An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term

We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one...

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Published inNonlinear analysis: real world applications Vol. 43; pp. 167 - 191
Main Authors Ghisi, Marina, Gobbino, Massimo, Haraux, Alain
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Ltd 01.10.2018
Elsevier BV
Elsevier
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ISSN1468-1218
1878-5719
DOI10.1016/j.nonrwa.2018.02.007

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Abstract We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one unstable, and all solutions are asymptotic for t large to one of these stationary solutions. We show that this pattern extends to the case where the external force is bounded and small enough, in the sense that solutions can exhibit only three different asymptotic behaviors.
AbstractList We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one unstable, and all solutions are asymptotic for t large to one of these stationary solutions. We show that this pattern extends to the case where the external force is bounded and small enough, in the sense that solutions can exhibit only three different asymptotic behaviors.
We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one unstable, and all solutions are asymptotic for t large to one of these stationary solutions. We show that this pattern extends to the case where the external force is bounded and small enough, in the sense that solutions can exhibit only three different asymptotic behaviors.
Author Haraux, Alain
Gobbino, Massimo
Ghisi, Marina
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  surname: Gobbino
  fullname: Gobbino, Massimo
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  organization: Università degli Studi di Pisa, Dipartimento di Ingegneria Civile e Industriale, PISA, Italy
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  givenname: Alain
  surname: Haraux
  fullname: Haraux, Alain
  email: haraux@ann.jussieu.fr
  organization: Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France
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Cites_doi 10.1016/0022-0396(83)90005-0
10.1215/S0012-7094-57-02412-2
10.1073/pnas.48.12.2039
10.1016/j.jfa.2013.07.019
10.1016/j.matpur.2009.08.005
10.1002/sapm1955341173
10.1007/BF00251249
10.1016/0022-460X(79)90520-0
10.3934/dcds.2013.33.211
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Keywords Dissipative hyperbolic equation
Duffing equation
Magneto-elastic oscillations
Asymptotic behavior
Language English
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Snippet We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or...
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SubjectTerms Analysis of PDEs
Asymptotic behavior
Asymptotic methods
Asymptotic properties
Dimensional analysis
Dissipative hyperbolic equation
Duffing equation
Dynamical Systems
Functional Analysis
Hilbert space
Linear algebra
Linear evolution equations
Magneto-elastic oscillations
Mathematics
Nonlinear equations
Title An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term
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