This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more generalized nonlinear functions at subcritical initial energy level. Further for arbitrarily positive initial energy we give some sufficient conditions ensuring finite time blow up.
L.J.An and A.Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math.54 (1994), 708–730. doi:10.1137/S0036139992238498.
2.
L.J.An and A.Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math.55 (1995), 136–155. doi:10.1137/S0036139993255327.
3.
H.Chen and S.Y.Tian, Initial boundary value problem for a class of semi-linear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations258 (2015), 4424–4442. doi:10.1016/j.jde.2015.01.038.
4.
S.T.Chen, B.L.Zhang and X.H.Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal.9 (2020), 148–167. doi:10.1515/anona-2018-0147.
5.
Y.X.Chen and R.Z.Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal.192 (2020), 35–75.
6.
I.Chueshov and I.Lasiecka, Global attractors for Mindlin–Timoshenko plates and for their Kirchhoff limits, Milan J. Math.74 (2006), 117–138. doi:10.1007/s00032-006-0050-8.
7.
I.Chueshov and I.Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff–Boussinesq models, Discrete Contin. Dyn. Syst.15 (2006), 777–809. doi:10.3934/dcds.2006.15.777.
8.
I.Chueshov and I.Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc.195 (2008), 978–1008.
9.
I.Chueshov and I.Lasiecka, On global attractor for 2D Kirchhoff–Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations36 (2011), 67–99. doi:10.1080/03605302.2010.484472.
10.
J.A.Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal.63 (2005), 331–343. doi:10.1016/j.na.2005.02.108.
11.
H.A.Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form , Trans. Amer. Math. Soc.192 (1974), 1–21. doi:10.1090/S0002-9947-1974-0344697-2.
12.
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.
13.
W.Lian, V.D.Rădulescu, R.Z.Xu, Y.B.Yang and N.Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Advances in Calculus of Variations. doi:10.1515/acv-2019-0039.
14.
W.Lian and R.Z.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.
15.
Y.C.Liu and R.Z.Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl.331 (2007), 585–607. doi:10.1016/j.jmaa.2006.09.010.
16.
Y.C.Liu and R.Z.Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations244 (2008), 200–228. doi:10.1016/j.jde.2007.10.015.
17.
T.F.Ma and M.L.Pelicer, Attractors for weakly damped beam equations with p-Laplacian, Discrete Contin. Dyn. Syst. (2013), 525–534. doi:10.3934/proc.2013.2013.525.
18.
N.S.Papageorgiou, V.D.Rădulescu and D.D.Repovs˘, Nonlinear Analysis – Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
19.
R.Racke and C.Y.Shang, Global attractors for nonlinear beam equations, Proc. Roy. Soc. Edinburgh Sect. A.142 (2012), 1087–1107. doi:10.1017/S030821051000168X.
20.
T.Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal.9 (2020), 1383–1401. doi:10.1515/anona-2020-0073.
21.
F.Shakeri and M.Dehghan, A hybrid Legendre tau method for the solution of a class of nonlinear wave equations with nonlinear dissipative terms, Numer. Methods Partial Differential Equations27 (2011), 1055–1071. doi:10.1002/num.20569.
22.
J.H.Shen, Y.B.Yang, S.H.Chen and R.Z.Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math.24 (2013), 35–77.
23.
E.Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations186 (2002), 259–298. doi:10.1016/S0022-0396(02)00023-2.
24.
M.Q.Xiang, B.L.Zhang and V.D.Rădulescu, Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal.9 (2020), 690–709. doi:10.1515/anona-2020-0021.
25.
R.Z.Xu, X.C.Wang, Y.B.Yang and S.H.Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys.59 (2018), 061503. doi:10.1063/1.5006728.
26.
Z.J.Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations187 (2003), 520–540. doi:10.1016/S0022-0396(02)00042-6.
27.
Z.J.Yang, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys.51 (2010), 092703. doi:10.1063/1.3477939.
28.
Z.J.Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys.53 (2012), 032702. doi:10.1063/1.3694730.
29.
M.Y.Zhang and M.Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal.9 (2020), 882–894. doi:10.1515/anona-2020-0031.
