@article {
author = {Papadopoulos, P. and Matiadou, N.L. and Pappas, A.},
title = {Global existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$},
journal = {International Journal of Nonlinear Analysis and Applications},
volume = {6},
number = {1},
pages = {85-95},
year = {2015},
publisher = {Semnan University},
issn = {2008-6822},
eissn = {2008-6822},
doi = {10.22075/ijnaa.2015.220},
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.},
keywords = {quasilinear hyperbolic equations,Global Solution,Blow-Up,Dissipation,Potential Well,Concavity Method,Unbounded Domains,Kirchhoff strings,generalised Sobolev spaces},
url = {https://ijnaa.semnan.ac.ir/article_220.html},
eprint = {https://ijnaa.semnan.ac.ir/article_220_f07e93852128c32222dc12dc8f60cab7.pdf}
}