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

Papadopoulos, Perikles; Matiadou, N.L.

doi:10.22075/ijnaa.2017.11600.1575

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

Document Type : Research Paper

Authors

Perikles Papadopoulos

N.L. Matiadou

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

10.22075/ijnaa.2017.11600.1575

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

quasilinear hyperbolic equations

Kirchhoff strings

global attractor

generalised Sobolev spaces

weighted $L^p$ Spaces