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The shadowing lemma for elliptic PDE. (English) Zbl 0653.35030

Dynamics of infinite dimensional systems, Proc. NATO Adv. Study. Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, 7-22 (1987).
[For the entire collection see Zbl 0623.00009.]
The author is concerned with the partial differential equation \((1)\quad \Delta u+f(x,u)=0,\) which are defined throughout \({\mathbb{R}}^ n.\) Here f(x,u) is a given smooth nonlinearity, and u: \({\mathbb{R}}\) \(n\to {\mathbb{R}}^ m \)is the unknown vector-valued variable. If the space dimension n equals 1 then (1) is actually an ordinary differential equation which may be rewritten as a first order system of equations \((2)\quad u'=v,\quad v'=- f(x,u).\)
If \(f(x+1,u)=f(x,u)\), then one can construct a mapping F: \({\mathbb{R}}^{2m}\to {\mathbb{R}}^{2m}\) of the system (2) as follows. Given a point (u(0),v(0)) in \({\mathbb{R}}^{2m}\), solve (2) with this point as initial data. Then F(u(0),v(0)) is defined to be (u(1),v(1)). The shadowing lemma in the title is a certain lemma (which is too complicated to be described here) concerning the above F and is applied to investigate (1).
Reviewer: H.Haruki

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
37-XX Dynamical systems and ergodic theory
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0623.00009