Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the...

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Bibliographic Details
Published in:Mathematische Nachrichten 2024-07, Vol.297 (7), p.2750-2785
Main Authors: Latushkin, Yuri, Pogan, Alin
Format: Article
Language:English
Subjects:
analytic semigroups
Banach spaces
bidomain equation
Decay rate
Electrophysiology
Lyapunov linear stability
Operators (mathematics)
Parameters
planar traveling waves
Reaction-diffusion equations
reaction–diffusion systems
Semigroups
Stability
Traveling waves
uniform exponential bounds
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Summary:We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.202300273