The existence of global attractors for a class nonlinear evolution equation. (English) Zbl 1132.35019

The authors deal with the long-time behaviour of the solutions for the following class of nonlinear evolution equations
\[ \begin{gathered} u_{tt}- \Delta_x u-\mu\Delta_x u_t- \Delta_x u_{tt}= f(u)\quad\text{in }\Omega,\\ u|_{t=0}= u_0,\quad u_t|_{t=0}= u_1\quad\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) with smooth boundary \(\partial\Omega\) and \(f\in C^0(\mathbb{R}, \mathbb{R})\) satisfying some natural growth assumptions. Due to the presence of the term \(\Delta_x u_{tt}\) and some critical growth exponent for the nonlinearity \(f\), (1) cannot be directly reformulated and studied in the framework of the wave equation. The key idea of the authors is to use the asymptotic a priori estimate method, which leads to an existence of a global attractor for (1) in \(H^1_0(\Omega)\times H^1_0(\Omega)\).


35B41 Attractors
35L82 Pseudohyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text: DOI


[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 mech. sinica, 1, 2, 247-253, (1980)
[5] Babin, A.V.; Vishik, M.I., Attractors of evolution equations, (1992), North-Holland Amsterdam · Zbl 0778.58002
[6] Cholewa, J.W.; Dlotko, T., Bi-spaces global attractors in abstract parabolic equations, Banach center publ., 60, 13-26, (2003) · Zbl 1024.35058
[7] 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
[8] Teman, R., Infinite dynamical system in mechanics and physics, (1997), Springer-Verlag New York
[9] Sell, G.R.; You, Y., Dynamics of evolutionary equations, (2002), Springer-Verlag New York · Zbl 1254.37002
[10] Zhong, C.K.; Yang, M.H.; Sun, C.Y., The existence of global attractors for the norm-to-weak continuous semigroup, J. differential equations, 223, 367-399, (2006) · Zbl 1101.35022
[11] 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)
[12] 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. (Chinese ser.), 23, 3, 385-393, (2000) · Zbl 0960.35059
[13] Zhang, H.W.; Hu, Q.Y., Existence of global weak solution and stability of a class nonlinear evolution equation, Acta math. sci., 24A, 3, 329-336, (2004) · Zbl 1138.35375
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.