Blow up result for a viscoelatic plate equation with nonlinear source
DOI:
https://doi.org/10.5269/bspm.51725Resumen
We consider a viscoelastic plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in finite time. The background of the problem comes from the modeling of the downward displacement of suspension bridge using a thin rectangular plate. This result shows that in the present of a nonlinear source such as the earthquake shocks, the bridge will collapse in finite timeReferencias
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14. S. E. Mukiawa, Existence and general decay estimate for a nonlinear plate problem, Boundary Value Problems (2018), 2018:11. https://doi.org/10.1186/s13661-018-0931-0
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16. S. E. Mukiawa, Decay Result for a Delay Viscoelastic Plate Equation, Bull. Braz. Math. Soc., New Series 51 (2), 333-356, (2020). https://doi.org/10.1007/s00574-019-00155-y
17. S. E. Mukiawa, Blow-up result for a plate equation with fractional damping and nonlinear source terms,Open Journal of Mathematical Analysis 4,32-41, (2020). https://doi.org/10.30538/psrp-oma2020.0060
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19. S. H. Park, M. J. Lee, J.R. Kang, Blow-up results for viscoelastic wave equations with weak damping, Apl. Math. Letters 80, 20-26, (2018). https://doi.org/10.1016/j.aml.2018.01.002
20. S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications 265, 296-308, (2002). https://doi.org/10.1006/jmaa.2001.7697
21. Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, Journal of Mathematical Analysis and Applications, 418, 713-733, (2014). https://doi.org/10.1016/j.jmaa.2014.04.015
22. Tacoma Narrows Bridge collapse, http://www.youtube.com/watch?v=3mclp9QmCGs, (1940).
2. W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equations, Nonlinear Anal. TMA 70(9), 3203-3208, (2009). https://doi.org/10.1016/j.na.2008.04.024
3. A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. 35, 5879-5908, (2015). https://doi.org/10.3934/dcds.2015.35.5879
4. F. Gazzola, Mathematical models for Suspension Bridges: Nonlinear Structural Instability, Modeling, Simulation and Applications 15, Springer-Verlag (2015).
5. F. Gazzola and Y. Wang, Modeling suspension bridges through the Von Karman quasilinear plate equations, Progress in Nonlinear Differential Equations and Their Applications, In Contributions to Nonlinear Differential Equations and Systems, a tribute to Djairo Guedes de Figueiredo on occasion of his 80th birthday, 269-297, (2015). https://doi.org/10.1007/978-3-319-19902-3_18
6. V. Georgiev, G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms , J. Differential Equations 109, 295-308, (1994). https://doi.org/10.1006/jdeq.1994.1051
7. T. G. Ha, Asymptotic stability of the semilinear wave equation with boundary damping and source term, C. R. Math.Acad. Sci. Paris 352, 213-218, (2014). https://doi.org/10.1016/j.crma.2014.01.005
8. T. Kawada, History of the modern suspension bridge: solving the dilemma between economy and stiffness, ASCE Press (2010). https://doi.org/10.1061/9780784410189
9. W. J. Liu, J. Yu, On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms, Nonlinear Anal. 74, 2175-2190, (2011). https://doi.org/10.1016/j.na.2010.11.022
10. S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl. 320, 902-915, (2006). https://doi.org/10.1016/j.jmaa.2005.07.022
11. S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr. 260, 58-66, (2003). https://doi.org/10.1002/mana.200310104
12. S. A. Messaoudi and S.E. Mukiawa, Existence and decay of solutions to a viscoelastic plate equation, Electronic J. Differ. Equ. 22, 1-14, (2016).
13. S. A. Messaoudi and S.E. Mukiawa, A Suspension Bridge Problem: Existence and Stability, Mathematics Across Contemporary Sciences. Springer, Geneva (2016). https://doi.org/10.1007/978-3-319-46310-0_9
14. S. E. Mukiawa, Existence and general decay estimate for a nonlinear plate problem, Boundary Value Problems (2018), 2018:11. https://doi.org/10.1186/s13661-018-0931-0
15. S. E. Mukiawa, Asymptotic behaviour of a suspension bridge problem, Arab Journal of Mathematical Sciences 24, 31-42, (2018). https://doi.org/10.1016/j.ajmsc.2017.07.002
16. S. E. Mukiawa, Decay Result for a Delay Viscoelastic Plate Equation, Bull. Braz. Math. Soc., New Series 51 (2), 333-356, (2020). https://doi.org/10.1007/s00574-019-00155-y
17. S. E. Mukiawa, Blow-up result for a plate equation with fractional damping and nonlinear source terms,Open Journal of Mathematical Analysis 4,32-41, (2020). https://doi.org/10.30538/psrp-oma2020.0060
18. J. Y. Park, T. G. Ha, Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term, J. Math. Phys. 49, (2008). https://doi.org/10.1063/1.2919886
19. S. H. Park, M. J. Lee, J.R. Kang, Blow-up results for viscoelastic wave equations with weak damping, Apl. Math. Letters 80, 20-26, (2018). https://doi.org/10.1016/j.aml.2018.01.002
20. S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications 265, 296-308, (2002). https://doi.org/10.1006/jmaa.2001.7697
21. Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, Journal of Mathematical Analysis and Applications, 418, 713-733, (2014). https://doi.org/10.1016/j.jmaa.2014.04.015
22. Tacoma Narrows Bridge collapse, http://www.youtube.com/watch?v=3mclp9QmCGs, (1940).
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2022-12-23
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