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Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation. (English) Zbl 1100.37052

The author investigates the asymptotic behaviour of solutions for the second-order lattice dynamical system \[ \ddot u_i+\delta\dot u_i+ \alpha(Au)_i+\beta (Bu)_i+\lambda u_i-\tfrac 13 k\bigl(D(D^*u)^3 \bigr)_i=f_i,\quad i\in\mathbb{Z},\tag{1} \] with the initial conditions \(u_i(b)=u_{i,0}, \dot u_i(0)=u_{1i,0}\), \(i\in\mathbb{Z}\), where \(\alpha,\delta,\lambda\) and \(k\) are positive constants, \(\beta\) is a real constant, \(f=(f_i)_{i\in \mathbb{Z}}\in\ell^2\), and \(\lambda>4|\beta|\). The linear operators \((Du)_i =u_{i+1}-u_i\), \((D^*u)_i=u_i-u_{i-1}\), \((Bu)_i=u_{i+1}-2u_i+ u_{i-1}\), \((Au)_i=u_{i+2}-4k_{i+1}+6u_i-4u_{i-2}\), \(i\in\mathbb{Z}\). The purpose of this work is to prove the existence of the global attractor of associated semiflows generated by (1).

MSC:

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
35B41 Attractors
35Q53 KdV equations (Korteweg-de Vries equations)
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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