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

Document Type : Research Paper


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


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.


Volume 8, Issue 2
December 2017
Pages 159-168
  • Receive Date: 09 June 2017
  • Revise Date: 17 September 2017
  • Accept Date: 26 September 2017