Approximate inertial manifolds for the sine-Gordon equation. (English) Zbl 0748.35039

Flahaut, Isabelle

Differ. Integral Equ. 4, No. 6, 1169-1193 (1991).

The study of the long time behaviour of the solution of the problem \[ u''+\alpha u'-\Delta u=j\sin u,\quad\text{ in } \Omega\subset R^ n,\quad n\leq 3, \]
\[ u(x,t)=0\quad \text{ on } \partial\Omega,\quad u(x,0)=u_ 0,\quad u'(x,0)=u_ 1 \] is considered. The approximate inertial manifold (AIM) is a finite dimensional smooth manifold

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, such that all the orbits enter — after a transient time — a very thin neighbourhood of the manifold. The aim of the paper is to construct an infinite family of AIM approximating the orbits. The instrument used is a set of eigenfunctions of the problem \(\Delta w_ j=\lambda_ jw_ j\), the projectors \(P,Q\), \(p=Pu\), \(q=Qu\), \((u=p+q)\) and then the estimate of \(q\) and of its derivatives with respect to time.

Reviewer: A.Haimovici (Iaşi)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35B40	Asymptotic behavior of solutions to PDEs
35L70	Second-order nonlinear hyperbolic equations

Keywords:

approximate inertial manifold