The stability of positive semigroups on 𝐿_{𝑝} spaces

Weis, Lutz

doi:10.1090/s0002-9939-1995-1273529-2

Cited by 20 publications

(8 citation statements)

References 10 publications

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“…Finally, as a direct corollary of our results on polynomial stability we recover in a unified manner various results on exponential stability from [26,37,47,49,64,65]. We also obtain a new stability result for positive semigroups, Theorem 5.8.…”

Section: Introductionsupporting

confidence: 64%

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Stability theory for semigroups using (L,L) Fourier multipliers

Rozendaal

Veraar²

2018

Journal of Functional Analysis

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We study polynomial and exponential stability for C 0 -semigroups using the recently developed theory of operator-valued (L p , L q ) Fourier multipliers. We characterize polynomial decay of orbits of a C 0 -semigroup in terms of the (L p , L q ) Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.2010 Mathematics Subject Classification. Primary: 47D06. Secondary: 34D05, 35B40, 42B15, 46B20.

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Section: Introductionsupporting

confidence: 64%

“…On the other hand, cf. [47,49,64,65], uniform bounds for the resolvent do imply exponential stability for orbits in fractional domains, with the fractional domain parameter depending on the geometry of the underlying space.…”

Section: Introductionmentioning

confidence: 99%

Stability theory for semigroups using (L,L) Fourier multipliers

Rozendaal

Veraar²

2018

Journal of Functional Analysis

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“…In particular, if s( A ) < 0, then S t converges uniformly to 0 as t → ∞. This result is due to Weis [18,19].…”

Section: Convergence To Equilibriummentioning

confidence: 92%

Positive irreducible semigroups and their long-time behaviour

Arendt

Glück

2020

Phil. Trans. R. Soc. A.

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The notion Perron–Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t , while being absent for smaller times. This article is part of the theme issue ‘Semigroup applications everywhere’.

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“…However, such (semi)groups cannot be exponentially stable. Note also that by a theorem due to Weis [15] the equality (2.2) holds for positive semigroups on L p -spaces and C ( K )-spaces. However, as noted in [16], there exists a positive group on the Banach lattice L 1 failing to satisfy even the WSMT.…”

Section: Exponential and Strong Stability Revisitedmentioning

confidence: 99%

Semi-uniform stability of operator semigroups and energy decay of damped waves

Chill

Seifert

Tomilov

2020

Phil. Trans. R. Soc. A.

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Only in the last 15 years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of C 0 -semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems. This article is part of the theme issue ‘Semigroup applications everywhere’.

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The stability of positive semigroups on 𝐿_{𝑝} spaces

Abstract: Abstract.For a positive semigroup Tt on LP(Q, m) with generator A , the growth bound of (Tt) equals the spectral bound of A . In particular, if s(A) < 0 , the mild solutions of the Cauchy problem u' = Au are asymptotically stable.

Cited by 20 publications

References 10 publications

Stability theory for semigroups using (L,L) Fourier multipliers

Stability theory for semigroups using (L,L) Fourier multipliers

Positive irreducible semigroups and their long-time behaviour

Semi-uniform stability of operator semigroups and energy decay of damped waves

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