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Time-dependent global attractors for the strongly damped wave equations with lower regular forcing term. (English) Zbl 1509.35063

Summary: In this paper, based on a new theoretical framework of time-dependent global attractors [M. Conti et al., J. Differ. Equations 255, No. 6, 1254–1277 (2013; Zbl 1288.35098)], we consider the strongly damped wave equations \(\varepsilon(t)u_{tt}-\Delta u_t-\Delta u+f(u)=g(x)\) and establish the existence of attractors in \(\mathcal{H}_t=H_0^1(\Omega)\times L^2(\Omega)\) and \(\mathcal{V}_t=H_0^1(\Omega)\times H_0^1(\Omega)\), respectively.

MSC:

35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations

Citations:

Zbl 1288.35098
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References:

[1] J. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations 17 (1992), 841-866. · Zbl 0815.35067
[2] A.N. Carvalho and J.W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002), 443-463. · Zbl 1020.35059
[3] A.N. Carvalho and J.W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math. 207 (2002), 287-310. · Zbl 1060.35082
[4] A.N. Carvalho, J.A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. · Zbl 1263.37002
[5] J.W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. 64 (2006), 174-187. · Zbl 1083.35066
[6] M. Conti and V. Pata, Asymptotic structure of the attractor for processes on timedependent spaces, Nonlinear Anal. Real World Appl. 19 (2014), 1-10. · Zbl 1297.35046
[7] M. Conti, V. Pata and M. Squassina, Strongly damped wave equations on R3 with critical nonlinearities, Commun. Appl. Anal. 9 (2005), 161-176. · Zbl 1096.35090
[8] M. Conti, V. Pata and R. Temam, Attractors for process on time-dependent spaces: Applications to wave equations, J. Differential Equations 255 (2013), 1254-1277. · Zbl 1288.35098
[9] F. Di Plinio, G.S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst. 29 (2011), 141-167. · Zbl 1223.37100
[10] Y.L. Du, X. Li and C.Y. Sun, On the asymptotic behavior of strongly damped wave equations, Topol. Methods Nonlinear Anal. 44 (2014), 161-175. · Zbl 1368.35186
[11] J.M. Ghidaglia and A. Marzocchi, Longtime behavior of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal. 22 (1991), 879-895. · Zbl 0735.35015
[12] F.J. Meng, M.H. Yang and C.K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 205-225. · Zbl 1343.35035
[13] V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), 511-533. · Zbl 1068.35077
[14] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), 1495-1506. · Zbl 1113.35023
[15] C.Y. Sun, D.M. Cao and J.Q. Duan, Non-autonomous wave dynamics with memoryasymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 743-761. · Zbl 1170.35026
[16] C.Y. Sun and M.H. Yang, Dynamics of the nonclassical diffusion equations, Asymptot. Anal. 59 (2008), 51-81. · Zbl 1154.35063
[17] Y. Sun and Z.J. Yang, Longtime dynamics for a nonlinear viscoelastic equation with time-dependent memory kernel, Nonlinear Anal. Real World Appl. 64 (2022), 103432. · Zbl 1479.35118
[18] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. · Zbl 0871.35001
[19] Y.L. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. (Engl. Ed.) 18 (2002), 273-276. · Zbl 1017.35025
[20] Y.Q. Xie, Q.S. Li and K.X. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl. 31 (2016), 23-37. · Zbl 1338.35066
[21] M.H. Yang and C.Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc. 361 (2009), 1069-1101. · Zbl 1159.37022
[22] M.H. Yang and C.Y. Sun, Attractors for strongly damped wave equations, Nonlinear Anal. Real World Appl. 10 (2009), 1097-1100. · Zbl 1167.35319
[23] M.H. Yang and C.Y. Sun, Exponential attractors for the strongly damped wave equations, Nonlinar Anal. Real World Appl. 11 (2010), 913-919. · Zbl 1188.37075
[24] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal. 3 (2004), 921-934. · Zbl 1197.35168
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