1
vote
2answers
37 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
1
vote
1answer
293 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
0
votes
0answers
126 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
2
votes
0answers
148 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
0
votes
2answers
49 views

Prove that $\left \|x(t_0) \right \|\exp \left (-\int_{t_0}^{t} \left \|A(t_1) \right \|\mathrm{d}t_1 \right )\le \left \| x(t) \right \|$

I have a problem: For $\dfrac{dx}{dt}=A(t)x$, where $A(t)\in C\left [t_0,+\infty \right )$. Prove that: $$\left \|x(t_0) \right \|\exp \left (-\int_{t_0}^{t} \left \|A(t_1) \right \|\mathrm{d}t_1 ...
5
votes
1answer
355 views

Are there any interpretations for the Gronwall's inequality in view of comparison theorem?

One form of the Gronwall's inequality is that If $\alpha(x),u(x)$ are non-negative continuous functions on $[0,1]$, and $$\forall x\in [0,1], u(x)\leq ...