TITLE: Property of solution to a Cauchy problem
QUESTION [0 upvotes]: Let $I\subseteq \Bbb R$ an interval, let $w$ differentiable on $I$ such that $$w'(t)\le L|w(t)|\qquad \forall t\in I$$ for some $L>0.$ Let $t_0\in I$.
Prove that $$w(t_0)\le 0\implies w(t)\le0\quad \forall t>t_0$$ and $$w(t_0)\ge0\implies w(t)\ge0\quad \forall t<t_0$$

REPLY [0 votes]: $w$ can increase no faster than $w'(t) = L |w(t)|$. 
If $w(t_0)<0$, then you have that $w'(t) \leq -L w(t)$ for $t\leq T$ where $T = \inf {s: w(s)\geq 0}$.
Define $u(t)$ as the solution of $u'(t) = -L u(t)$ and $u(t_0) = w(t_0)$. Then for $t\leq T$ you have that $w(t)\leq u(t)$.
$u(t)$ can be easily found as 
$$u(t) = u(t_0)\,e^{-L (t-t_0)}\leq0\,.$$
Since $u(t)$ is always $\leq 0$ for all $t>t_0$ then $w(t)\leq 0$ for all $t>t_0$, i.e. $T=\infty$. 
The case for $t<t_0$ is analogous.
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