Maximum principle and its application to multi-index Hadamard fractional diffusion equation

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.


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 spacetime 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: (1.1) Here, Lv is a uniformly elliptic operator Moreover, we suppose that the functions ϕ i , φ i,j (i, j = 1, 2, . . . , n) are continuous on¯ × Clearly, the matrix A = (φ i,j ) n×n is positive definite and symmetric. P( H D t ) is a multi-term Hadamard fractional derivative defined by Besides this, we also suppose that C and ϑ i are continuous on × (1, T] equipped with ϑ i (x, t) ≥ 0 and C(x, t) ≤ 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.

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

Maximum principle
In this subsection, we develop some maximum principle of IBVP for the multi-index where ∈ R N is an open domain with a smooth boundary ∂ . Denote b(x, t), 0 .
Proof First of all, suppose that the statement is violated, then there exists (x 0 , t 0 ) ∈ × (1, T] such that v(x, t) attains the maximum value v(x 0 , t 0 ) and satisfies From the definition of ζ , we get

It follows from Lemma 2.3 that
According to Lemma 2.2 and ϑ i (x, t) ≥ 0, we know By the definition of ζ (x, t) and Lemma 2.1, we obtain which is not in accordance with (x * , t * ) ≤ 0.
In the same way, we can prove the following. b(x, t), 0 . holds, where N 0 = a C 2 (¯ ) ,

Application of the maximum principle
Proof For ∀(x, t) ∈¯ × [1, T], set the auxiliary function t) is a solution of (1.1) with the function instead of (x, t) and b(x, t), respectively. Since 1 (x, t) ≤ 0, we apply the maximum principle (Theorem 3.1) to ψ(x, t), we can get Again, set another auxiliary function and applying the minimum principle (Theorem 3.2), we obtain for the corresponding classical solution v(x, t) andv(x, t) holds true.
The last inequality Then P satisfies the equation  It follows from the assumptions on that whereṽ = λv 1 + (1λ)v 2 for some 0 ≤ λ ≤ 1.
It is obvious to observe from the proof process of Theorem 4.5.