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Global existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $\mathbb{R}^{N}$ | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 13، دوره 6، شماره 1، خرداد 2015، صفحه 85-95 اصل مقاله (412.65 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2015.220 | ||
| نویسندگان | ||
| P. Papadopoulos* 1؛ N.L. Matiadou2؛ A. Pappas3 | ||
| 1adepartment of electronics engineering, school of technological applications, technological educational institution (tei) of piraeus, gr 11244, egaleo, athens, Greece | ||
| 2Department of Electronics Engineering, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece | ||
| 3Civil Engineering Department, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece. | ||
| تاریخ دریافت: 16 مرداد 1392، تاریخ بازنگری: 27 خرداد 1393، تاریخ پذیرش: 16 آذر 1393 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| quasilinear hyperbolic equations؛ Global Solution؛ Blow-Up؛ Dissipation؛ Potential Well؛ Concavity Method؛ Unbounded Domains؛ Kirchhoff strings؛ generalised Sobolev spaces | ||
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