During my research I came across a weak gronwall-type inequality of the following type:
$$-\int_0^T f'(t)(u(t)-u_0) \leq \int_0^T f(t)u(t)$$ for non-negative $f\in C_c^\infty(0,T)$, $u\in L^1(0,T)$ and $u_0$ a number.
From such a "weak Gronwall inequality" one can conclude that $$ u(t^*) \leq u_0 +\int_0^{t*} u(t)$$ Then we can derive a strong Gronwall inequality for $u$ with the classical "strong" Gronwall inequality (see Evans book on pde). My question is: When does one need such a weak type of Gronwall inequality for ODEs? Or to be more precise, For which interesting problems is the classical Gronwall inequality not applicable and the weak one is. Unfortunately, I know not much/nothing about weak theories of ODEs. I'm "just" doing PDEs.
Suppose $u$ is not differentiable in the classical sense. Then an inequality like
$$ u'(t) \leq u(t)$$
doesn't make sense. So what is usually done in these weak formulations is to derive an identity using integration by parts that would hold if the function were differentiable.
For example, if the function were differentiable, then
$$ \int_0^T u'(t) f(t) \ dt = - \int_0^T u(t) f'(t) \ dt$$
(since $f$ is compactly supported, the boundary term is zero). So, if $u$ isn't differentiable, then
$$ - \int_0^T u(t) f'(t) \ dt \leq \int_0^T u(t) f(t) \ dt$$
for all non-negative, smooth, compactly supported $f$ is the next best thing we can say.
See also weak derivatives and Sobolev spaces.