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Loss of Energy Concentration in Nonlinear Evolution Beam Equations

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Abstract

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation

$$\begin{aligned} u_{tt} + u_{xxxx} + f(u)= g(x, t) \end{aligned}$$

in bounded space–time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities f and for some forcing terms g, highlighting some of their structural properties and performing some numerical simulations.

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Acknowledgements

The second author is partially supported by the PRIN project Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni. Both authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are grateful to the anonymous referees, whose valuable comments allowed to considerably improve the paper and its readability.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, 20126, Milan, Italy

    Maurizio Garrione

  2. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Filippo Gazzola

Authors
  1. Maurizio Garrione
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  2. Filippo Gazzola
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Correspondence to Filippo Gazzola.

Additional information

Communicated by Gabor Stepan.

Appendix

Appendix

We state here the result that we have used in Sect. 6 for the computation of some integral terms when dealing with the nonlinearity \(f(u)=u^3\). The proof follows from the prosthaphaeresis formulas.

Lemma 16

For all \(p\in {\mathbb {N}}\), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^4(px)\, \mathrm{d}x=3 . \end{aligned}$$

For all \(p,q\in {\mathbb {N}}\) (\(q\ne p\)), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^2(px)\sin ^2(qx)\, \mathrm{d}x=2 . \end{aligned}$$

For all \(p,q\in {\mathbb {N}}\) (\(q\ne p\)), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^3(px)\sin (qx)\, \mathrm{d}x=\left\{ \begin{array}{ll}-1\ &{}\quad \hbox {if }q=3p\\ 0\ &{}\quad \hbox {if }q\ne 3p . \end{array}\right. \end{aligned}$$

For all \(p,q,r\in {\mathbb {N}}\) (all different and \(q<r\)), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin ^2(px)\sin (qx)\sin (rx)\, \mathrm{d}x=\left\{ \begin{array}{ll}1\ &{}\quad \hbox {if }r+q=2p\\ -1\ &{}\quad \hbox {if }r-q=2p\\ 0\ &{}\quad \hbox {if }r\pm q\ne 2p . \end{array}\right. \end{aligned}$$

For all \(p,q,r,s\in {\mathbb {N}}\) (all different and \(p<q\), \(r<s\)), we have

$$\begin{aligned} \frac{8}{\pi }\int _0^\pi \sin (px)\sin (qx)\sin (rx)\sin (sx)\, \mathrm{d}x=\left\{ \begin{array}{ll}1\ &{}\quad \hbox {if }q\pm p=s\pm r\\ -1\ &{}\quad \hbox {if }q\pm p=s\mp r\\ 0\ &{}\quad \hbox {otherwise.} \end{array}\right. \end{aligned}$$

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Garrione, M., Gazzola, F. Loss of Energy Concentration in Nonlinear Evolution Beam Equations. J Nonlinear Sci 27, 1789–1827 (2017). https://doi.org/10.1007/s00332-017-9386-1

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