Opuscula Math. 39, no. 2 (2019), 297-313
https://doi.org/10.7494/OpMath.2019.39.2.297

Opuscula Mathematica

# Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations

Yang Yanbing
Md Salik Ahmed
Qin Lanlan
Xu Runzhang

Abstract. Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time.

Keywords: fourth-order nonlinear wave equation, strong damping, blow up, global existence.

Mathematics Subject Classification: 35B44, 35L35, 35L05.

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1. D. Ang, A. Dinh, On the strongly damped wave equation: $$u_{tt}-\Delta u -\Delta u_t +f(u)=0$$, SIAM Journal on Mathematical Analysis 19 (1988), 1409-1418.
2. P. Aviles, J. Sandefur, Nonlinear second order equations with applications to partial differential equations, Journal of Differential Equations 58 (1985), 404-427.
3. V. Belleri, V. Pata, Attractors for semilinear strongly damped wave equation on $$R^3$$, Discrete Continues Dynamic Systems 7 (2001), 719-735.
4. A. Carvalho, J. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bulletin of the Australian Mathematical Society 66 (2002), 443-463.
5. L. Fatoria, M. Silva, T. Ma, Z. Yang, Long-time behavior of a class of thermoelastic plates with nonlinear strain, Journal of Differential Equations 259 (2015), 4831-4862.
6. A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete and Continuous Dynamical Systems 35 (2015), 5879-5908.
7. Q. Fu, P. Gu, J. Wu, Iterative learning control for one-dimensional fourth order distributed parameter systems, Science China Information Sciences 60 (2017) 01220.
8. F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Annales de l'Institut Henri Poincaré Analyse Non Linéaire 23 (2006), 185-207.
9. Q. Lin, Y. Wu, S. Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonlinear Analysis, Theory, Methods & Applications 69 (2008), 4340-4351.
10. Y. Liu, R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, Journal of Differential Equations 244 (2008), 200-228.
11. V. Pata, M. Squassina, On the strongly damped wave equation, Communications in Mathematical Physics 253 (2005), 511-533.
12. V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), 1495-1506.
13. T. Saanouni, Fourth-order damped wave equation with exponential growth nonlinearity, Annales Henri Poincaré 18 (2017), 345-374.
14. J. Shen, Y. Yang, S. Chen, R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, International Journal of Mathematics 24 (2013), 1350043.
15. G. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian Journal of Mathematics 32 (1980), 631-643.
16. R. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Mathematical Methods in Applied Science 33 (2010), 831-844.
17. R. Xu, Y. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, International Journal of Mathematics 23 (2012), 1250060.
18. R. Xu, Y. Yang, Global existence and asymptotic behaviour of solution for a class of fourth order strongly damped nonlinear wave equations, Quarterly of Applied Mathematics 71 (2013), 401-415.
19. R. Xu, Y. Yang, B. Liu, J. Shen, S. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Zeitschrift fur Angewandte Mathematik und Physik 66 (2015), 955-976.
20. Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, Journal of Differential Equations 187 (2003), 520-540.
21. Z. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, Journal of Mathematical Physics 53 (2012) 032702.
22. Z. Yang, Z. Liu, Global attractor for a strongly damped wave equation with fully supercritical nonlinearities, Discrete Continues Dynamic Systems 37 (2017), 2181-2205.
• Yang Yanbing
• Harbin Engineering University, College of Science, 150001, People's Republic of China
• Md Salik Ahmed
• Harbin Engineering University, College of Science, 150001, People's Republic of China
• Qin Lanlan
• Harbin Engineering University, College of Science, 150001, People's Republic of China
• Communicated by Marius Ghergu.
• Accepted: 2018-11-03.
• Published online: 2018-12-07.