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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The uniqueness of the damped Sine-Gordon.
The damped Sine-Gordon equation given :
\begin{equation}
\partial_{tt} u + \alpha \partial_t u - \Delta u + \beta \sin(u) = 0
\end{equation}
for an unknown $u : D \times [0, \infty) \rightarrow \...
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Variant of Gronwall’s inequality
Gronwall inequality typically is used to bound a function $u(t)$ if it satisfies $u(t)\leq \alpha(t)+ \int_{0}^{t}\beta(s) u(s)ds $ with the condition that $\beta$ is non-negative.
I want to use these ...
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Uniqueness and continuous dependence on the data of Heat equation.
Let two smooth $v_1$ and $v_2$ both satisfy the system
$$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$
$$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$
for some ...
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Gronwall inequality for backward (linear) differential equation
In the differential form of the Gronwall's Lemma, we have the following:
$$
\frac{d}{dt} \phi(t) \leq \psi(t) \phi(t),
$$
for all $t\geq t_0$. Then, we get
$$
\phi(t) \leq \phi(t_0) exp\left(\int_{t_0}...
- 109
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About Gronwall's inequality
I knew the following Gronwall's inequality (Integral form)
If $\alpha$ is non-negative and $H(t)$ satisfies the integral inequality
\begin{align*}
H(t) \leq c+ \int_0^t \alpha(s) H(s)ds \quad \text{(c:...
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Using Gronwall to prove bi-Lipschitz
I am working through a proof which, for fixed $x \in \mathbb{R}^k$ considers an initial value problem of the form:
$$\frac{d}{d t}u_t(x)=v_t(u_t(x)), \quad u_t(0)=x$$
where $u_t:\mathbb{R}^k \...
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1
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Gronwall's Inequality Application
Let say we have
$$ \frac{\partial |F|}{\partial t} \le K |F| + K \alpha^{2},\:\:\:\:\: t \in [t_{n}, t_{n+1}]$$
with $F(x, t_{n})=0$ and $t_{n+1}-t_{n}=\triangle t$. The above is the same thing as
$$ \...
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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
$$|u(t)|\leq AC$$
...
- 2,655
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
- 21
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1
answer
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Generalized Gronwall Inequality covering many different applications
Recently, I needed some generalized version of Gronwall's Lemma, which I couldn't find in a quick search. However, I discovered that MSE is full of questions differing only in details on this very ...
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"Nonlinear" Gronwall-type inequality
Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
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Gronwall inequality
This question concerns a proof of a theorem involving Gronwall type inequality. We have the following:
The question is: how did we apply Gronwall type inequality to get estimate (3.6)?
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Dynamical systems proof that $ f(t)$ is less than or equal to $g(t)$
Suppose that $f'(t) \le F(f(t),t)$ where $F$ is continuously differentiable. If $g(t)$ is a solution to the equation $g'(t) = F(g(t),t)$ and $g(a) = f(a)$, then prove that $f(t) \le g(t)$ for all $t \...
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Where can I find this Gronwall's inequality proof?
I'm looking for a proof of the following result:
Let I denote an interval of the real line of the form $[a,b]$ with $ a<b $. Let $\beta$ and $u$ be real-valued continuous functions defined on I. ...
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Nonlinear Gronwall inequality
Suppose a (continuous, non-negative) function $f$ satisfies
$$ 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) \...
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