# Maximum Principle for the Space-Time Fractional Conformable Differential System Involving the Fractional Laplace Operator

Authors: Tingting Guan; Guotao Wang
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the authors consider a IBVP for the time-space fractional PDE with the fractional conformable derivative and the fractional Laplace operator. A fractional conformable extremum principle is presented and proved. Based on the extremum principle, a maximum principle for the fractional conformable Laplace system is established. Furthermore, the maximum principle is applied to the linear space-time fractional Laplace conformable differential system to obtain a new comparison theorem. Besides that, the uniqueness and continuous dependence of the solution of the above system are also proved.


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## Body

## 1. Introduction

Many fractional partial differential equations were used for modeling complex dynamic systems of engineering, physics, biology, and many other fields [1–4]. As a significant tool, the maximum principle plays an important role in the study of the complex dynamic systems without certain knowledge of the solutions [5–13]. In 2016, by using the maximum principle, Luchko and Yamamoto [14] obtained the uniqueness of both the strong and the weak solutions of the IBVP for a general time-fractional distributed order diffusion equation. In 2016, Jia and Li [15] applied the maximum principle to the classical solution and weak solution of a time-space fractional diffusion equation. Furthermore, they also deduced the maximum principle for a full fractional diffusion equation other than time-fractional and spatial-integer order diffusion equations. In 2019, Wang et al. [16] investigated the IBVP for Hadamard fractional differential equations with fractional Laplace operator −Δβ by using the maximum principle.There are diverse fractional derivatives, such as the Riemann–Liouville derivative, the Caputo fractional derivative, the left and right conformable derivatives, and other fractional derivatives [17–40]. In 2015, Abdeljawad [34] defined the left and right conformable derivatives. Depending on [34], Jarad et al. [35] introduced the fractional conformable derivatives and presented the fractional conformable derivative in the sense of Caputo. The extremum principle of the Caputo fractional conformable derivative is seldom regarded in the existing literature. In addition, the papers which mentioned the fractional conformable derivative do not include the fractional Laplace operator.Motivated by the above works, in this context, the authors investigate the IBVP for a space-time Caputo fractional conformable diffusion system with the fractional Laplace operator. First, we provide a detailed proof of the Caputo fractional conformable extremum principle. Then, the new maximum principle is obtained by applying the extreme principle. As some applications of the maximum principle, a comparison principle for the space-time fractional Laplace conformable differential system is developed, and the properties of the solution of the system are given, such as the uniqueness and continuous dependence on the initial and boundary condition.The article is organized as follows: in Section2, the extremum principle for the Caputo fractional conformable derivative is established. In Section 3, the maximum principle of the space-time fractional Laplace conformable differential system is derived, which is used to obtain the comparison principle for the space-time fractional Laplace conformable differential system, and the properties of the solution of the above system are given in Section 4.


## 2. Problem Formulation and Extremum Principles

In this paper, we focus on a space-time Caputo fractional conformable system with the fractional Laplace operator:(1)aCβDtαux,t+−Δγux,t−ax,tux,t=gx,t,x,t∈Ω×a,b,ux,t=0,x∈ℝN\Ω¯,t≥a,where Ω represents an open and bounded domain in ℝNN≥1 in which boundary Γ is smooth and ax,t∈Ω¯×a,b is a bounded function. Here, aCβDtα is the left Caputo fractional conformable derivative. For a function f∈Cα,an, the left Caputo fractional conformable derivative of order β is defined by(2)aCβDtαft=1Γn−β∫att−aα−τ−aααn−β−1anTαfττ−a1−αdτ,with 0<β<1, 0<α<1, n=β+1, Tαaft=t−a1−αf′t, Tαan=TαaTαa,…,Tαa︸ntimes, and Cα,ana,b=f:a,b⟶ℝ|an−1Tαf∈Iαa,b (where Iαa,b is defined in Definition 1 of [34]). For detailed information of the Caputo fractional conformable derivative, see [35].Whenϕ∈Cloc1,1ℝN∩Lγ, the fractional Laplace operator could be given by(3)−Δγϕx=CN,γ∫ℝNϕx−ϕyx−yN+2γdy,with CN,γ=γ22γΓN+2γ/2/πN/2Γ1−γ, and(4)Lγ=ϕ:ℝN⟶ℝ|∫ℝNϕx1+xN+γdx<∞..Denote(5)HΩ¯=ux,t|ux,t∈C2,1Ω×a,b,ux,t∈CΩ¯×a,b.Firstly, we can state two Caputo fractional conformable extremum principles.Lemma 1.
Iff∈Cα,a1a,b reaches its maximum at a point t0∈a,b, then(6)aCβDt0αft0≥0,holds.Proof.
First, we introduce an auxiliary function(7)gt=ft0−ft≥0,t∈a,b.
Concurrently,gt∈Cα,a1a,b, gt0=0, and aCβDtαgt=−aCβDtαft.
By calculation, we notice that(8)aCβDαt0gt0=1Γ1−β∫at0t0−aα−τ−aαα−βτ−a1−αg′ττ−a1−αdτ=1Γ1−β∫at0t0−aα−τ−aαα−βg′τdτ=1Γ1−βt0−aα−τ−aαα−βgτ|at0−βΓ1−β∫at0t0−aα−τ−aαα−β−1τ−aα−1gτdτ.
This is because(9)limτ⟶t01Γ1−βt0−aα−τ−aαα−βgτ=αβΓ1−βlimτ⟶t0g′τβt0−aα−τ−aαβ−1−ατ−aα−1=0.
Therefore, formula (9) becomes(10)aCβDt0αgt0=−1Γ1−βt0−aαα−βga−βΓ1−β∫at0t0−aα−τ−aαα−β−1τ−aα−1gτdτ≤0.
We can obtainaCβDt0αft0≤0.
The lemma is proved.
Using the same method, it is easy to obtain the following lemma.Lemma 2.
Iff∈Cα,a1a,b reaches its minimum at a point t0∈a,b, then(11)aCβDt0αft0≤0,holds.


## 3. Maximum Principle

In this section, we focus on linear space-time Caputo fractional conformable Laplace system (1) with the initial-boundary condition:(12)ux,a=φx,x∈Ω,(13)ux,t=μx,t,x,t∈Γ×a,b.Theorem 1.
Let a functionu∈HΩ¯ satisfy linear space-time Caputo fractional conformable Laplace system (1), (12), and (13). Suppose gx,t≤0, ∀x,t∈Ω×a,b. Then, we have(14)ux,t≤maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t,0,∀x,t∈Ω¯×a,b.Proof.
We first suppose that inequality (14) is false; then, there exists a point x0,t0∈Ω×a,b such that(15)ux0,t0>maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t,0=M>0.
Denoteε=ux0,t0−M>0 and(16)wx,t=ux,t+ε2b−t−ab,x,t∈Ω¯×a,b.
Besides,w implies(17)wx,t≤ux,t+ε2,x,t∈Ω¯×a,b,wx0,t0≥ux0,t0=ε+M≥ε+ux,t≥ε+wx,t−ε2≥ε2+wx,t,x,t∈Γ×a,b∪Ω×a.
The latter property implies that the maximum ofw cannot be attained on Γ×a,b∪Ω×a. Let wx1,t1=maxx,t∈Ω¯×a,bwx,t; then,(18)wx1,t1≥ux0,t0≥ε+M>ε,(19)−Δγwx,t|x1,t1=CN,γ∫ℝNwx1,t1−wx,t1x1−xN+2γdx≥0.
By Lemma1, we know(20)aCβDtαwx,t|x1,t1≥0.
By calculation, we can show(21)aCβDtαε2b−t−ab=−1Γ1−βε2b∫att−aα−τ−aαα−βdτ.
Assumingu=τ−a/t−aα and substituting into formula (21), we get(22)aCβDtαε2b−t−ab=−1Γ1−βε2bαβ−1∫01t−a1−αβ1−u−βu1−αdu=−αβ−1t−a1−αβε2bΓ2−αΓ3−α−β.
Applying (19)–(22), it holds that(23)aCβDtαux,t|x1,t1+−Δγux,t|x1,t1−ax1,t1ux1,t1−gx1,t1=aCβDtαwx,t|x1,t1−aCβDtαε2b−t1−ab+−Δγwx1,t1−ax1,t1wx1,t1−ε2b−t1−ab−gx1,t1≥αβ−1t1−a1−αβε2bΓ2−αΓ3−α−β−ax1,t1ε1−b−t1−a2b≥0.
Equation (23) is in contradiction with (1).
The proof of the theorem is completed.
Similarly, the minimum principle can be obtained as follows.Theorem 2.
Let a functionu∈HΩ¯ satisfy linear space-time Caputo fractional conformable Laplace system (1), (12), and (13). Suppose gx,t≥0, ∀x,t∈Ω×a,b. Then, we have(24)ux,t≥minminx∈Ωφx,minx,t∈Γ×a,bμx,t,0,∀x,t∈Ω¯×a,b.


## 4. Some Applications of the Maximum Principle

Theorem 3.
Letux,t∈HΩ¯ be a solution of system (1) with initial-boundary values (12) and (13). Then,(25)uCΩ¯×a,b≤maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t+2MΓ2+αβ−α−ββαβΓ1+αβ−αb−aαβ,where(26)M=gCΩ¯×a,b.Proof.
We first present a function(27)wx,t=ux,t−MΓ2+αβ−α−ββαβΓ1+αβ−αt−aαβ,x,t∈Ω¯×a,b.
Ifux,t is a solution of system (1), (12), and (13), then wx,t is a solution of problem (1) with(28)g1x,t=gx,t−MΓ2+αβ−α−ββαβΓ1+αβ−αaCβDtαt−aαβ=gx,t−M,μ1x,t=μx,t−MΓ2+αβ−α−ββαβΓ1+αβ−αt−aαβ.
Substituteg1x,t and μ1x,t for gx,t and μx,t, respectively. Owing to g1x,t≤0, applying Theorem 1 (maximum principle), we have(29)wx,t≤maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t+MΓ2+αβ−α−ββαβΓ1+αβ−αb−aαβ.
Therefore,(30)ux,t≤maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t+2MΓ2+αβ−α−ββαβΓ1+αβ−αb−aαβ.
In a similar manner, we can get(31)ux,t≥−maxmaxx∈Ωφx,maxx,t∈Γ×a,bμx,t−2MΓ2+αβ−α−ββαβΓ1+αβ−αb−aαβ.
Combining (30) and (31), the theorem is proved.Theorem 4.
Letux,t satisfy IBVP (1), (12), and (13). ux,t is continuous depending on the data given. That is, if(32)g−g1CΩ¯×a,b≤ε,φx−φ1xCΩ¯≤ε0,μx,t−μ1x,tCΓ×a,b≤ε1,then the estimation of the classical solution of ux,t and u1x,t,(33)u−u1CΩ¯×a,b≤maxε0,ε1+2Γ2+αβ−α−ββαβΓ1+αβ−αb−aαβε,holds.
The demonstration process is similar to Theorem3.Theorem 5.
Letu∈HΩ¯ be a solution of IBVP (1), (12), and (13). Assume gx,t≤0 and ax,t≤0,∀x,t∈Ω×a,b. Then, it follows that(34)ux,t≤0,x,t∈Ω¯×a,b,if φx≤0, and μx,t≤0.Theorem 6.
Letu∈HΩ¯ satisfy IBVP (1), (12), and (13). Assume gx,t≥0 and ax,t≥0,∀x,t∈Ω×a,b. Then, it follows that(35)ux,t≥0,x,t∈Ω¯×a,b,if φx≥0, and μx,t≥0.
The conclusion of Theorem5 and Theorem 6 is obtained by Theorem 1.Remark 1.
Letu∈HΩ¯ satisfy IBVP (1), (12), and (13). Assume gx,t=ax,t=0,∀x,t∈Ω×a,b. Then, it follows that(36)ux,t=0,∀x,t∈Ω¯×a,b,if φx=μx,t=0.Theorem 7.
(comparison theorem). Supposeax,t≥0, bx,t≥0, and bx,t>ax,t, ∀x,t∈Ω×a,b. Assume u,v∈HΩ¯×HΩ¯ satisfies the following linear space-time fractional Laplace conformable differential system:(37)aCβDtαux,t+−Δγux,t−ax,tvx,t−bx,tux,t≥0,x,t∈Ω×a,b,aCβDtαvx,t+−Δγvx,t−ax,tux,t−bx,tvx,t≥0,x,t∈Ω×a,b,ux,t=0,vx,t=0,x∈ℝN\Ω¯,t≥a,ux,a≥0,vx,a≥0,x∈Ω,ux,t≥0,vx,t≥0,x,t∈Γ×a,b.Then, it follows that(38)ux,t≥0,vx,t≥0,x,t∈Ω¯×a,b.Proof.
Letpx,t=ux,t+vx,t,∀x,t∈Ω¯×a,b. Then, by (37), we have(39)aCβDtαpx,t+−Δγpx,t−ax,tpx,t+bx,tpx,t≥0,x,t∈Ω×a,b,px,t=0,x∈ℝN\Ω¯,t≥a,px,a≥0,x∈Ω,px,t≥0,x,t∈Γ×a,b.
Thus, by (39) and Theorem 6, we obtain(40)px,t≥0,∀x,t∈Ω¯×a,b,i.e.ux,t+vx,t≥0,x,t∈Ω¯×a,b.
Using (37) and (40), we have that(41)aCβDtαux,t+−Δγux,t−bx,t−ax,tux,t≥0,x,t∈Ω×a,b,ux,t=0,x∈ℝN\Ω¯,t≥a,ux,a≥0,x∈Ω,ux,t≥0,x,t∈Γ×a,b,(42)aCβDtαvx,t+−Δγvx,t−bx,t−ax,tvx,t≥0,x,t∈Ω×a,b,vx,t=0,x∈ℝN\Ω¯,t≥a,vx,a≥0,x∈Ω,vx,t≥0,x,t∈Γ×a,b.
Applying Theorem6 to (41) and (42), we can get(43)ux,t≥0,vx,t≥0,x,t∈Ω¯×a,b.
Thus, the conclusion holds.
Similarly, the following theorem holds.Theorem 8.
Supposeax,t≤0, bx,t≤0, and bx,t<ax,t, ∀x,t∈Ω×a,b. Assume u,v∈HΩ¯×HΩ¯ satisfies the following linear space-time fractional Laplace conformable differential system:(44)aCβDtαux,t+−Δγux,t−ax,tvx,t−bx,tux,t≤0,x,t∈Ω×a,b,aCβDtαvx,t+−Δγvx,t−ax,tux,t−bx,tvx,t≤0,x,t∈Ω×a,b,ux,t=0,vx,t=0,x∈ℝN\Ω¯,t≥a,ux,a≤0,vx,a≤0,x∈Ω,ux,t≤0,vx,t≤0,x,t∈Γ×a,b.Then, it follows that(45)ux,t≤0,vx,t≤0,x,t∈Ω¯×a,b.Remark 2.
Letu,v∈HΩ¯×HΩ¯ satisfy the following linear space-time fractional Laplace conformable differential system:(46)aCβDtαux,t+−Δγux,t−ax,tvx,t−bx,tux,t=0,x,t∈Ω×a,b,aCβDtαvx,t+−Δγvx,t−ax,tux,t−bx,tvx,t=0,x,t∈Ω×a,b,ux,t=0,vx,t=0,x∈ℝN\Ω¯,t≥a,ux,a=φ1x,vx,a=φ2x,x∈Ω,ux,t=μ1x,t,vx,t=μ2x,t,x,t∈Γ×a,b.
Supposeax,t=bx,t=0,∀x,t∈Ω×a,b. Then, it follows that(47)ux,t=0,vx,t=0,∀x,t∈Ω¯×a,b,if φ1x=φ2x=μ1x,t=μ2x,t=0.
Next, we focus on the following linear space-time fractional Laplace conformable differential system:(48)aCβDtαux,t+−Δγux,t−ax,tvx,t−bx,tux,t=g1x,t,x,t∈Ω×a,b,aCβDtαvx,t+−Δγvx,t−ax,tux,t−bx,tvx,t=g2x,t,x,t∈Ω×a,b,ux,t=0,vx,t=0,x∈ℝN\Ω¯,t≥a,ux,a=φ1x,vx,a=φ2x,x∈Ω,ux,t=μ1x,t,vx,t=μ2x,t,x,t∈Γ×a,b.Theorem 9.
Supposeax,t≤0, bx,t≤0, bx,t<ax,t, g1x,t≤0, and g2x,t≤0, ∀x,t∈Ω×a,b; then, IBVP (48) has a unique solution on HΩ¯×HΩ¯.Proof.
Letu1,v1andu2,v2 be two solutions of IBVP (48). Denote(49)ux,t=u1x,t−u2x,t,vx,t=v1x,t−v2x,t,∀x,t∈Ω¯×a,b,satisfies the system(50)aCβDtαux,t+−Δγux,t−ax,tvx,t−bx,tux,t=0,x,t∈Ω×a,b,aCβDtαvx,t+−Δγvx,t−ax,tux,t−bx,tvx,t=0,x,t∈Ω×a,b,ux,t=0,vx,t=0,x∈ℝN\Ω¯,t≥a,ux,a=0,vx,a=0,x∈Ω,ux,t=0,vx,t=0,x,t∈Γ×a,b.
Letpx,t=ux,t+vx,t,∀x,t∈Ω¯×a,b. By (50), we have(51)aCβDtαpx,t+−Δγpx,t−bx,t−ax,tpx,t=0,x,t∈Ω×a,b.px,t=0,x∈ℝN\Ω¯,t≥a,px,a=0,x∈Ω,px,t=0,x,t∈Γ×a,b.
Applying Theorem8, we get(52)ux,t≤0,vx,t≤0,x,t∈Ω¯×a,b.
By the same way, using Theorem8 to −ux,t and −vx,t, we have(53)ux,t≥0,vx,t≥0,x,t∈Ω¯×a,b.
Combining (52) and (53), we can get(54)ux,t=0,vx,t=0,∀x,t∈Ω¯×a,b.
Thus, the conclusion holds.



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