Maximal solution

Definition

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous. Let $I$ be an open interval and let $\gamma : I \to \mathbb{R}^n$ be a solution of $x' = f(t, x)$ whose graph lies in $D$. The solution $\gamma$ is maximal if there is no solution $\tilde{\gamma} : \tilde{I} \to \mathbb{R}^n$ on an open interval $\tilde{I}$ with $I \subsetneq \tilde{I}$ and $\tilde{\gamma}(t) = \gamma(t)$ for each $t \in I$. The interval $I$ of a maximal solution is its maximal interval of existence.

Remarks

If $f$ is locally Lipschitz in $x$, each initial value problem $x(t_0) = x_0$ with $(t_0, x_0) \in D$ has exactly one maximal solution. Each other solution of the problem is a restriction of it. The maximal interval $(t_-, t_+)$ can depend on the initial value, and it can be smaller than $\mathbb{R}$. If $t_+ < \infty$, then $(t, \gamma(t))$ leaves every compact subset of $D$ as $t \to t_+$. For $D = \mathbb{R} \times \mathbb{R}^n$, this means $\|\gamma(t)\| \to \infty$. See blow-up in finite time.

Explanations

L1 human

A maximal solution is a solution that we cannot continue any further. Its interval ends only where the solution runs away to infinity or to the edge of the domain of the field.