## Abstract

This study establishes some new maximum principle which will help to investigate an IBVP for multi-index Hadamard fractional diffusion equation. With the help of the new maximum principle, this paper ensures that the focused multi-index Hadamard fractional diffusion equation possesses at most one classical solution and that the solution depends continuously on its initial boundary value conditions.

### Similar content being viewed by others

## 1 Introduction

As is known, the maximum principle is one of the most effective tools to investigate ordinary (partial, evolution, fractional) differential equations. In the absence of any clear information about the solution, some properties of the solution can be obtained using the maximum principle. Recently, the maximum principle and its effective application in investigating fractional differential equations have received great attention from scholars. In [1], the authors studied the IBVP for the single-term and the multi-term as well as the distributed order time-fractional diffusion equations with Riemann–Liouville and Caputo type time-fractional derivatives. Meanwhile, they proved the weak maximum principle and established the uniqueness of solutions to the IBVP with Dirichlet boundary conditions. The maximum principles for classical solution and weak solution of a time-space fractional diffusion equation with the fractional Laplacian operator were considered in [2]. In [3], Korbol and Luchko generalized the mathematical model of variable-order space-time fractional diffusion equation to analyze some financial data and considered the option pricing as an application of this model. In [4], the authors established the maximum principle for the multi-term time-space Riesz–Caputo fractional differential equation, uniqueness and continuous dependence of the solution as well as presented a numerical method for the specified equation. In recent years, the study of maximum principle has attracted a lot of attention, we refer the reader to papers [5,6,7,8,9,10] and the references therein.

The importance of Hadamard fractional calculus has risen. For its recent study and development, we refer to [11,12,13,14,15,16,17,18,19,20]. The maximum principle for IBVP with the Hadamard fractional derivative has just been awakened. Only in [21], Kirane and Torebek obtained the extreme principles for the Hadamard fractional derivative and applied the extreme principles to develop some Hadamard fractional maximum principles, by which the authors show the uniqueness and continuous dependence of the solution of a class of Hadamard time-fractional diffusion equations.

In this article, we study the following multi-index Hadamard fractional diffusion equation:

Here, \(\mathbb{L}v\) is a uniformly elliptic operator

Moreover, we suppose that the functions \(\varphi _{i}\), \(\phi _{i,j}\) (\(i,j=1,2, \ldots , n\)) are continuous on \(\bar{\Omega } \times [1,T]\) and equipped with \(\phi _{i,j}=\phi _{j,i}\) on \(\Omega \times (1,T]\). In addition, for a positive constant *η*,

Clearly, the matrix \(A=(\phi _{i,j})_{n\times n}\) is positive definite and symmetric. \(\mathbb{P}({}^{H}D_{t})\) is a multi-term Hadamard fractional derivative defined by

Besides this, we also suppose that *C* and \(\vartheta _{i}\) are continuous on \(\Omega \times (1,T]\) equipped with \(\vartheta _{i}(x,t) \geq 0\) and \(C(x,t)\leq 0\).

The structure of the article is as follows: In Sect. 2 we give basic concepts and the definitions of Hadamard fractional calculus, and also give some lemmas, which will be needed in our subsequent proof. Further, the maximum principle of IBVP for the multi-index Hadamard fractional differential equation is derived in Sect. 3. In Sect. 4, some applications are demonstrated, i.e., the uniqueness and continuous dependence of solution to the multi-index linear (nonlinear) Hadamard fractional diffusion equations are discussed.

## 2 Preliminaries

Now, we list some basic definitions and lemmas needed in our subsequent proof.

From paper [22], Hadamard fractional integral and derivative of order *p* are defined as

and

where \(n=[p]+1\) and \(\log (\cdot )=\log _{e}(\cdot )\), respectively.

### Lemma 2.1

([22])

*If*\(a, p, q>0\), *then*

### Lemma 2.2

([21])

*For*\(0< p<1\), *if*\(g\in C^{1}([1,T])\)*attains its maximum at*\(t_{0}\in [1,T]\), *then*

*holds*. *Further*, *if*\(g(t_{0})\geq 0\), *then*

### Lemma 2.3

([10])

*Suppose that a function*\(g\in C^{2}(\bar{\Omega })\)*attains its maximum at*\(x_{0}\in \Omega \), *then*

*and*

*hold*.

## 3 Maximum principle

In this subsection, we develop some maximum principle of IBVP for the multi-index Hadamard fractional diffusion equation, by means of which we shall show the uniqueness and continuous dependence of the solution of the multi-index Hadamard fractional diffusion equation.

First, consider the multi-index Hadamard fractional diffusion equation (1.1) with the initial-boundary conditions:

where \(\Omega \in \mathbb{R}^{N}\) is an open domain with a smooth boundary *∂*Ω. Denote

### Theorem 3.1

*Let*\(\Psi (x,t)\), \(C(x,t)\)*be nonpositive on*\(\Omega \times (1,T]\)*and*\(v(x,t)\in W_{*}\)*be a solution of IBVP* (1.1) *and* (3.1)*–*(3.2). *It follows that*

### Proof

First of all, suppose that the statement is violated, then there exists \((x_{0},t_{0})\in \Omega \times (1,T]\) such that \(v(x,t)\) attains the maximum value \(v(x_{0},t_{0})\) and satisfies

Let \(\delta =v(x_{0},t_{0})-N>0\). For \(\forall (x,t)\in \bar{\Omega } \times [1,T]\), let us introduce the auxiliary function

From the definition of *ζ*, we get

and

The last inequality means that \(\zeta (x,t)\) cannot get the maximum on \(\Omega \times \{1\}\cup \partial \Omega \times [1,T]\). Without loss of generality, put \((x^{*},t^{*})\) to be a maximum point of \(\zeta (x,t)\) on \(\bar{\Omega }\times [1,T]\), then we have

It follows from Lemma 2.3 that

According to Lemma 2.2 and \(\vartheta _{i}(x,t)\geq 0\), we know

By the definition of \(\zeta (x,t)\) and Lemma 2.1, we obtain

which is not in accordance with \(\Psi (x^{*},t^{*})\leq 0\). □

In the same way, we can prove the following.

### Theorem 3.2

*Let functions* Ψ, *C**be nonnegative on*\(\Omega \times (1,T]\)*and*\(v(x,t)\in W_{*}\)*be a solution of IBVP* (1.1) *and* (3.1)*–*(3.2), *it follows that*

## 4 Application of the maximum principle

### Theorem 4.1

*Let*\(C(x,t)\)*be nonpositive on*\(\Omega \times (1,T]\)*and*\(v(x,t) \in W_{*}\)*be a solution of IBVP* (1.1) *and* (3.1)*–*(3.2). *Then*

*holds*, *where*

### Proof

For \(\forall (x,t)\in \bar{\Omega }\times [1,T]\), set the auxiliary function

then \(\psi (x,t)\) is a solution of (1.1) with the function

instead of \(\Psi (x,t)\) and \(b(x,t)\), respectively. Since \(\Psi _{1}(x,t) \leq 0\), we apply the maximum principle (Theorem 3.1) to \(\psi (x,t)\), we can get

Therefore,

Again, set another auxiliary function

and applying the minimum principle (Theorem 3.2), we obtain

Inequalities (4.2) and (4.3) together complete the proof of the theorem. □

### Theorem 4.2

*The solution of problem* (1.1) *and* (3.1)*–*(3.2) *depends continuously on the data given*. *That is*, *if*

*then the estimate*

*for the corresponding classical solution*\(v(x,t)\)*and*\(\bar{v}(x,t)\)*holds true*.

The last inequality (4.4) is a simple consequence of norm estimate (4.1). Applying Theorem 4.1 and replacing Ψ, *a*, and *b* by \(\Psi -\bar{\Psi }\), \(a-\bar{a}\), and \(b-\bar{b}\) in problem (1.1), (3.1), and (3.2), respectively, one can easily prove Theorem 4.2.

### Theorem 4.3

*Assume that*\(\Psi (x,t)\leq 0\), \(C(x,t)\leq 0\), \(\forall (x,t)\in \bar{ \Omega }\times [1,T]\), *and*\(v(x,t)\in W_{*}\)*is a solution of IBVP* (1.1) *and* (3.1)*–*(3.2). *If*\(a(x)\leq 0\), \(x\in \Omega \), *and*\(b(x,t)\leq 0\), \((x,t)\in \partial \Omega \times [1,T]\), *then*

### Theorem 4.4

*If the inequality is reversed in Theorem *4.3, *then the inequality of the conclusion is also reversed*.

From Theorems 4.3 and 4.4, the following remark holds.

### Remark 4.1

If functions Ψ, *C*, *a*, *b* are zero in Theorem 4.3 (or 4.4), then \(v(x,t)\) is also zero on \(\bar{\Omega }\times [1,T]\).

Now, let us consider the uniqueness of solution for the multi-index nonlinear Hadamard fractional diffusion equation

with initial boundary value conditions (3.1)–(3.2).

### Theorem 4.5

*If the smooth function*\(\Psi (x,t,v)\)*of diffusion equation* (4.5) *is nonincreasing with respect to the third variable and*\(C(x,t)\leq 0\), *then the multi*-*index nonlinear Hadamard fractional diffusion problem* (4.5) *and* (3.1)*–*(3.2) *has at most one solution*\(v(x,t)\in W_{*} \).

### Proof

Let \(v_{1}, v_{2}\in W_{*}\) be two solutions of Eq. (4.5) with initial boundary value conditions (3.1)–(3.2). Define an auxiliary function on \(\bar{\Omega }\times [1,T]\)

Then \(\mathfrak{P}\) satisfies the equation

It follows from the assumptions on Ψ that

where \(\tilde{v}=\lambda v_{1}+(1-\lambda )v_{2}\) for some \(0\leq \lambda \leq 1\).

Since Ψ is nonincreasing with respect to the third variable, i.e., \(\frac{\partial \Psi }{\partial v}\le 0\), it follows from Theorem 4.3 that, for the multi-index nonlinear Hadamard fractional diffusion problem (4.6),

In the same way, applying Theorem 4.3 to function \(- \mathfrak{P}(x,t)\), for \((x,t)\in \bar{\Omega }\times [1,T]\), the inequality

holds. Thus, (4.8) and (4.9) imply \(\mathfrak{P}(x,t)=0\). This completes the proof. □

It is obvious to observe from the proof process of Theorem 4.5.

### Remark 4.2

If \(\frac{\partial \Psi }{\partial v}(\tilde{v})+C\leq 0\), then the conclusion of Theorem 4.5 holds.

### Corollary 4.1

*If the function**C**is nonpositive on*\(\bar{\Omega }\times [1,T]\), *then IBVP* (1.1) *and* (3.1)*–*(3.2) *has at most one solution on*\(W_{*}\).

## References

Luchko, Y., Yamamoto, M.: General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems. Fract. Calc. Appl. Anal.

**19**(3), 676–695 (2016)Jia, J., Li, K.: Maximum principle for a time-space fractional diffusion equation. Appl. Math. Lett.

**62**, 23–28 (2016)Korbel, J., Luchko, Y.: Modeling of financial processes with a space-time fractional diffusion equation of varying order. Fract. Calc. Appl. Anal.

**19**(6), 1414–1433 (2016)Ye, H., Liu, F., Anh, V., Turner, I.: Maximum principle and numerical method for the multi-term time-space Riesz–Caputo fractional differential equations. Appl. Math. Comput.

**227**, 531–540 (2014)Al-Refai, M., Luchko, Y.: Maximum principle for the fractional diffusion equations with the Riemann–Liouville fractional derivative and its application. Fract. Calc. Appl. Anal.

**17**(2), 483–498 (2014)Al-Refai, M., Luchko, Y.: Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives. Appl. Math. Comput.

**257**, 40–51 (2015)Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl.

**351**(1), 218–223 (2009)Liu, Z., Zeng, S., Bai, Y.: Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications. Fract. Calc. Appl. Anal.

**19**(1), 188–211 (2016)Zhang, L., Ahmad, B., Wang, G.: Analysis and application of diffusion equations involving a new fractional derivative without singular kernel. Electron. J. Differ. Equ.

**2017**, 289 (2017)Walter, W.: On the strong maximum principle for parabolic differential equations. Proc. Edinb. Math. Soc.

**29**, 93–96 (1986)Hadamard, J.: Essai sur l’ tude des fonctions, données par leur developpement de Taylor. J. Math. Pures Appl.

**8**, 101–186 (1892)Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham (2017)

Yukunthorn, W., Ahmad, B., Ntouyas, S.K., Tariboon, J.: On Caputo–Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions. Nonlinear Anal. Hybrid Syst.

**19**, 77–92 (2016)Ahmad, B., Ntouyas, S.K., Tariboom, J.: A study of mixed Hadamard and Riemann–Liouville fractional integro-differential inclusions via endpoint theory. Appl. Math. Lett.

**52**, 9–14 (2016)Pei, K., Wang, G., Sun, Y.: Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain. Appl. Math. Comput.

**312**, 158–168 (2017)Wang, G., Pei, K., Agarwal, R.P., Zhang, L., Ahmad, B.: Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J. Comput. Appl. Math.

**343**, 230–239 (2018)Ma, Q., Wang, R., Wang, J., Ma, Y.: Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative. Appl. Math. Comput.

**257**, 436–445 (2015)Wang, G., Pei, K., Baleanu, D.: Explicit iteration to Hadamard fractional integro-differential equations on infinite domain. Adv. Differ. Equ.

**2016**, 299 (2016)Wang, T., Wang, G., Yang, X.: On a Hadamard-type fractional turbulent flow model with deviating arguments in a porous medium. Nonlinear Anal., Model. Control

**22**, 765–784 (2017)Wang, G., Pei, K., Chen, Y.: Stability analysis of nonlinear Hadamard fractional differential system. J. Franklin Inst. (2019). https://doi.org/10.1016/j.jfranklin.2018.12.033

Kirane, M., Torebek, B.T.: Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations. Fract. Calc. Appl. Anal.

**22**(2), 358–378 (2019)Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, pp. 110–120. Elsevier, Amsterdam (2006)

### Acknowledgements

The authors would like to express gratitude to the anonymous referees for their hard work, helpful comments, and suggestions.

### Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

## Funding

The work is supported by NSFC (No. 11501342), NSF of Shanxi, China (No. 201701D221007), Science and Technology Innovation Project of Shanxi Normal University (No. 01053006) and STIP (Nos. 201802068 and 201802069).

## Author information

### Authors and Affiliations

### Contributions

All authors equally contributed to this manuscript and approved the final version.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Additional information

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## About this article

### Cite this article

Ren, X., Wang, G., Bai, Z. *et al.* Maximum principle and its application to multi-index Hadamard fractional diffusion equation.
*Bound Value Probl* **2019**, 182 (2019). https://doi.org/10.1186/s13661-019-01299-y

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13661-019-01299-y