Global existence‎, ‎stability results and compact invariant sets‎ ‎for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R} {N}, ,tgeq 0;,] with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3,; delta geq 0$ and $(phi (x))^{-1} =g (x)$ is a positive function lying in $L^ {N/2}(mathbb{R}^{N})cap L^{infty}(mathbb{R}^{N})$. It is proved that, when the initial energy $ E(u_{0},u_{1})$, which corresponds to the problem, is non-negative and small, there exists a unique global solution in time in the space ${cal{X}}_{0}=:D(A) times {cal{D}}^{1,2}(mathbb{R}^{N})$. When the initial energy $E(u_{0},u_{1})$ is negative, the solution blows-up in finite time. For the proofs, a combination of the modified potential well method and the concavity method is used. Also, the existence of an absorbing set in the space ${cal{X}}_{1}=:{cal{D}}^{1,2}(mathbb{R}^{N}) times L {2}_{g}(mathbb{R}^{N})$ is proved and that the dynamical system generated by the problem possess an invariant compact set ${cal {A}}$ in the same space. Finally, for the generalized dissipative Kirchhoff''s String problem [u_{tt}=-||A^{1/2}u||^{2}_{H} Au-delta Au_{t}+f(u), ;; x in mathbb{R}^{N},; ; t geq 0;,]with the same hypotheses as above, we study the stability of the trivial solution $uequiv 0$. It is proved that if $f''(0)>0$, then the solution is unstable for the initial Kirchhoff''s system, while if $f''(0)<0$ the solution is asymptotically stable. In the critical case, where $f''(0)=0$, the stability is studied by means of the central manifold theory. To do this study we go through a transformation of variables similar to the one introduced by R. Pego.