Blow-up" of bounded solutions of differential equations
Résumé
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 $D$. Dieudonne 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 all infinite-dimensional Banach spaces.