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Refined blow-up results for nonlinear fourth order differential equations
1. | Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano |
2. | School of Mathematics, Trinity College, Dublin 2 |
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. |
[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. |
[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. |
[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. |
[5] |
J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engineering and Structural Dynamics, 23 (1994), 1351-1367. |
[6] |
F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Diff. Equ., 211 (2013), 1-47. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
I. V. Ivanov, D. S. Velchev, M. Kneć and T. Sadowski, Computational models of laminated glass plate under transverse static loading, In Shell-like Structures, Non-classical Theories and Applications (H. Altenbach and V. Eremeyev eds.), |
[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. |
[14] |
W. Lacarbonara, Nonlinear Structural Mechanics, Springer, 2013.
doi: 10.1007/978-1-4419-1276-3. |
[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. |
[16] |
M. A. Peletier, Sequential buckling: a variational analysis, SIAM J. Math. Anal., 32 (2001), 1142-1168.
doi: 10.1137/S0036141099359925. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engineering and Structural Dynamics, 23 (1994), 1351-1367. |
[6] |
F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Diff. Equ., 211 (2013), 1-47. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
I. V. Ivanov, D. S. Velchev, M. Kneć and T. Sadowski, Computational models of laminated glass plate under transverse static loading, In Shell-like Structures, Non-classical Theories and Applications (H. Altenbach and V. Eremeyev eds.), |
[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. |
[14] |
W. Lacarbonara, Nonlinear Structural Mechanics, Springer, 2013.
doi: 10.1007/978-1-4419-1276-3. |
[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. |
[16] |
M. A. Peletier, Sequential buckling: a variational analysis, SIAM J. Math. Anal., 32 (2001), 1142-1168.
doi: 10.1137/S0036141099359925. |
[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. |
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