We consider a Cauchy's problem : $x'(t) = x(t)^3$, $x(0)=a \in \mathbb{R}$.

I have to determine if the maximal solution is global (defined on $\mathbb{R}$) or not.

To begin, I know $x^3$ is positive on $]0,+\infty[$ and negative on $]-\infty,0[$. The solution $u(t)=0$ is a constant solution. A non constant solution $u$ is strictly greater than $0$ or strictly less than $0$.

Let $u(t) > 0$. In this case, $u(t)$ is increasing on its definition interval $]T^-, T^+[$. Then, $u(t)$ is bounded lower by $0$. Then, $T^- = -\infty$.

Let $u(t) < 0$. In this case, $u(t)$ is decreasing on its definition interval $]T^-,T^+[$. In more $u(t)$ is bounded upper by $0$. Then, $T^- = -\infty$.

My question is : how to study this problem, in the two cases, for $T^+$ ? We have no enough informations...