Bound-narrowing 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 bound-narrowing if it is both max-decreasing and min-increasing or equivalently if it satisfies the following condition:

$C_1\le u(0,x)\le C_2\mathrm{\forall }x\in M⟹C_1\le u(t,x)\le C_2\mathrm{\forall }x\in M$