Semiflows generated by lipschitz perturbations of non-densely defined operators

Horst Thieme, Glenn Webb

Research output: Contribution to journalArticle

171 Citations (Scopus)

Abstract

A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not densely defined but satisfies the resolvent estimates of the Hille and Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato and Sinestrari. We derive a variation of constants formula which allows us to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.

Original languageEnglish (US)
Pages (from-to)1035-1066
Number of pages32
JournalDifferential and Integral Equations
Volume3
Issue number6
StatePublished - 1990

Fingerprint

Semiflow
Integral Solution
Lipschitz
Mathematical operators
Perturbation
Operator
Semilinear
Evolution Equation
Cauchy Problem
Variation of Constants Formula
Resolvent Estimates
Abstract Differential Equations
Closed Operator
Sufficient Conditions
Volterra Integral Equations
Local Stability
Population Model
Functional Differential Equations
Closed set
Convex Sets

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Semiflows generated by lipschitz perturbations of non-densely defined operators. / Thieme, Horst; Webb, Glenn.

In: Differential and Integral Equations, Vol. 3, No. 6, 1990, p. 1035-1066.

Research output: Contribution to journalArticle

@article{0f82069eed44415eba464965708765dc,
title = "Semiflows generated by lipschitz perturbations of non-densely defined operators",
abstract = "A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not densely defined but satisfies the resolvent estimates of the Hille and Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato and Sinestrari. We derive a variation of constants formula which allows us to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.",
author = "Horst Thieme and Glenn Webb",
year = "1990",
language = "English (US)",
volume = "3",
pages = "1035--1066",
journal = "Differential and Integral Equations",
issn = "0893-4983",
publisher = "Khayyam Publishing, Inc.",
number = "6",

}

TY - JOUR

T1 - Semiflows generated by lipschitz perturbations of non-densely defined operators

AU - Thieme, Horst

AU - Webb, Glenn

PY - 1990

Y1 - 1990

N2 - A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not densely defined but satisfies the resolvent estimates of the Hille and Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato and Sinestrari. We derive a variation of constants formula which allows us to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.

AB - A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not densely defined but satisfies the resolvent estimates of the Hille and Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato and Sinestrari. We derive a variation of constants formula which allows us to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.

UR - http://www.scopus.com/inward/record.url?scp=84972549029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972549029&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84972549029

VL - 3

SP - 1035

EP - 1066

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 6

ER -