3.2 Global existence
Theorem 3.2.1. Let \(f\) be continuous on \(I \times \mathbb R^n\), \(I\) an open interval, locally Lipschitz in \(x\), and of at most linear growth: \[ \|f(t, x)\| \leq a(t) + b(t)\|x\| \] with \(a, b \geq 0\) continuous on \(I\). Then every maximal solution is defined on the whole of \(I\). In particular, the solutions of the linear system \(x' = A(t)x + g(t)\), with \(A\) and \(g\) continuous on \(I\), exist on all of \(I\).
Proof. Suppose \(\omega _+\) is less than the right endpoint of \(I\). On \([t_0, \omega _+]\), \(a \leq A\) and \(b \leq B\) for constants \(A, B\). From the integral equation, \(\|\phi (t)\| \leq \|x_0\| + A(\omega _+ - t_0) + B\int _{t_0}^t\|\phi (s)\|\,ds\), and Gronwall’s inequality gives \(\|\phi (t)\| \leq (\|x_0\| + A(\omega _+ - t_0))e^{B(\omega _+ - t_0)}\) for \(t_0 \leq t < \omega _+\). The solution is bounded, contradicting Theorem 3.1.3. The same argument applies to \(\omega _-\). For the linear system, \(\|A(t)x + g(t)\| \leq \|g(t)\| + \|A(t)\|\|x\|\). □
Linear growth is sharp in a precise sense: Example 2.3.2 and Problem 1.8.2 show quadratic growth producing blow-up. Two other routes to global existence are common. A conserved quantity, such as the energy of the pendulum, \(E = \frac 12\theta '^2 - (g/\ell )\cos \theta \), confines a solution to a level set, and if the level set is bounded the solution exists for all time. A trapping region, a compact set that solutions cannot leave because \(f\) points inward on its boundary, does the same; it is the idea behind Lyapunov functions in a later section.
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