Conditions for the Absence of Blowing Up Solutions to Fractional Differential Equations

Abstract

When addressing ordinary differential equations in infinite dimensional Banach spaces, an interesting question that arises concerns the existence (or non existence) of blowing up solutions in finite time. In this manuscript we discuss this question for the fractional differential equation $cD_{t}^{\alpha}u=f(t,u)$ proving that when $f$ is locally Lipschitz in the second variable, uniformly with respect to the first variable, however does not maps bounded sets into bounded sets, we can construct a maximal local solution that does not “blow up” in finite time.