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

## TL;DR

This paper investigates conditions under which solutions to fractional differential equations in Banach spaces do not blow up in finite time, showing that certain properties of the nonlinearity prevent finite-time blow-up.

## Contribution

It provides new conditions related to the properties of the nonlinear term that guarantee the absence of finite-time blow-up solutions in fractional differential equations.

## Key findings

- Constructed maximal local solutions that do not blow up in finite time.
- Identified that non-boundedness of the nonlinear term on bounded sets prevents blow-up.
- Extended understanding of solution behavior for 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, however does not maps bounded sets into bounded sets, we can construct a maximal local solution that not "blows up" in finite time.