Complexity of Blowup Problems

Consider the initial value problem of the first-order ordinary differential equationddtx(t)=f(t,x(t)),x(t0)=x0 where the locally Lipschitz continuous function f:Rl+1→Rl with open domain and the initial datum (t0,x0)∈Rl+1 are given. It is shown that the solution operator producing the maximal “time” interval of existence and the solution on it is computable. Furthermore, the complexity of the blowup problem is studied for functions f defined on the whole space. For each such function f the set Z of initial conditions (t0,x0) for which the positive solution does not blow up in finite time is a Gδ-set. There is even a computable operator determining Z from f. For l⩾2 this upper Gδ-complexity bound is sharp. For l=1 the blowup problem is simpler.