American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$
  • CPAA Home
  • This Issue
  • Next Article
    Hopf bifurcation in an age-structured population model with two delays
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

[1]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[2]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[3]

Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577
[4]

P. J. McKenna. Oscillations in suspension bridges, vertical and torsional. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 785-791. doi: 10.3934/dcdss.2014.7.785
[5]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189
[6]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
[7]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[8]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[9]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
[10]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037
[11]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[12]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564
[13]

Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021141
[14]

Elvise Berchio, Filippo Gazzola. The role of aerodynamic forces in a mathematical model for suspension bridges. Conference Publications, 2015, 2015 (special) : 112-121. doi: 10.3934/proc.2015.0112
[15]

Alberto Ferrero, Filippo Gazzola. A partially hinged rectangular plate as a model for suspension bridges. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5879-5908. doi: 10.3934/dcds.2015.35.5879
[16]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[17]

Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188
[18]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[19]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284
[20]

Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences