Maximal Solutions and Continuation

The local existence theorem guarantees solutions on short intervals. The theory of maximal solutions addresses how far solutions can be extended and what happens at the boundary of their existence interval.

Maximal Interval of Existence

Definition8.5Maximal solution

A solution $\mathbf{x}: (t^-, t^+) \to \mathbb{R}^n$ of $\mathbf{x}' = \mathbf{F}(t, \mathbf{x})$ is maximal (or non-continuable) if it cannot be extended to a larger interval. The interval $(t^-, t^+)$ is called the maximal interval of existence. When $t^+ < \infty$ , we say the solution has a finite-time blowup or reaches the boundary of the domain $\Omega$ .

Theorem8.3Continuation theorem

Let $\mathbf{F}$ be continuous and locally Lipschitz on an open set $\Omega \subset \mathbb{R} \times \mathbb{R}^n$ . Then every solution of the IVP can be extended to a maximal solution. Moreover, if $(t^-, t^+)$ is the maximal interval and $t^+ < \infty$ , then for every compact set $K \subset \Omega$ , there exists $t_K < t^+$ such that $(t, \mathbf{x}(t)) \notin K$ for all $t \in (t_K, t^+)$ .

In other words, as $t \to t^+$ , the solution either leaves every compact subset of $\Omega$ (blows up or reaches the boundary).

Finite-Time Blowup

ExampleFinite-time blowup

$x' = x^2$ , $x(0) = 1$ : Solution $x(t) = 1/(1-t)$ blows up at $t^+ = 1$ . Here $x(t) \to +\infty$ .
$x' = 1 + x^2$ , $x(0) = 0$ : Solution $x(t) = \tan t$ blows up at $t^+ = \pi/2$ .
The system $x' = y$ , $y' = x + x^2$ with appropriate initial conditions can exhibit finite-time blowup in the norm $\|\mathbf{x}(t)\| \to \infty$ .

RemarkCriteria for global existence

If $\mathbf{F}$ satisfies a linear growth condition $\|\mathbf{F}(t, \mathbf{x})\| \leq A + B\|\mathbf{x}\|$ on $\mathbb{R} \times \mathbb{R}^n$ , then by Gronwall's inequality, solutions satisfy:

$\|\mathbf{x}(t)\| \leq (\|\mathbf{x}_0\| + A|t-t_0|)e^{B|t-t_0|}$

which is finite for all finite $t$ . Therefore all solutions exist for all time. This is a sufficient condition for global existence.

Gronwall's Inequality

Theorem8.4Gronwall's inequality

Let $u, \alpha, \beta: [t_0, T] \to \mathbb{R}$ be continuous with $\beta \geq 0$ . If

$u(t) \leq \alpha(t) + \int_{t_0}^{t} \beta(s)u(s)\,ds \quad \text{for all } t \in [t_0, T],$

then

$u(t) \leq \alpha(t) + \int_{t_0}^{t} \alpha(s)\beta(s)\exp\!\left(\int_s^t \beta(r)\,dr\right)ds.$

In particular, if $\alpha$ is non-decreasing, then $u(t) \leq \alpha(t)\exp\!\left(\int_{t_0}^t \beta(s)\,ds\right)$ .

ExampleContinuous dependence via Gronwall

Let $\mathbf{x}(t)$ and $\mathbf{y}(t)$ solve $\mathbf{x}' = \mathbf{F}(t,\mathbf{x})$ with initial conditions $\mathbf{x}_0$ and $\mathbf{y}_0$ respectively. If $\mathbf{F}$ is Lipschitz with constant $L$ , then:

$\|\mathbf{x}(t) - \mathbf{y}(t)\| \leq \|\mathbf{x}_0 - \mathbf{y}_0\| + \int_{t_0}^{t} L\|\mathbf{x}(s) - \mathbf{y}(s)\|\,ds.$

By Gronwall: $\|\mathbf{x}(t) - \mathbf{y}(t)\| \leq \|\mathbf{x}_0 - \mathbf{y}_0\| e^{L(t-t_0)}$ . This proves continuous dependence on initial conditions: nearby initial data produce nearby solutions on bounded time intervals.

RemarkSensitivity to initial conditions

The exponential factor $e^{L(t-t_0)}$ in the continuous dependence estimate shows that errors can grow exponentially. This is the mathematical basis of sensitive dependence on initial conditions in chaotic systems. The bound is sharp: for the linear system $x' = Lx$ , the ratio $|x(t; x_0) - x(t; y_0)| / |x_0 - y_0| = e^{Lt}$ exactly.