In this paper, we study the long-time behavior of the following plate equation
where the coefficient ε depends explicitly on time, the nonlinear damping and the nonlinearity both have critical growths.
M.M.Al-Gharabli and S.A.Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ.18 (2018), 105–125. doi:10.1007/s00028-017-0392-4.
2.
I.Chueshov and I.Lasiecka, Inertial manifolds for von Karman plate equations, Appl. Math. Optim46 (2002), 179–207(special issue dedicated to J. L. Lions.
3.
I.Chueshov and I.Lasiecka, Attractor for second-order evolution equations with damping, J. Dynam. Differential Equations16 (2004), 469–512. doi:10.1007/s10884-004-4289-x.
4.
I.Chueshov and I.Lasiecka, Global attractors for von Karman evolutions with a nonlinear, J. Dynam. Differential Equations198 (2004), 196–231. doi:10.1016/j.jde.2003.08.008.
5.
I.Chueshov and I.Lasiecka, Long-time dynamics of semilinear wave equations with a nonlinear interior-boundary damping and sources of critial exponents, in: Contemp. Math., Vol. 426, Amer. Math. Soc., Providence, RI, 2007, pp. 153–192.
6.
M.Conti and V.Pata, On the time-dependent Cattaneo law in space dimension one, Applied Mathematic andn Computations259 (2015), 32–44. doi:10.1016/j.amc.2015.02.039.
7.
M.Conti, V.Pata and R.Temam, Attrators for processes on time-dependent space. Application to Wave equation, J. Differential Equations255 (2013), 1254–1277. doi:10.1016/j.jde.2013.05.013.
8.
F.Di Plinio, G.S.Duane and R.Temam, Time dependent attracor for the oscillon equation, Discrete Conti. Dyn. Syst.29 (2011), 141–167. doi:10.3934/dcds.2011.29.141.
9.
A.K.Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in unbounded domain, Appl. Math. Lett.18 (2005), 827–832. doi:10.1016/j.aml.2004.08.013.
10.
A.K.Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl.318 (2006), 92–101. doi:10.1016/j.jmaa.2005.05.031.
11.
A.K.Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations225 (2006), 528–548. doi:10.1016/j.jde.2005.12.001.
12.
I.Lasiecka, Finite dimensionality and compactness of atrractors for von Karmam equations with nonlinear dissipation, NoDEA Nonlinear Differential Equations Appl.6 (1999), 437–472. doi:10.1007/s000300050012.
13.
I.Lasiecka and A.R.Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation, J. Math Anal. Appl.270 (2002), 16–50. doi:10.1016/S0022-247X(02)00006-9.
14.
F.J.Meng, J.Wu and C.X.Zhao, Time-dependent global attractor for extensible Berger equation, J. Math Anal. Appl.469 (2019), 1045–1069. doi:10.1016/j.jmaa.2018.09.050.
15.
F.J.Meng, M.H.Yang and C.K.Zhong, Attractors for wave equtions with nonlinear damping on time-dependent space, Discrete Conti. Dyn. Syst.21 (2016), 205–225. doi:10.3934/dcdsb.2016.21.205.
16.
L.Yang and C.K.Zhong, Global attractor for plate eqution with nonlinear damping, Nonliear Anal.69 (2008), 3802–3810. doi:10.1016/j.na.2007.10.016.
