Characterization of Log-convex decay in non-selfadjoint dynamics

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doi:10.3934/era.2018.25.008

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2018, Volume 25: 72-86. Doi: 10.3934/era.2018.25.008

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Characterization of Log-convex decay in non-selfadjoint dynamics

Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark

Received: June 28, 2018

Published: March 2018

Supported by the Danish Research Council, Natural Sciences grant no. 4181-00042

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The short-time and global behavior are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

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Mathematics Subject Classification: Primary: 35E15; Secondary: 47D06.

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