Non-existence for fractionally damped fractional differential problems

TL;DR

This paper proves the non-existence of non-trivial global solutions for a class of fractional differential inequalities with damping and source terms, highlighting how the blow-up range depends solely on the lower order derivative.

Contribution

It introduces a non-existence result for fractional differential problems with damping and polynomial sources, extending understanding of solution behavior in fractional dynamics.

Findings

Non-trivial solutions do not exist globally under certain conditions.

The blow-up range depends only on the lower order fractional derivative.

Solutions tend to converge to parabolic solutions in the damped case.

Abstract

In this paper, we are concerned with a fractional differential inequality containing a lower order fractional derivative and a polynomial source term in the right hand side. A non-existence of non-trivial global solutions result is proved in an appropriate space by means of the test-function method. The range of blow up is found to depend only on the lower order derivative. This is in line with the well-known fact for an internally weakly damped wave equation that solutions will converge to solutions of the parabolic part.