TY - JOUR
ID - 220
TI - Global existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN -
AU - Papadopoulos, P.
AU - Matiadou, N.L.
AU - Pappas, A.
AD - adepartment of electronics engineering, school of technological applications, technological educational institution (tei) of piraeus, gr 11244, egaleo, athens, greece
AD - Department of Electronics Engineering, School of Technological Applications, Technological Educational Institution (TEI) of Piraeus, GR 11244, Egaleo, Athens, Greece
AD - Civil Engineering Department, School of Technological Applications, Technological Educational Institution (TEI) of
Piraeus, GR 11244, Egaleo, Athens, Greece.
Y1 - 2015
PY - 2015
VL - 6
IS - 1
SP - 85
EP - 95
KW - quasilinear hyperbolic equations
KW - Global Solution
KW - Blow-Up
KW - Dissipation
KW - Potential Well
KW - Concavity Method
KW - Unbounded Domains
KW - Kirchhoff strings
KW - generalised Sobolev spaces
DO - 10.22075/ijnaa.2015.220
N2 - 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.
UR - https://ijnaa.semnan.ac.ir/article_220.html
L1 - https://ijnaa.semnan.ac.ir/article_220_f07e93852128c32222dc12dc8f60cab7.pdf
ER -