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Global attractor for a nonlocal hyperbolic problem on ${\mathcal{R}}^{N}$ | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 15، دوره 8، شماره 2، اسفند 2017، صفحه 159-168 اصل مقاله (405.47 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2017.11600.1575 | ||
| نویسندگان | ||
| 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 | ||
| تاریخ دریافت: 19 خرداد 1396، تاریخ بازنگری: 26 شهریور 1396، تاریخ پذیرش: 04 مهر 1396 | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| quasilinear hyperbolic equations؛ Kirchhoff strings؛ global attractor؛ generalised Sobolev spaces؛ weighted $L^p$ Spaces | ||
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