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.