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March  2015, 14(2): 677-693. doi: 10.3934/cpaa.2015.14.677

Refined blow-up results for nonlinear fourth order differential equations

Filippo Gazzola 1, and  Paschalis Karageorgis 2,

1. 

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

School of Mathematics, Trinity College, Dublin 2

Received  May 2014 Revised  October 2014 Published  December 2014

We study a class of nonlinear fourth order differential equations which arise as models of suspension bridges. When it comes to power-like nonlinearities, it is known that solutions may blow up in finite time, if the initial data satisfy some positivity assumption. We extend this result to more general nonlinearities allowing exponential growth and to a wider class of initial data. We also give some hints on how to prevent blow-up.
Keywords: fourth-order ODE, Suspension bridges, blow up.
Mathematics Subject Classification: 34A12, 65L05, 34C10, 35B0.
Citation: Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
References:
[1]

C. J. Amick and J. F. Toland, Homoclinic orbits in the dynamic phase-space analogy of an elastic strut, Eur. J. Appl. Math., 3 (1992), 97-114. doi: 10.1017/S0956792500000735.  Google Scholar

[2]

E. Berchio, A. Ferrero, F. Gazzola and P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations, J. Diff. Eq., 251 (2011), 2696-2727. doi: 10.1016/j.jde.2011.05.036.  Google Scholar

[3]

D. Bonheure, Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 319-340. doi: 10.1016/S0294-1449(03)00037-4.  Google Scholar

[4]

D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Diff. Eq., Vol. III (2006), Elsevier Science, 103-202. doi: 10.1016/S1874-5725(06)80006-4.  Google Scholar

[5]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engineering and Structural Dynamics, 23 (1994), 1351-1367. Google Scholar

[6]

F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Diff. Equ., 211 (2013), 1-47.  Google Scholar

[7]

F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x.  Google Scholar

[8]

F. Gazzola and R. Pavani, Blow up oscillating solutions to some nonlinear fourth order differential equations, Nonlinear Analysis, 74 (2011), 6696-6711. doi: 10.1016/j.na.2011.06.049.  Google Scholar

[9]

F. Gazzola and R. Pavani, Blow-up oscillating solutions to some nonlinear fourth order differential equations describing oscillations of suspension bridges, IABMAS12, $6^{th}$ International Conference on Bridge Maintenance, Safety, Management, Resilience and Sustainability, 3089-3093, Stresa 2012, Biondini & Frangopol (Editors), Taylor & Francis Group, London (2012). doi: 10.1016/j.na.2011.06.049.  Google Scholar

[10]

F. Gazzola and R. Pavani, Wide oscillations finite time blow up for solutions to nonlinear fourth order differential equations, Arch. Rat. Mech. Anal., 207 (2013), 717-752. doi: 10.1007/s00205-012-0569-5.  Google Scholar

[11]

G. W. Hunt, H. M. Bolt and J. M. T. Thompson, Localisation and the dynamical phase-space analogy, Proc. Roy. Soc. London A, 425 (1989), 245-267.  Google Scholar

[12]

I. V. Ivanov, D. S. Velchev, M. Kneć and T. Sadowski, Computational models of laminated glass plate under transverse static loading,, In \emph{Shell-like Structures, ().   Google Scholar

[13]

P. Karageorgis and P. J. McKenna, The existence of ground states for fourth-order wave equations, Nonlinear Analysis, 73 (2010), 367-373. doi: 10.1016/j.na.2010.03.025.  Google Scholar

[14]

W. Lacarbonara, Nonlinear Structural Mechanics, Springer, 2013. doi: 10.1007/978-1-4419-1276-3.  Google Scholar

[15]

L. A. Peletier and W. C. Troy, Spatial Patterns. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and their Applications 45, Birkhäuser Boston Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0135-9.  Google Scholar

[16]

M. A. Peletier, Sequential buckling: a variational analysis, SIAM J. Math. Anal., 32 (2001), 1142-1168. doi: 10.1137/S0036141099359925.  Google Scholar

[17]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound and Vibration, 307 (2007), 894-905. Google Scholar

show all references

References:
[1]

C. J. Amick and J. F. Toland, Homoclinic orbits in the dynamic phase-space analogy of an elastic strut, Eur. J. Appl. Math., 3 (1992), 97-114. doi: 10.1017/S0956792500000735.  Google Scholar

[2]

E. Berchio, A. Ferrero, F. Gazzola and P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations, J. Diff. Eq., 251 (2011), 2696-2727. doi: 10.1016/j.jde.2011.05.036.  Google Scholar

[3]

D. Bonheure, Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 319-340. doi: 10.1016/S0294-1449(03)00037-4.  Google Scholar

[4]

D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Diff. Eq., Vol. III (2006), Elsevier Science, 103-202. doi: 10.1016/S1874-5725(06)80006-4.  Google Scholar

[5]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engineering and Structural Dynamics, 23 (1994), 1351-1367. Google Scholar

[6]

F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Diff. Equ., 211 (2013), 1-47.  Google Scholar

[7]

F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x.  Google Scholar

[8]

F. Gazzola and R. Pavani, Blow up oscillating solutions to some nonlinear fourth order differential equations, Nonlinear Analysis, 74 (2011), 6696-6711. doi: 10.1016/j.na.2011.06.049.  Google Scholar

[9]

F. Gazzola and R. Pavani, Blow-up oscillating solutions to some nonlinear fourth order differential equations describing oscillations of suspension bridges, IABMAS12, $6^{th}$ International Conference on Bridge Maintenance, Safety, Management, Resilience and Sustainability, 3089-3093, Stresa 2012, Biondini & Frangopol (Editors), Taylor & Francis Group, London (2012). doi: 10.1016/j.na.2011.06.049.  Google Scholar

[10]

F. Gazzola and R. Pavani, Wide oscillations finite time blow up for solutions to nonlinear fourth order differential equations, Arch. Rat. Mech. Anal., 207 (2013), 717-752. doi: 10.1007/s00205-012-0569-5.  Google Scholar

[11]

G. W. Hunt, H. M. Bolt and J. M. T. Thompson, Localisation and the dynamical phase-space analogy, Proc. Roy. Soc. London A, 425 (1989), 245-267.  Google Scholar

[12]

I. V. Ivanov, D. S. Velchev, M. Kneć and T. Sadowski, Computational models of laminated glass plate under transverse static loading,, In \emph{Shell-like Structures, ().   Google Scholar

[13]

P. Karageorgis and P. J. McKenna, The existence of ground states for fourth-order wave equations, Nonlinear Analysis, 73 (2010), 367-373. doi: 10.1016/j.na.2010.03.025.  Google Scholar

[14]

W. Lacarbonara, Nonlinear Structural Mechanics, Springer, 2013. doi: 10.1007/978-1-4419-1276-3.  Google Scholar

[15]

L. A. Peletier and W. C. Troy, Spatial Patterns. Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and their Applications 45, Birkhäuser Boston Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0135-9.  Google Scholar

[16]

M. A. Peletier, Sequential buckling: a variational analysis, SIAM J. Math. Anal., 32 (2001), 1142-1168. doi: 10.1137/S0036141099359925.  Google Scholar

[17]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound and Vibration, 307 (2007), 894-905. Google Scholar

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