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4 votes
0 answers
122 views

Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
Jae Ho Cho's user avatar
3 votes
1 answer
243 views

Existence and uniqueness of solutions for continuous and directionally differentiable ODE

Given $f:\mathbb{R}^n \to \mathbb{R}^n$ continuous and directionally differentiable (i.e., such that the directional derivative of $f$ exists for any direction) at a neighborhood $N$ of $x_0\in\mathbb{...
Todd Chavez's user avatar
3 votes
2 answers
376 views

Is the converse of Osgood criterion for ODEs also true?

Namely, Assuming that $f$ is a continuous real function and $f(0)=0$ , $f(x)>0 $ when $x\neq 0$, Consider the differential equation $x'= f(x)$ with the initial value $x(0)=0$ , is it true that if ...
NotaChoice's user avatar
2 votes
1 answer
155 views

Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations: $$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$ has a unique Caratheodory solution ...
Bogdan's user avatar
  • 1,759
2 votes
2 answers
136 views

Uniqueness of a second order linear ode

I am currently confronted with the following equation $$ 0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t $$ for $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in ...
mhmmm1997's user avatar
0 votes
0 answers
133 views

Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$

I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
G. Ander's user avatar
  • 151
0 votes
0 answers
41 views

Existence and Uniqueness of lifting Hele-Shaw problem

I am researching for the existence and uniqueness of solutions for the equation in figure below enter image description here $$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$ The ...
fayez ahmed's user avatar