Asymptotics of evolution equations beyond Banach spaces

B. Jacob, S. A. Wegner

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup’s speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fréchet function spaces.

Original languageEnglish
Pages (from-to)347-377
Number of pages31
JournalSemigroup Forum
Volume91
Issue number2
DOIs
Publication statusPublished - Dec 2 2014
Externally publishedYes

Fingerprint

Evolution Equation
Banach space
Strongly Continuous Semigroups
Semigroup
Zero
Operator Semigroups
Converge
Locally Convex Space
Stability Theory
Heat Equation
Function Space
Multiplication
Asymptotic Behavior
Polynomial

Keywords

  • Asymptotic behavior
  • Exponential stability
  • Semigroups on locally convex space

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Asymptotics of evolution equations beyond Banach spaces. / Jacob, B.; Wegner, S. A.

In: Semigroup Forum, Vol. 91, No. 2, 02.12.2014, p. 347-377.

Research output: Contribution to journalArticle

Jacob, B. ; Wegner, S. A. / Asymptotics of evolution equations beyond Banach spaces. In: Semigroup Forum. 2014 ; Vol. 91, No. 2. pp. 347-377.
@article{fa79a9b34a0146fe803d3e9c6e9fe7ea,
title = "Asymptotics of evolution equations beyond Banach spaces",
abstract = "We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup’s speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fr{\'e}chet function spaces.",
keywords = "Asymptotic behavior, Exponential stability, Semigroups on locally convex space",
author = "B. Jacob and Wegner, {S. A.}",
year = "2014",
month = "12",
day = "2",
doi = "10.1007/s00233-014-9659-z",
language = "English",
volume = "91",
pages = "347--377",
journal = "Semigroup Forum",
issn = "0037-1912",
publisher = "Springer New York",
number = "2",

}

TY - JOUR

T1 - Asymptotics of evolution equations beyond Banach spaces

AU - Jacob, B.

AU - Wegner, S. A.

PY - 2014/12/2

Y1 - 2014/12/2

N2 - We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup’s speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fréchet function spaces.

AB - We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup’s speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fréchet function spaces.

KW - Asymptotic behavior

KW - Exponential stability

KW - Semigroups on locally convex space

UR - http://www.scopus.com/inward/record.url?scp=84942830128&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84942830128&partnerID=8YFLogxK

U2 - 10.1007/s00233-014-9659-z

DO - 10.1007/s00233-014-9659-z

M3 - Article

AN - SCOPUS:84942830128

VL - 91

SP - 347

EP - 377

JO - Semigroup Forum

JF - Semigroup Forum

SN - 0037-1912

IS - 2

ER -