I 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.