Global attractor for a nonlocal hyperbolic problem on RN

We consider the quasilinear Kirchhoff's problem
utt−ϕ(x)||∇u(t)||2Δu(u)=0,x∈RN,t≥0,
with the initial conditions u(x,0)=u0(x) and ut(x,0)=u1(x), in the case where \ N≥3,f(u)=|u|au \ and (ϕ(x))−1∈LN/2(RN)∩L∞(RN) is a positive function. The purpose of our work is to study the long time behaviour of the solution of this equation. Here, we prove the existence of a global attractor for this equation in the strong topology of the space X1=:D1,2(RN)×L2g(RN).
We succeed to extend some of our earlier results concerning the asymptotic behaviour of the solution of the problem.