# Questions tagged [gronwall-type-inequality]

Questions on inequalities similar to the classical Gronwall's lemma. Typically, a function's derivative is bounded by (some variation of) itself. This may also be given in the corresponding integral version instead. From there an estimate for the original function can be derived.

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• 4,835
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### Gronwall lemma with highly oscillatory kernel

As a toy model for a larger problem, I want to show that if $A,B\geq 0$ and $u(t)$ satisfies $$|u(t)|\leq A+\left|\int_0^t B\cos(s^2)u(s)ds\right|$$ then $u$ satisfies a bound like $$|u(t)|\leq AC$$ ...
• 2,271
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1 vote
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### Gronwall-type inequality in higher dimension

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.
• 2,703
1 vote
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Could anyone see how the following two lines follow from Gronwall lemma? I use the usual differential form gronwall lemma from in Evans book. I do not know how to deal with the term involves the ...
• 3,065
276 views

### Kind of Gronwall Inequality

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 ...
• 119
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### Intuition of Gronwall lemma

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 ...
• 1,069
287 views

### Gronwall type inequality

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 $\alpha>1$ ?
• 51
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If I have that $$||\eta_u(t)||\leq 1+C_1\int_0^t \frac{1}{||\eta(s)||}||\eta_u(s)||ds$$ and $$\sqrt{1-\frac{2\varepsilon}{C}}||u||\leq ||\eta(s)||\leq 2||u||$$ how to obtain using the Gronwall ...
• 5,183
454 views

### Reference Request: Weak ODEs and weak Gronwall inequality

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 $f\in C_c^\infty(0,T)$, $u\in L^1(0,T)$ ...
• 2,119
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) +...