%0 Journal Article
%T Global existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$
%J International Journal of Nonlinear Analysis and Applications
%I Semnan University
%Z 2008-6822
%A Papadopoulos, P.
%A Matiadou, N.L.
%A Pappas, A.
%D 2015
%\ 04/13/2015
%V 6
%N 1
%P 85-95
%! Global existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$
%K quasilinear hyperbolic equations
%K Global Solution
%K Blow-Up
%K Dissipation
%K Potential Well
%K Concavity Method
%K Unbounded Domains
%K Kirchhoff strings
%K generalised Sobolev spaces
%R 10.22075/ijnaa.2015.220
%X 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.
%U https://ijnaa.semnan.ac.ir/article_220_f07e93852128c32222dc12dc8f60cab7.pdf