- - - Department of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR ۱۱۲۴۴, Egaleo, Athens, Greece
- - - Department of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR ۱۱۲۴۴, Egaleo, Athens, Greece

We consider the quasilinear Kirchhoff's problem u_{tt}-\phi (x)||\nabla u(t)||^{۲}\Delta u+f(u)=۰ ,\;\; x \in {\mathcal{R}}^{N}, \;\; t \geq ۰,with the initial conditions  u(x,۰) = u_۰ (x)  and u_t(x,۰) = u_۱ (x), in the case where \ N \geq ۳, \;  f(u)=|u|^{a}u \ and (\phi (x))^{-۱} \in L^{N/۲}({\mathcal{R}}^{N})\cap L^{\infty}({\mathcal{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 {\cal X}_{۱}=:{\cal D}^{۱,۲}({\mathcal{R}}^{N}) \times L^{۲}_{g}({\mathcal{R}}^{N}). We succeed to extend some of our earlier results concerning the asymptotic behaviour of the solution of the problem.

quasilinear hyperbolic equations, Kirchhoff strings, global attractor, generalised Sobolev spaces, weighted L^p Spaces