Bounded solutions of finite lifetime to differential equations in Banach spaces

Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $γ\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $γ$ is bounded, then $γ$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is replaced with an infinite-dimensional Banach space.

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