Consider the following initial value problem:
$$ \begin{cases} y'(t) = f(t, y(t)) \\ y(t_0) = y_0 \end{cases} $$
where $f$ is a continuous and locally $y$-Lipschitz function defined on a vertical strip $]a,b[ \times \Bbb R$ such that $t_0 \in ]a,b[$.
Now, I know that by Cauchy's Existence and Uniqueness Theorem there exists a local solution in a neighborhood of the initial condition $(t_0, y_0)$. Moreover, I also know that if $f$ is sub-linear (i.e.: $\forall (t.y) \in ]a,b[ \times \Bbb R, |f(t,y)| \leq L_1|y| + L_2$ for some non negative constants $L_1,L_2$) then the solution is global in that we can use the bound given by Peano to extend the solution up to $]a,b[$.
My questions are thus the following: what can we say when $f$ is sub-linear but it is not defined on a vertical strip? and what can we say when $f$ is sub-linear but not locally $y$-Lipschitz and hence we may only invoke Peano's Existence Theorem?
I tried proving the existence of a global solution in the first case but I couldn't do it given that now I have to be careful that the bound on the solution is also compatible with the domain of $f$. Moreover I don't even know where to start in the second case since I can not use the uniqueness of a solution to past two solutions together and get a new continuously differentiable solution.
As always any comment or answer is welcome and let me know if I can explain myself clearer!