Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 77, pp. 1-10.

Title: Strong global attractor for a quasilinear nonlocal
       wave equation on $\mathbb{R}^N$

Authors: Perikles G. Papadopoulos (National Technical Univ., Greece)
         Nikolaos M. Stavrakakis  (National Technical Univ., Greece)
	 
Abstract: 
 We study the long time behavior of solutions to the nonlocal quasilinear
 dissipative wave equation
 $$
 u_{tt}-\phi (x)\|\nabla u(t)\|^{2}\Delta u+\delta u_{t}+|u|^{a}u=0,
 $$
 in $\mathbb{R}^N$, $t \geq 0$,  with
 initial conditions  $ u(x,0) = u_0 (x)$  and  $u_t(x,0) = u_1(x)$.
 We consider  the case  $N \geq 3$, $\delta> 0$,  and
 $(\phi (x))^{-1}$ a positive function in
 $L^{N/2}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N )$.
 The existence of a global attractor is proved in the strong topology of the
 space  $\mathcal{D}^{1,2}(\mathbb{R}^N) \times L^{2}_{g}(\mathbb{R}^N)$.
 
Submitted May 10, 2006. Published Juy 12, 2006.
Math Subject Classifications: 35A07, 35B30, 35B40, 35B45, 35L15, 35L70, 35L80.
Key Words: Quasilinear hyperbolic equations; Kirchhoff strings;
    global attractor; unbounded domains; generalized Sobolev spaces;
    weighted $L^p$ spaces.