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We know from the Picard-Lindelof Theorem that given an IVP with $f(x)\in C(E)$ where $E$ is an open subset of $\mathbb{R}^n$ then we have a local solution define in some interval $]t_0-\alpha,t_0+\alpha[$. Now suppose we want the maximal interval of existence , I am interest in figuring out what are the factors that stop us from this maximal interval being $\mathbb{R}$. One thing I think could happen is that when we try to apply the Picard Lindelof again to a point in $]t_o-\alpha,t_0+\alpha[$ we obtain a new interval but the point is that this sum of lenghts of the intervals could be converging, and so we could not get to the whole space of $\mathbb{R}$ ?

Is there anything more that can stop us from getting a solution defined in the whole line ? We need to have control over the values of $x(t)$ but I think the picard-Lindelof theorem gives us that $x(t)$ is in $E$ for all $t$ in the interval given by picard lindelof. I guess a problem could be that if we try to use picard-lindelof in $t_0-\alpha$ we need to check that $x(t_0-\alpha)$ is defined and so this could become a problem too.

Also I have seen both version where the interval where the solution is defined is closed and sometimes I have seen it open so I guess I am confused about that too, I can see why they could equivalent but I don't understand why we jut don't fix one.

Is there anthing I am missing or missunderstanding? Thanks in advance.

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Try $x(0) = 0,$ $$ \dot{x} = 1 + x^2 $$

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  • $\begingroup$ Yeah we get $tan(x)$ and we will have the problem that it's going to $\infty$ as we aprroach $\frac{\pi}{2}$ and so our solution won't be defined. I still don't quite understand if what I talked about makes sense or not :) $\endgroup$
    – Something
    Oct 13 '20 at 21:54
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If a solution $y(t)$ exists in an open interval $(a,b)$ where $b$ is finite, but not in any larger open interval, then either $\lim_{t \to b-} y(t)$ does not exist at all or it is a point outside $E$. Either of these is possible.

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  • $\begingroup$ Oh yeah that's true I forgot about that , thanks. $\endgroup$
    – Something
    Oct 14 '20 at 6:03

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