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This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

Fractional derivatives have proved to be very efficient and adequate to describe many phenomena with memory and hereditary processes. These phenomena are abundant in science, engineering (viscoelasticity, control, porous media, mechanics, electrical engineering, electromagnetism, etc.) as well as in geology, rheology, finance, and biology. Unlike the classical derivatives, fractional derivatives have the ability to characterize adequately processes involving a past history. We are witnessing a huge development of fractional calculus and methods in the theory of differential equations. Indeed, after the appearance of the papers by Bagley and Torvik [

We will consider the problem:

A nonexistence result for global solutions of the problem (

The existence and uniqueness of solutions for problem (

We also point out here that the case where

The rest of the paper is divided into three sections. In Section

In this section, we present some background material for the forthcoming analysis. For more details, see [

The space

Let

Let

Let

Let

The left-sided Hadamard fractional derivative of order

The right-sided Hadamard fractional derivative of order

If

Let

If

If

This lemma justifies the following one

Let

Let

Let

The fractional derivative

Now, motivated by the Hilfer fractional derivative introduced in [

If

Before we state and prove our main result, we will start with the following lemma.

If

Since

The proof of the next result is based on the test function method developed by Mitidieri and Pokhozhaev in [

Assume that

Assume that a nontrivial solution exists for all time

An integration by parts in (

Therefore, Lemma

In the case

If

Although we are concerend here about nonexistence of solutions, using standard techniques, one may show the existence of local solutions of Problem (

For our example, we need the following lemma.

The following result holds for the fractional derivative operator

We observe from Lemma

Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order

The authors wish to express their thanks to the referees for their suggestions. The authors are also very grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through the Project no. In101003.