Global attractor for a nonlocal hyperbolic problem on ${\mathcal{R}}^{N}$

Document Type: Research Paper

Authors

Department of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, Greece

Abstract

We consider the quasilinear Kirchhoff's problem
$$ u_{tt}-\phi (x)||\nabla u(t)||^{2}\Delta u+f(u)=0 ,\;\; x \in {\mathcal{R}}^{N}, \;\; t \geq 0,$$
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.

Keywords