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December  2021, 14(12): 4521-4550. doi: 10.3934/dcdss.2021135

## Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

 1 Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam 2 Department of Mathematics and Statistics, University of Economics Ho Chi Minh City, Vietnam

* Corresponding author: Quang-Minh Tran

Received  July 2021 Revised  October 2021 Published  December 2021 Early access  November 2021

The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.

Citation: Quang-Minh Tran, Hong-Danh Pham. Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4521-4550. doi: 10.3934/dcdss.2021135
##### References:
 [1] G. Autuori, F. Colasuonno and P. Pucci, Lifespan estimates for solutions of polyharmonic kirchhoff systems, Math. Models Methods Appl. Sci., 22 (2012), 1150009, 36 pp. doi: 10.1142/S0218202511500096. [2] H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051. [3] Y. Chen, X. Qiu, R. Xu and Y. Yang, Global existence and blowup of solutions for a class of nonlinear wave equations with linear pseudo-differential operator, Eur. Phys. J. Plus, 135 (2020), 573. [4] Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal. Theory Methods Appl., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664. [5] X. Dai, C. Yang, S. Huang, T. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006. [6] H. Di, Y. Shang and J. Yu, Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source, Electron. Res. Arch., 28 (2020), 221-261.  doi: 10.3934/era.2020015. [7] J. Fernandes and L. Maia, Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium, Discrete Contin. Dyn. Syst., 41 (2021), 1297-1318.  doi: 10.3934/dcds.2020318. [8] A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879. [9] F. Gazzola and M. Squassina., Global solutions and fiite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007. [10] F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ. Integr. Equations, 18 (2005), 961-990. [11] A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear aalysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [12] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form putt = −au+f(u), Trans. Am. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814. [13] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015. [14] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal. Theory Methods Appl., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015. [15] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111. [16] W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047. [17] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016. [18] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066. [19] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [20] X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with hardy-hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032. [21] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407. [22] L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595. [23] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [24] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171. [25] X. Wang, Y. Chen, Y. Yang, J. Li and R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal. Theory Methods Appl., 188 (2019), 475-499.  doi: 10.1016/j.na.2019.06.019. [26] X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141. [27] R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0. [28] R. Xu and Y. Niu, Addendum to ''global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations'', [J. Func. Anal. 264 (12) (2013), 2732–2763], J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026. [29] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010. [30] R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728. [31] Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 8 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065. [32] M. Zhang, Q. Zhao, Y. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021. [33] J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

show all references

##### References:
 [1] G. Autuori, F. Colasuonno and P. Pucci, Lifespan estimates for solutions of polyharmonic kirchhoff systems, Math. Models Methods Appl. Sci., 22 (2012), 1150009, 36 pp. doi: 10.1142/S0218202511500096. [2] H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051. [3] Y. Chen, X. Qiu, R. Xu and Y. Yang, Global existence and blowup of solutions for a class of nonlinear wave equations with linear pseudo-differential operator, Eur. Phys. J. Plus, 135 (2020), 573. [4] Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal. Theory Methods Appl., 192 (2020), 111664, 39 pp. doi: 10.1016/j.na.2019.111664. [5] X. Dai, C. Yang, S. Huang, T. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006. [6] H. Di, Y. Shang and J. Yu, Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source, Electron. Res. Arch., 28 (2020), 221-261.  doi: 10.3934/era.2020015. [7] J. Fernandes and L. Maia, Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium, Discrete Contin. Dyn. Syst., 41 (2021), 1297-1318.  doi: 10.3934/dcds.2020318. [8] A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879. [9] F. Gazzola and M. Squassina., Global solutions and fiite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007. [10] F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differ. Integr. Equations, 18 (2005), 961-990. [11] A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear aalysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [12] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form putt = −au+f(u), Trans. Am. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814. [13] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015. [14] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal. Theory Methods Appl., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015. [15] W. Lian, M. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111. [16] W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047. [17] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016. [18] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066. [19] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [20] X. Liu and J. Zhou, Initial-boundary value problem for a fourth-order plate equation with hardy-hénon potential and polynomial nonlinearity, Electron. Res. Arch., 28 (2020), 599-625.  doi: 10.3934/era.2020032. [21] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407. [22] L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595. [23] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [24] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171. [25] X. Wang, Y. Chen, Y. Yang, J. Li and R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal. Theory Methods Appl., 188 (2019), 475-499.  doi: 10.1016/j.na.2019.06.019. [26] X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141. [27] R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0. [28] R. Xu and Y. Niu, Addendum to ''global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations'', [J. Func. Anal. 264 (12) (2013), 2732–2763], J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026. [29] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010. [30] R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728. [31] Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 8 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065. [32] M. Zhang, Q. Zhao, Y. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021. [33] J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.
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