Scalar weak maximum principle

Definition

Consider a differential equation involving one dependent variable $u$ that is a function of $M\times \mathbb{R}$ where $M$ denotes a manifold (whose generic point will be denoted as $x$) and $\mathbb{R}$ corresponds to the tiem axis (whose generic point will be denoted as $t$).

An initial value problem corresponding to this differential equation is a specification of $u(x,0)$ for each $x\in M$.

Such a differential equation is said to satisfy the maximum principle if for any solution $u$, if there are constants $C_1$ and $C_2$ such that $C_1\le u(x,0)\le C_2$ for all $x\in M$, then $C_1\le u(x,t)\le C_2$ for all $x\in M,t\in \mathbb{R}$.