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 $C_{1}\leq u(x,0)\leq C_{2}$ for all $x\in M$, then $C_{1}\leq u(x,t)\leq C_{2}$ for all $x\in M,t\in \mathbb {R}$.