# American Institute of Mathematical Sciences

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

## Characterization of Log-convex decay in non-selfadjoint dynamics

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

Received  June 29, 2018 Published  December 2018

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

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.

Citation: Jon Johnsen. Characterization of Log-convex decay in non-selfadjoint dynamics. Electronic Research Announcements, 2018, 25: 72-86. doi: 10.3934/era.2018.25.008
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##### References:
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