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فهرست مقالات

Global existence‌, ‌stability results and compact invariant sets‌ ‌for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$


علمی-پژوهشی (وزارت علوم)/ISC (11 صفحه - از 85 تا 95)


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.

کلیدواژه ها:

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

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