# Tagged Questions

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### 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 ...
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### 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 ...
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 ... 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*} - ... 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 ...
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 ...