Here is the statement :
Let $f \in \mathcal{C}^1([0,T],\mathbb{R}_+)$, $T>0$ such that : for all $t \in [0,T]$, $\displaystyle f(t) \le c + \int_{0}^{t} \eta(f(x))\mathrm{d}x$ where $\eta : \mathbb{R}_+ \to \mathbb{R}_+$ is continuous, non-decreasing and checks $\displaystyle \int_{0}^{1}\frac{\mathrm{d}s}{\eta(s)}=+\infty$.
$\bullet$ If $c=0$ then $f \equiv 0$.
$\bullet$ If $c >0$ then for all $t \in [0,T]$, $\mathcal{M}(c)\le t + \mathcal{M}(f(t))$ with $\displaystyle \mathcal{M}(x)=\int_{x}^{1}\frac{\mathrm{d}s}{\eta(s)}$ .
I heard that this result could have been proved by Yudovich but I did not find anything relevant about it.
So if anyone has references about this statement, it would be nice to share.
Edit : I managed to find the following statement which I guess generalizes the previous one :
However I still found nothing about Yudovich's result...
Thanks in advance !
Your result appears to be a special case of the Bihari-LaSalle inequality, which generalizes to the analogous problem $$f(t)\leq c+\int_0^t{\eta(f(s))\cdot v(s)\,ds}$$ when integrating against measures other than Lebesgue.
As of writing, the linked Wikipedia article only cites LaSalle and Bihari's original papers.
The only missing case is when $c=0$, but that can be seen by taking a pointwise limit.