Does Continuity of F Affect the Maximal Solution Theorem?

[Calabi]

Hello, I know a theorem that say that if ##F : \mathbb{R} \times \Omega \rightarrow E## is continuous and local lispchitziann in is seconde set value(where ##\Omega## is an open of a Banach space E.). we have that the maximum solution ##(\phi, J)##(where J is an open intervall and ##\phi : J \rightarrow \Omega## is ##C^{1}## .). of ##\phi'(t) = F(t, \phi(t))## diverge if ##sup(J) < + \infty##(##lim_{t \rightarrow sup(J)} \phi(t) = +\infty##.).

Is there the same results if F is just continuos please?

Thank you in advance and have a nice aftrenoon.

 

[Calabi]

Hello and thanks. In fact I recently knew that the local lipschitz condition is necessar for the uniqueness of a local solution to a diffential equation. I can give more if you want.

 

[math771]

Hi there!

It's great that you know a theorem about continuous and locally Lipschitzian functions! To answer your question, the same result may not hold if F is only continuous. The key here is the locally Lipschitzian property, which guarantees the existence of a unique solution for each initial condition. Without this property, the solution may not exist or may not be unique, so it's possible that the maximum solution may not diverge in the same way.

I hope that helps! Have a great afternoon as well.