3.1 Maximal solutions
Lemma 3.1.1. If \(\phi \) and \(\psi \) are solutions of \(x' = f(t, x)\) on intervals \(I\) and \(J\) with \(\phi (t_0) = \psi (t_0)\) for some \(t_0 \in I \cap J\), then \(\phi = \psi \) on \(I \cap J\).
Proof. The set \(S = \{t \in I \cap J : \phi (t) = \psi (t)\}\) contains \(t_0\), and it is closed in \(I \cap J\) by continuity. It is also open: if \(t_1 \in S\), both functions solve the initial value problem with data \((t_1, \phi (t_1))\), and by the uniqueness part of Theorem 2.3.1 they agree on a neighbourhood of \(t_1\). A non-empty subset of an interval that is both open and closed is the whole interval. □
Theorem 3.1.2. For every \((t_0, x_0) \in D\) there is a unique maximal solution \(\phi \) on an open interval \((\omega _-, \omega _+) \ni t_0\): every solution of the initial value problem is a restriction of \(\phi \). It is also called the solution that cannot be continued.
Proof. Let \((\omega _-, \omega _+)\) be the union of the domains of all solutions through \((t_0, x_0)\), which is an interval containing \(t_0\). Define \(\phi (t)\) as the value at \(t\) of any solution defined there; by Lemma 3.1.1 all such values agree, so \(\phi \) is well defined, and it is a solution because near each \(t\) it coincides with one. Every solution is a restriction of it by construction. The interval is open: if it contained an endpoint, say \(\omega _+\), the Picard–Lindelöf theorem at \((\omega _+, \phi (\omega _+))\) would give a solution on a larger interval, contradicting the definition of \(\omega _+\). □
Theorem 3.1.3 (Escape from compact sets). Let \(\phi \) be the maximal solution on \((\omega _-, \omega _+)\). If \(\omega _+ < \infty \), then \((t, \phi (t))\) eventually leaves every compact set \(K \subset D\): there is \(t_K < \omega _+\) with \((t, \phi (t)) \notin K\) for \(t_K < t < \omega _+\). In particular, if \(D = (\alpha , \beta ) \times \mathbb R^n\) and \(\omega _+ < \beta \), then \(\|\phi (t)\| \to \infty \) as \(t \to \omega _+\).
Proof. Suppose not: then there is a compact \(K \subset D\) and times \(t_k \to \omega _+\) with \((t_k, \phi (t_k)) \in K\). Since \(K\) is compact and \(D\) open, there are \(a, b > 0\) such that the closed cylinders \(R_{(s, y)} = \{|t - s| \leq a, \|x - y\| \leq b\}\) centred at points of \(K\) all lie in a fixed compact set \(K' \subset D\), on which \(\|f\| \leq M\) and \(f\) is Lipschitz in \(x\). By Theorem 2.3.1, the solution through any point of \(K\) exists for a time at least \(\alpha = \min (a, b/M)\), the same for every point. Choose \(k\) with \(t_k > \omega _+ - \alpha \). The solution through \((t_k, \phi (t_k))\) exists on \([t_k, t_k + \alpha ]\), beyond \(\omega _+\), and by Lemma 3.1.1 it continues \(\phi \), which contradicts maximality.
For the second statement, the sets \(K_r = [t_0, \omega _+] \times \bar B(0, r)\) are compact subsets of \(D\) when \(\omega _+ < \beta \), so for each \(r\) the solution eventually has \(\|\phi (t)\| > r\). □
A maximal solution can therefore end in finite time only by running off to infinity, or by approaching the boundary of \(D\). It cannot simply stop, and it cannot oscillate in a bounded region of \(D\) as it approaches \(\omega _+\). The theorem turns existence questions into questions about bounds: to show that a solution exists for all time, it is enough to show that it stays in a compact subset of \(D\).
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