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)