^{1}Department of Electronics Engineering, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece.

^{2}Civil Engineering Department, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece.

Abstract

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type \[ u_{tt}-\phi (x)||\nabla u(t)||^{2}\Delta u+\delta u_{t}=|u|^{a}u,\, x \in \mathbb{R}^{N} ,\,t\geq 0\;,\]

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 $u\equiv 0$. 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.