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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391062, 17 pages
http://dx.doi.org/10.1155/2012/391062
Research Article

On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 27 December 2011; Revised 10 April 2012; Accepted 14 April 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 M. D. Qassim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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 [13], 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 [436], and for some applications, one can consult [13, 8, 25, 26, 26, 27, 2731, 33, 34, 3749] to cite but a few.

We will consider the problem: where is a new type of fractional derivative we will define below and 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 for some and . That is we consider the Cauchy problem: where and show that no solutions can exist for all time for certain values of and . 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 and in some appropriate space.

The existence and uniqueness of solutions for problem (1.1) has been discussed in [50] in the space defined by where for and .

We also point out here that the case where 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 .

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.

2. 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 consists of those real-valued Lebesgue measurable functions on for which , where In particular, when , we see that .

Definition 2.2. Let be a finite interval and , we introduce the weighted space of continuous functions on : In the space , we define the norm:

Definition 2.3. Let be the -derivative, for , we denote by () the Banach space of functions which have continuous -derivatives on up to order and have the derivative of order on such that : with the norm: When , we set

Definition 2.4. Let be a finite or infinite interval of the half-axis and let . The Hadamard left-sided fractional integral of order is defined by provided that the integral exists. When , we set

Definition 2.5. Let be a finite or infinite interval of the half-axis and let . The Hadamard right-sided fractional integral of order is defined by provided that the integral exists. When , we set

Definition 2.6. The left-sided Hadamard fractional derivative of order on is defined by that is, When , we set

Definition 2.7. The right-sided Hadamard fractional derivative of order on is defined by that is, When , we set

Lemma 2.8. If and , then In particular, if , then the Hadamard fractional derivative of a constant is not equal to zero: when .

Lemma 2.9. Let , and . If , then is bounded from into . In particular, is bounded in .If , then is bounded from into . In particular, is bounded in .

This lemma justifies the following one

Lemma 2.10 (the semigroup property of the fractional integration operator ). Let , and . If , then, for , holds at any point . When , this relation is valid at any point .

Lemma 2.11. Let and . If , then the fractional derivatives and exist on and , respectively, ) and can be represented in the forms: respectively.

Lemma 2.12 (fractional integration by Parts). Let and . If and , then where .

Definition 2.13. The fractional derivative of order on defined by where , is called the Hadamard-Caputo fractional derivative of order .

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 and type : 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 , we find the Hadamard fractional derivative as defined in Definition 2.6 and, for , we find its Caputo type counterpart (Definition 2.13).

Theorem 2.14 (Young’s inequality). If and are nonnegative real numbers and and are positive real numbers such that , then one has Equality holds if and only if .

3. Nonexistence Result

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

Lemma 3.1. If and , then

Proof. Since , then on we have for some positive constant . Therefore, As , 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].

Theorem 3.2. Assume that and . Then, Problem (1.2) does not admit global nontrivial solutions in , where when .

Proof. Assume that a nontrivial solution exists for all time . Let be a test function satisfying is non-increasing and such that for some and some () such that . Multiplying the inequality in (1.2) by and integrating over , we get Observe that the integral in left-hand side exists and the one in the right-hand side exists for when . Moreover, from the definition of , we can rewrite (3.6) as By virtue of Lemma 2.12 (after extending by zero outside ), we may deduce from (3.7) that Notice that Lemma 2.12 is valid in our case since ( implies that on for some positive constant ) Let , then by the definition of the Gamma function, Hence, (and ) for some .
An integration by parts in (3.8) yields or because (see Lemma 3.1) and Multiplying by inside the integral in the left hand side of (3.12), we see that It appears from Definition 2.7 that and from Lemma 2.11, we see that Since and the last equality becomes Note that and by the same argument as the one used at the beginning of the proof we may show that since .
Therefore, Lemma 2.12 again allows us to write and by the semigroup property Lemma 2.10 On the other hand, As is nonincreasing, we have for all and , . Also, it is clear that Therefore, Definition 2.4 allows us to write By the same argument as the one used at the beginning of the proof, we may show that . Moreover, it is easy to see that (for, otherwise, we consider with some sufficiently large ). Thus, we can apply Lemma 2.12 to get Next, we multiply by inside the integral in the right-hand side of (3.25): For , we have (because and ). It follows that By using the Young inequality (see Theorem 2.14), with and such that , in the right-hand side of (3.27), we find Clearly, from (3.14) and (3.28), we see that or Therefore, by Definition 2.5, we have The change of variable yields Another change of variable gives We may assume that the integral term in the right-hand side of (3.33) is convergent, that is, for some positive constant , for otherwise we consider with some sufficiently large . Therefore If , then as . Finally, from (3.35), we obtain We reach a contradiction since the solution is not supposed to be trivial.
In the case we have and the relation (3.35) ensures that Moreover, it is clear that This relation, together with (3.27) (relations (3.28) and (3.31) also are used without ), implies that for some positive constant , with due to the convergence of the integral in (3.38). This is again a contradiction.
If , we have (because and ). Then, the change of variables and in (3.27) yields or The expression may be assumed bounded (or else we use with a large value of ). Hence, for some positive constant .

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 . However, according to Theorem 3.2, such a solution cannot be continued for all time in . 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.

4. Example

For our example, we need the following lemma.

Lemma 4.1. The following result holds for the fractional derivative operator : where and .

Proof. We observe from Lemma 2.8 that Therefore, which, in light of the definition of , yields From Lemma 2.8 again, we have The proof is complete.

Example 4.2. Consider the following differential equation of Hilfer-Hadamard-type fractional derivative of order and type : with real , (). Suppose that the solution has the following form: Our aim next is to find the values of and . By using Lemma 4.1 we have Therefore, It can be directly shown that and . If , that is, , then (4.6) has the exact solution: This solution satisfies the initial condition when . Note that there is an overlap of the interval of existence in this example and the interval of nonexistence in the previous theorem. This may be explained by the fact that this solution is in but not in .

Acknowledgments

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.

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