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

Papadopoulos, Perikles G. ; Stavrakakis, Nikolaos M.

Electronic Journal of Differential Equations, Tome 2006 (2006).

Résumé

Summary: 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 $\mathcal{D}^{1,2}(\mathbb{R}^N) \times L^{2}_{g}(\mathbb{R}^N)$.

Détail

Classification : 35A07, 35B30, 35B40, 35B45, 35L15, 35L70, 35L80
Keywords: quasilinear hyperbolic equations, Kirchhoff strings, global attractor, unbounded domains, generalized Sobolev spaces, weighted $L^p$ spaces