×

Asymptotic behavior of a class of nonlinear evolution equations. (English) Zbl 1180.35114

Summary: Using a new method (or technology), we prove the existence and upper semi-continuity of the global attractors \({\mathcal A}_\omega\) for a class of nonlinear evolution equations in \(D(A)\times D(A)\), where the nonlinear term \(f\) satisfies a critical exponential growth condition.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
47J35 Nonlinear evolution equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bogolubsky, I.L., Some examples of inelastic soliton interaction, Comput. phys. comm., 13, 149-155, (1977)
[2] Clarkson, P.A.; Leveque, R.J.; Saxton, R.A., Solitary-wave interaction in elastic rods, Stud. appl. math., 75, 95-122, (1986) · Zbl 0606.73028
[3] Seyler, C.E.; Fanstermacher, D.L., A symmetric regularized long wave equation, Phys. fluids, 27, 1, 58-66, (1984) · Zbl 0544.76170
[4] Zhu, W.G., Nonlinear waves in elastic rods, Acta solid mechanica sinica, 1, 2, 247-253, (1980)
[5] Shang, Y.D., Initial boundary value problem of equation \(u_{t t} - \Delta u - \Delta u_t - \Delta u_{t t} = f(u)\), Acta math. appl. sin. chin. ser., 23, 3, 385-393, (2000) · Zbl 0960.35059
[6] Xie, Y.Q.; Yang, L.; Qin, G.X., Strain solitary waves in a nonlinear elastic rod, J. hunan university, 34, 5, 58-61, (2007)
[7] Xie, Y.Q.; Zhong, C.K., The existence of global attractors for a class nonlinear evolution equation, J. math. anal. appl., 336, 54-69, (2007) · Zbl 1132.35019
[8] Pata, V.; Zelik, S., Smooth attractors for strongly wave equations, Nonlinearity, 19, 1495-1506, (2006) · Zbl 1113.35023
[9] Pata, V.; Squassina, M., On the strongly damped wave equation, Comm. math. phys., 253, 511-533, (2005) · Zbl 1068.35077
[10] Sell, G.R.; You, Y., Dynamics of evolutionary equations, (2002), Springer-Verlag New York · Zbl 1254.37002
[11] Teman, R., Infinite dynamical system in mechanics and physics, (1997), Springer-Verlag New York
[12] Ma, Q.F.; Wang, S.H.; Zhong, C.K., Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, J. indiana univ. math. J., 51, 1541-1559, (2002) · Zbl 1028.37047
[13] Sun, C.Y.; Wang, S.Y.; Zhong, C.K., Global attractors for a nonclassical diffusion equation, Acta math. sin. (engl. ser.), 26B, 3, 1-8, (2005)
[14] Zhong, C.K.; Yang, M.H.; Sun, C.Y., The existence of global attractors for the norm-to-weak continuous semigroup, J. differntial equations, 223, 367-399, (2006) · Zbl 1101.35022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.