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 \[ u_{tt}-\phi (x)||\nabla u(t)||^{۲}\Delta u+\delta u_{t}=|u|^{a}u,\, x \in \mathbb{R}^{N} ,\,t\geq ۰\;,\]with initial conditions u(x,۰) = u_۰ (x) and u_t(x,۰) = u_۱ (x), in the case where N \geq ۳, \; \delta \geq ۰ and (\phi (x))^{-۱} =g (x)  is a positive function lying in L^{N/۲}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N}). It is proved that, when the initial energy \ E(u_{۰},u_{۱}), which corresponds to the problem, is non-negative and small, there exists a unique global solution in time in the space \ {\cal{X}}_{۰}=:D(A) \times {\cal{D}}^{۱,۲}(\mathbb{R}^{N}). When the initial energy E(u_{۰},u_{۱}) 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}}_{۱}=:{\cal{D}}^{۱,۲}(\mathbb{R}^{N}) \times L^{۲}_{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\[ u_{tt}=-||A^{۱/۲}u||^{۲}_{H} Au-\delta Au_{t}+f(u) ,\; \; x \in \mathbb{R}^{N}, \;\; t \geq ۰\;,\]with the same hypotheses as above, we study the stability of the trivial solution u\equiv ۰. It is proved that if f'(۰)>۰, then the solution is unstable for the initial Kirchhoff's system, while if f'(۰)<۰ the solution is asymptotically stable. In the critical case, where f'(۰)=۰, 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.

کلیدواژه ها:

quasilinear hyperbolic equations ، Global Solution ، Blow-Up ، Dissipation ، Potential Well ، Concavity Method ، Unbounded Domains ، Kirchhoff strings ، generalised Sobolev spaces