Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.7

Nonautonomous Linear Systems

Mathematics — Calculus · maximal-existence-interval-boundary

Problem

Let Ω⊂R1+n\Omega \subset \mathbb{R}^{1+n} be a nonempty, bounded, open set, with ∂Ω\partial\Omega denoting its boundary. For p∈Ωp \in \Omega, define δ(p)=dist(p,∂Ω)=inf⁡{∥p−q∥:q∈∂Ω}.\delta(p) = \mathrm{dist}(p, \partial\Omega) = \inf\{\|p - q\| : q \in \partial\Omega\}. (a) Prove that δ\delta is continuous on Ω\Omega. (b) Let ε>0\varepsilon > 0. Prove that the set {p∈Ω:δ(p)≥ε}\{p \in \Omega : \delta(p) \geq \varepsilon\} is compact. (c) Now suppose that f:Ω→Rnf : \Omega \to \mathbb{R}^n is a vector field which satisfies the hypotheses of the Picard existence theorem. Given a point (t0,x0)∈Ω(t_0, x_0) \in \Omega, let x(t)x(t) be the solution of the IVP x′(t)=f(t,x(t)),x(t0)=x0,x'(t) = f(t, x(t)), \quad x(t_0) = x_0, defined on its maximal interval of existence (α,β)(\alpha, \beta). Prove that lim⁡t→β−δ(t,x(t))=0.\lim_{t \to \beta^-} \delta(t, x(t)) = 0.

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