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

• Autorzy: Yanbing Yang , Ahmed Md Salik , Lanlan Qin , Runzhang Xu

Identyfikatory

DOI

10.7494/OpMath.2019.39.2.297

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

fourth-order nonlinear wave equation strong damping; blow up; global existence

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2019
• Tom: Vol. 39, no. 2
• Strony: 297--313
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Yanbing Yang

• Harbin Engineering University College ol Science 150001, People's Republic of China

autor

Ahmed Md Salik

• Harbin Engineering University College ol Science 150001, People's Republic of China

autor

Lanlan Qin

• Harbin Engineering University College ol Science 150001, People's Republic of China

autor

Runzhang Xu

• Harbin Engineering University College ol Science 150001, People's Republic of China

Bibliografia

• [1] D. Ang, A. Dinh, On the strongly damped wave equation:[formula], 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 R3, 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 •es, 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 PInstitut Henri Poincare Analyse Non Lineaire 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 ol 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 Poincare 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 ol Mathematics 24 (2013), 1350043.
• [15] G. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian Journal ol 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 ol 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 ol 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, Zeitschrilt lur 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 ol Differential Equations 187 (2003), 520-540.
• [21] Z. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, Journal ol 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.