Banach空间上的有界强连续半群，且是它的生成元.我们说当S（t）（A+ 1）− 1在算子范数下趋于0时，S（t）是半一致稳定的。渐近稳定性的概念比点态稳定性强，但严格弱于一致稳定性，推广了已知的对数稳定性、多项式稳定性和指数稳定性，本文证明了如果S是半一致稳定的，则A的谱不与虚轴相交.相反的情形是已知的，但是我们给出了S（t）（A+ 1）−1的衰变率的估计，将衰变与A在虚轴上的预解式的行为联系起来。这推广了Lebeau和Burq（在对数稳定性的情况下）以及Liu-Rao和Bátkai-Engel-Prüss-Schnaubelt（在多项式稳定性的情况下）的结果。
LetS(t) be a bounded strongly continuous semi-group on a Banach spaceBand –Abe its generator. We say thatS(t) is semi-uniformly stable whenS(t)(A+ 1)−1tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities.In this note we show that ifSis semi-uniformly stable then the spectrum ofAdoes not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay ofS(t)(A+ 1)−1, linking the decay to the behaviour of the resolvent ofAon the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability).