Understanding why an IVP has a solution, using uniqueness and existence theory

Question

Given the existence and uniqueness theory:

If $f$ is Lipschitz continuous over some region $D$, then there is a unique solution to the initial value problem (IVP):

$u'(t) = f(u,t), \hspace{5mm} u(0)=\eta$

at least up to time $T^*=min(t_1,t_0+a/S)$ where,

$ S = \max\limits_{(u,t)\in D} |{f(u,t)}| $.

What I'd like to do (riffing off a previous question I asked, here), is to use the E&U theorem to show that:

$u'(t) = log(u(t))$

$u(0) = 2$

Actually has a unique solution for any $t \geq 0$.

My strategy would be to show that $f$ is Lipschitz continuous inside a domain that the solution never leaves, but I'm not quite sure how.

Answer 1

The function $f(u,t)=\log u$ is $C^\infty$, and thus Lipschitz continuous, with respect to $u$, for $u>0$. Also, $f(2)=\log 2>0$, and $f(u)>0$ for $u\ge 2$. Thus, the solution of the IVP is strictly increasing, and is globally defined. (The last statement requires a bit more work.)