Abstract. By the classical Cauchy--Lipschitz theory of ordinary differential equations, no maximal solution of $x'=f(t,x)$ can belong to some compact subset of the domain of definition $D$ of $f$. In the finite dimensional case it follows that the maximal solutions are defined up to the boundary of $D$. Dieudonné and later Deimling gave counterexamples in some infinite dimensional spaces: the maximal solution can remain bounded while it blows up in finite time. We give a complete, elementary and natural proof of this result for {\it all} infinite dimensional Banach spaces.

AMS Subject Classification (1991): 34K30, 34K35

Keyword(s): Ordinary differential equation, blow-up, bounded solutions

Received July 5, 2002, and in the final form April 18, 2003. (Registered under 2922/2009.)