Computability, noncomputability and undecidability of maximal intervals of IVPs

Let $(\alpha ,\beta )\subseteq \mathbb {R}$ denote the maximal interval of existence of solutions for the initial-value problem \[ \left \{ \begin {array} [c]{l}\frac {dx}{dt}=f(t,x), x(t_{0})=x_{0}, \end {array} \right . \] where $E$ is an open subset of $\mathbb {R}^{m+1}$, $f$ is continuous in $E$ and $(t_{0},x_{0})\in E$. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable $f$ and $(t_{0},x_{0})$ whose maximal interval of existence $(\alpha ,\beta )$ is noncomputable. The fact that $f$ may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that $(\alpha ,\beta )$ is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.