Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228220171201Global attractor for a nonlocal hyperbolic problem on ${\mathcal{R}}^{N}$159168279310.22075/ijnaa.2017.11600.1575ENPerikles PapadopoulosDepartment of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, GreeceN.L. MatiadouDepartment of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, GreeceJournal Article20170609We consider the quasilinear Kirchhoff's problem<br />$$ u_{tt}-\phi (x)||\nabla u(t)||^{2}\Delta u+f(u)=0 ,\;\; x \in {\mathcal{R}}^{N}, \;\; t \geq 0,$$<br />with the initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where \ $N \geq 3, \; f(u)=|u|^{a}u$ \ and $(\phi (x))^{-1} \in L^{N/2}({\mathcal{R}}^{N})\cap L^{\infty}({\mathcal{R}}^{N} )$ 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 ${\cal X}_{1}=:{\cal D}^{1,2}({\mathcal{R}}^{N}) \times L^{2}_{g}({\mathcal{R}}^{N}).$ We succeed to extend some of our earlier results concerning the asymptotic behaviour of the solution of the problem.https://ijnaa.semnan.ac.ir/article_2793_ef30a57e5aaa4eb687c61b37a80ea4d1.pdf