Positivity-preserving trajectory

This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold

Definition

Let $M$ be a manifold and $u=u(t,x)$ be a function $\mathbb{R}\times M\to \mathbb{R}$, where:

• $t$ denotes the time parameter, and varies in $\mathbb{R}$
• $x$ denotes the spatial parameter, and varies in $M$

In other words, $u$ is a trajectory (or path) in the space of all functions from $M$ to $\mathbb{R}$.

Then, $u$ is said to be positivity-preserving if the function:

$t\mapsto {inf}_{x\in M}u(t,x)$

has the property that once it crosses zero, it never becomes negative again. In other words, if there is a $t_0$ such that $u(t_0,x)\ge 0$ for all $x\in M$, then $u(t,x)\ge 0$ for all $t\ge t_0,x\in M$.