Recently, I needed some generalized version of Gronwall's Lemma, which I couldn't find in a quick search. However, I discovered that MSE is full of questions differing only in details on this very topic. Hence, I was wondering if there is a general version of Gronwall's inequality which covers mo...

Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?

This question concerns a proof of a theorem involving Gronwall type inequality. We have the following: The question is: how did we apply Gronwall type inequality to get estimate (3.6)?

Suppose a (continuous, non-negative) function $f$ satisfies $$ f'(t) \leq f(t)^2 $$ for $t \in [0,1]$. Set $$ g(t) = \exp\left(-\frac{1}{f(t)}\right). $$ Then $$ g'(t) = \frac{f'(t)}{f(t)^2} g(t) \leq g(t), $$ so applying the standard Gronwall inequality one has $g(t) \leq g(0) e^t$. Since $\log...

I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that. $$ f^2(t) \leqslant g^2(t) + \int_0^t (f(s) +c) f(s) ds \;\;\;\; (t \in [0,T]) $$ So please help!

I am looking for a proof or refrence for the following Gronwall-type inequality: Let $ \varphi (t,s) $ is a continous function for $0 \leq s < t \leq T$. If the following inequality holds: $$ \varphi (t,s) \leq A + B \int_s^t ( t - \sigma)^{\alpha -1} \varphi ( \sigma , s) d \sigma $$ for som...

Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$. Suppose that we know $$\frac{d}{dt} \alpha\int_{\mathbb R} u(t,x) dx + A(t) \le \int_{\mathbb R}...

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as necessary and nonnegative. Thanks.

The Gronwall lemma is a well known and very useful statement which is used in many situations, in particular in the theory of differential equations. I have seen it so many times and even the proof is very easy to understand. But at the end of the day it is seems to me a very technical 'thing', I...

I would like to know if there is any kind of Gronwall inequality for a smooth function $u \colon \mathbb{R}^n \to \mathbb{R}$ satisfying $$ |\nabla u | \le K u, $$ where $K$ is a constant.

Resolved: Tag has been removed. I was going to remove the tag gronwall-inequality from this question until I realized that there were a hundred questions related to this inequality. I am not too familiar with DEs: do people who are think this could use a tag?