On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

and Applied Analysis 3 Definition 2.1. The space X c a, b c ∈ R, 1 ≤ p ≤ ∞ consists of those real-valued Lebesgue measurable functions g on a, b for which ‖g‖Xp c <∞, where


Introduction
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 1-3 , researchers started to deal directly with differential equations containing fractional derivatives instead of ignoring them as it used to be the case.For analytical treatments, we may refer the reader to 4-36 , and for some applications, one can consult 1-3, 8, 25, 26, 26, 27, 27-31, 33, 34, 37-49 to cite but a few.
We will consider the problem: where D α,β a u is a new type of fractional derivative we will define below and u 0 is a given constant.This new fractional derivative interpolates the Hadamard fractional derivative and its Caputo counterpart 26, 34 , in the same way the Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative.That is why we are naming it after Hilfer and Hadamard.
A nonexistence result for global solutions of the problem 1.1 will be proved when f t, u t ≥ log t/a μ |u t | m for some m > 1 and μ ∈ R.That is we consider the Cauchy problem: where γ α β−αβ and show that no solutions can exist for all time for certain values of μ and m.Clearly, sufficient conditions for nonexistence provide necessary conditions for existence of solutions.In addition, we construct an example for which there exist solutions for some powers m and in some appropriate space.
The existence and uniqueness of solutions for problem 1.We also point out here that the case where D α,β a is the usual Riemann-Liouville fractional derivative has been studied in 26 see also references therein .There are very few papers 26, 29 dealing with the pure Hadamard case, that is, when β 0.
The rest of the paper is divided into three sections.In Section 2, we present some definitions, notations, and lemmas which will be needed later in our proof.Section 3 is devoted to the nonexistence result and Section 4 contains an example of existence of solutions.

Preliminaries
In this section, we present some background material for the forthcoming analysis.For more details, see 25, 26, 33, 42, 51, 52 .Definition 2.1.The space X p c a, b c ∈ R, 1 ≤ p ≤ ∞ consists of those real-valued Lebesgue measurable functions g on a, b for which g X p c < ∞, where ess sup a≤x≤b x c g x , c ∈ R.

2.1
In particular, when c 1/p, we see that X In the space C γ,log a, b , we define the norm: Definition 2.3.Let δ x d/dx be the δ-derivative, for n ∈ N, we denote by C n δ,γ a, b 0 ≤ γ < 1 the Banach space of functions g which have continuous δ-derivatives on a, b up to order n − 1 and have the derivative δ n g of order n on a, b such that δ n g ∈ C γ,log a, b : with the norm: When n 0, we set Definition 2.4.Let a, b 0 ≤ a < b ≤ ∞ be a finite or infinite interval of the half-axis R and let α > 0. The Hadamard left-sided fractional integral J α a f of order α > 0 is defined by provided that the integral exists.When α 0, we set Definition 2.5.Let a, b 0 ≤ a < b ≤ ∞ be a finite or infinite interval of the half-axis R and let α > 0. The Hadamard right-sided fractional integral J α b − f of order α > 0 is defined by provided that the integral exists.When α 0, we set Definition 2.6.The left-sided Hadamard fractional derivative of order 0 ≤ α < 1 on a, b is defined by When α 0, we set Definition 2.7.The right-sided Hadamard fractional derivative of order α 0 ≤ α < 1 on a, b is defined by When α 0, we set

2.17
In particular, if β 1, then the Hadamard fractional derivative of a constant is not equal to zero: This lemma justifies the following one Lemma 2.10 the semigroup property of the fractional integration operator J α a .Let α > 0, β > 0, and respectively.
Lemma 2.12 fractional integration by Parts .Let α > 0 and where 1/p 1/q 1. Definition 2.13.The fractional derivative c D α a f of order α 0 < α < 1 on a, b defined by where δ x d/dx , is called the Hadamard-Caputo fractional derivative of order α.Now, motivated by the Hilfer fractional derivative introduced in 41, 42 , we introduce the new fractional derivative which we call Hilfer-Hadamard fractional derivative of order 0 < α < 1 and type 0 ≤ β ≤ 1: The Hilfer fractional derivative interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative.This new one interpolates the Hadamard fractional derivative and its caputo counterpart.Indeed, for β 0, we find the Hadamard fractional derivative as defined in Definition 2.6 and, for β 1, we find its Caputo type counterpart Definition 2.13 .Theorem 2.14 Young's inequality .If a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p 1/q 1, then one has ab ≤ a p p b q q .2.24 Equality holds if and only if a p b q .

Nonexistence Result
Before we state and prove our main result, we will start with the following lemma.
Proof.Since f ∈ C a, b , then on a, b we have |f t | < M for some positive constant M. Therefore,

3.2
As α > 0, we see that In a similar manner, we prove the second part of the lemma.
The proof of the next result is based on the test function method developed by Mitidieri and Pokhozhaev in 52 .Proof.Assume that a nontrivial solution exists for all time t > a.Let ϕ t ∈ C 1 a, ∞ be a test function satisfying ϕ t ≥ 0, ϕ t is non-increasing and such that for some T > a and some θ θ < 1 such that a < θT < T. Multiplying the inequality in 1.2 by ϕ t /t and integrating over a, T , we get Observe that the integral in left-hand side exists and the one in the right-hand side exists for 3.9 Let s p − 1 log t/a , then by the definition of the Gamma function, ϕ T 0 see Lemma 3.1 and Multiplying by t/t inside the integral in the left hand side of 3.12 , we see that

3.14
It appears from Definition 2.7 that and from Lemma 2.11, we see that ϕ s ds dt t .

3.16
Since ϕ T 0 and δϕ t , 3.17 the last equality becomes

3.18
Note that δϕ ∈ L p and by the same argument as the one used at the beginning of the proof we may show that J

3.21
As ϕ is nonincreasing, we have ϕ s ≥ ϕ t for all t ≥ s and 1/ϕ 1/m s ≤ 1/ϕ for, otherwise, we consider ϕ λ t with some sufficiently large λ .Thus, we can apply Lemma 2.12 to get Next, we multiply by log t/a μ/m .log t/a −μ/m inside the integral in the right-hand side of 3.25 :

3.27
By using the Young inequality see Theorem 2.14 , with m and m such that 1/m 1/m 1, in the right-hand side of 3.27 , we find

3.28
Clearly, from 3.14 and 3.28 , we see that Therefore, by Definition 2.5, we have

3.31
The change of variable σT t yields

3.32
Another change of variable r s/T gives

3.33
We may assume that the integral term in the right-hand side of 3.33 is convergent, that is,

3.43
The expression |ϕ r |/ϕ 1/m r may be assumed bounded or else we use ϕ λ r with a large value of λ .Hence, for some positive constant C.
Although we are concerend here about nonexistence of solutions, using standard techniques, one may show the existence of local solutions of Problem 1.1 with 1 < m < 1 μ / 1 − γ .However, according to Theorem 3.2, such a solution cannot be continued for all time in C γ 1−γ,log a, b .This is a phenomenon which occurs often in parabolic and hyperbolic problems with sources of polynomial type.In the absence of strong dissipations, these sources are the cause of blowup in finite time of local solutions .For this reason, they are called blowup terms.

Example
For our example, we need the following lemma.Proof.We observe from Lemma 2.8 that

4.4
From Lemma 2.8 again, we have

4.5
The proof is complete.

p 1 /
p a, b L p a, b .Definition 2.2.Let Ω a, b 0 < a < b < ∞ be a finite interval and 0 ≤ γ < 1, we introduce the weighted space C γ,log a, b of continuous functions g on a, b : log a, b , then the fractional derivatives D α a and D α b − exist on a, b and a, b , respectively, a > 0 and can be represented in the forms:
p and ϕ ∈ L p for some p > 1/γ.An integration by parts in 3.8 yields t | m dt t ≤ log θT/a −μm /m Γ m 1 − α
log a, b .Moreover, from the definition of D γ a u ∈ C a, T implies that