An unbounded attractor of a hyperbolic equation (Q1117104)

Consider the second order quasilinear hyperbolic equation \[ (1)\quad u_{tt}+\epsilon u_ t=\Delta u+\lambda u+f(u),\quad \epsilon >0 \] with periodic boundary conditions and initial conditions \[ u|_{t=0}=u_ 0\in H'(T^ n),\quad u_ t|_{t=0}=p|_{t=0}=p_ 0\in L_ 2(T^ n). \] \textit{V. V. Chepyzhov} [Mosc. Univ. Math. Bull. 41, No.6, 38-40 (1986); translation from Vestn. Mosk. Univ. Ser. I 1986, No.6, 52-54 (1986; Zbl 0626.35046)] first considered an unbounded attractor for the case of a parabolic equation. In this paper the author studies an unbounded attractor for the hyperbolic equation (1). The basic difficulty is absence of smoothness of the solution.