Global Existence, Stability Results and Compact Invariant Sets for a Quasilinear Nonlocal Wave Equation on R^N

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of kirchhoff type u_tt - φ(x)||nabla u(t)||^2 delta u + δut = |u|^a u, x in r^n, t ≥ 0 ,with initial conditions u(x,0) = u_0 (x) and u_t(x,0) = u_1 (x), in the case where n ≥ 3, δ ≥0 and (φ (x))^-1 =g (x) is a positive function lying in l^n/2 (mathbb{r}^{n})cap l^∞}(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 - δau_t+f(u) , x in mathbb{r}^{n}, t ≥ 0]with the same hypotheses as above, we study the stability of the trivial solution u=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.