Global attractor for a nonlocal hyperbolic problem on R^N

We consider the quasilinear kirchhoff’s problem utt − φ(x)||∇u(t)||^2 ∆u + f (u) = 0, x ∈ r^n , 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 ∈ l^n/2( r^ n ) n l∞( r^ n ) 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 =:d^ 1,2( r^ n ) * l^2_g( r^ n ). we succeed to extend some of our earlier results concerning the asymptotic behaviour of the solution of the problem.