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 in | Nonlinear analysis: real world applications Vol. 43; pp. 167 - 191 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier Ltd
01.10.2018
Elsevier BV Elsevier |
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Online Access | Get full text |
ISSN | 1468-1218 1878-5719 |
DOI | 10.1016/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. |
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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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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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References | Haraux (b9) 2005; 29 Fitouri, Gasmi (b10) 2012; 3 Haraux (b12) 1981; vol. 841 Gasmi, Haraux (b5) 2012; 97 Holmes, Marsden (b1) 1981; 76 Moon, Holmes (b2) 1979; 65 Loud (b7) 1955; 34 Bochner (b13) 1962; 48 Loud (b3) 1957; 24 Cazenave, Haraux (b11) 1998; vol. 13 Nakajima, Seifert (b6) 1983; 49 Fitouri, Haraux (b4) 2013; 33 Aloui, Haraux (b14) 2013; 265 Loud (b8) 1959; 31 Gasmi (10.1016/j.nonrwa.2018.02.007_b5) 2012; 97 Loud (10.1016/j.nonrwa.2018.02.007_b8) 1959; 31 Loud (10.1016/j.nonrwa.2018.02.007_b7) 1955; 34 Nakajima (10.1016/j.nonrwa.2018.02.007_b6) 1983; 49 Aloui (10.1016/j.nonrwa.2018.02.007_b14) 2013; 265 Haraux (10.1016/j.nonrwa.2018.02.007_b12) 1981; vol. 841 Haraux (10.1016/j.nonrwa.2018.02.007_b9) 2005; 29 Cazenave (10.1016/j.nonrwa.2018.02.007_b11) 1998; vol. 13 Fitouri (10.1016/j.nonrwa.2018.02.007_b4) 2013; 33 Fitouri (10.1016/j.nonrwa.2018.02.007_b10) 2012; 3 Holmes (10.1016/j.nonrwa.2018.02.007_b1) 1981; 76 Moon (10.1016/j.nonrwa.2018.02.007_b2) 1979; 65 Loud (10.1016/j.nonrwa.2018.02.007_b3) 1957; 24 Bochner (10.1016/j.nonrwa.2018.02.007_b13) 1962; 48 |
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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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