# On Fractional Diffusion Equation with Caputo-Fabrizio Derivative and Memory Term

Authors: Binh Duy Ho; Van Kim Ho Thi; Long Le Dinh; Nguyen Hoang Luc; Phuong Nguyen
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we examine a nonlinear fractional diffusion equation containing viscosity terms with derivative in the sense of Caputo-Fabrizio. First, we establish the local existence and uniqueness of lightweight solutions under some assumptions about the input data. Then, we get the global solution using some new techniques. Our main idea is to combine theories of Banach’s fixed point theorem, Hilbert scale theory of space, and some Sobolev embedding.


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## Body

## 1. Introduction

The fractional calculation has a long history and plays an important role in the simulation of physical phenomena or real life, for example, mechanics, electricity, chemistry, biology, economics, notably control theory, and images. It should be noted that the standard mathematical models of integer derivatives, including nonlinear models, do not work fully in many cases. Therefore, the advent of fractional calculus was significant in modeling physical and engineering processes, and it can be said that it is one of the best descriptors using fractional differential equations. In a series of research directions on fractional differential equations (FDE), the most prominent is the appearance of two derivatives: Caputo derivative and Riemann-Liouville derivative. Some works are attracting the attention of the community, like Debbouche and his group [1–3], Karapinar et al. [4–11], Inc and his group [12–16], Tuan and his group [17–20], and the references as follows: [21–23].In this paper, we consider the fractional Sobolev equation:(1)DtαuCF=Δu+Gu+∫0tψt−zKuzdz,x,t∈M×0,T,u=0,x,t∈∂M×0,T,ux,0=u0x,where CFDtα is the Caputo-Fabrizio operator for fractional derivatives of order α which is defined as (see [24])(2)CFDtαvt=Hα1−α∫0tDαt−ν∂vν∂νdν,fort≥0,where we denote by the kernel Dαz=exp−α/1−αz and Hα satisfies H0=H1=1 (see, e.g., [25, 26]).The Caputo-Fabrizio fractional derivative was presented in 2015 [25] with the aim of avoiding singular kernels. It is also the convolution of the exponential function and the first-order derivative. The Caputo-Fabrizio fraction derivative is an operator that has been widely applied to several derivative modes in many fields, such as biology, physics, control systems, materials science, dynamics, and liquid learning [27–32].Our main aim in this paper is to provide the local and global existence for problem (1) under some various assumptions on the input data. The difficulty in studying this problem is from the memory viscoelastic model appearing in the main equation. This term makes some of the assessments more complicated. Another difficulty is the study of the existence of global solutions. The topic of the existence of global solutions is still challenging for many mathematicians today. In the paper, we have to use a new norm in weighted space, thanks to the work of [33], to establish the global solution. The two main results in the paper are shown as follows:(i)
The first result is related to the existence of local solutions. The main technique is to apply Banach’s fixed point theorem(ii)
The second result is very interesting, proving the existence of a global solution. To do this, we have to thank a lemma in [33], where we have chosen suitable assumptions for functions G and K, to obtain our purposeThe paper is organized as follows. In Section2, we give preliminaries which are useful for the next results. Section 3 shows the local existence results. In Section 4, we provide global existence results.


## 2. Preliminaries

We recall the Hilbert scale space, which is given as follows:(3)HrM=f∈L2M,∑n=1∞λnrf,enL2M2<∞,for any r≥0. Here, the symbol ·,·L2M denotes the inner product in L2M. It is well known that HrM is a Hilbert space corresponding to the following norm:(4)fHrM=∑n=1∞λnrf,enL2M2,f∈HrM.HνΩ≡D−Lν is a Hilbert space. Then, D−L−ν is a Hilbert space with the norm:(5)vD−L−ν=∑j=1∞v,ej2λj−2ν1/2,where ·,· in the latter equality denotes the duality between D−L−ν and D−Lν.Definition 1.
The functionv is called a mild solution of problem (1) if it satisfies that(6)θt=Pαtu0+∫0tPαt−τGθτdτ+∫0tPαt−τ∫0τψτ−ξKθξdξdτ,where Pαt is defined by(7)Pαtw=1+α¯λn−1exp−αλn1+α¯λntw,enL2Menx,α¯=1−α,for any w∈L2M.Lemma 1.
Letθ∈Hr−2M∩Hr−2−2βM. Then,(8)PαtθHrM≤C¯1,α,βt−βθHr−2M+C¯2,α,βt−βθHr−2−2βM,for any 0<β<1.Proof.
By the definitions of the norm inHrM and using the inequality e−y≤C−βy−β for β>0, we get the following confirmation:(9)PαtθHrM=∑n=1∞λnr1+α¯λn−2exp−2αλn1+α¯λntθ,enL2M2≤Cβ1−α−1t−β∑n=1∞λnr−21+1−αλnαλn2βθ,enL2M2.
Since0<β<1, we know that(10)1+1−αλnαλn2β≤1−α2α2+1α2λn2β≤1−α2βα2β+α−2βλn−2β.
This follows from (9) that(11)PαtθHrM≤Cβ1−α−1t−β1−αβαβ∑n=1∞λnr−2θ,enL2M2+Cβ1−α−1α−βt−β∑n=1∞λnr−2−2βθ,enL2M2≤C¯1,α,βt−βθHr−2M+C¯2,α,βt−βθHr−2−2βM.☐


## 3. Local Existence Results

In this section, we give the following theorem which shows the local existence result.Theorem 1.
Let the two functionsG and K be(12)Gθ1−Gθ2L2M≤BGθ1−θ2L2M,Kθ1−Kθ2L2M≤BKθ1−θ2L2M,for constants BG,BK≥0. Let us assume that there exists δ such that(13)ψz≤Dz−δ,δ<1.
Letu0∈Hp−2M∩Hp−2−2βM. Then, problem (1) has a local mild solution:(14)u∈Lϑ∞0,T;HpM,where(15)0<β≤ϑ<1,0≤p≤2.Proof.
Let the functionF be as follows:(16)Fθt=Pαtu0+∫0tPαt−τGθτdτ+∫0tPαt−τ∫0τψτ−ξKθξdξdτ=Pαtu0+F1θt+F2θt.
Step 1. EstimateF1θ1−F1θ2HpM for any θ1,θ2 that belongs to the space HpM.
From the definition ofF1 as in (16), we find that(17)F1θ1−F1θ2HpM=∫0tPαt−τGθ1τdτ−∫0tPαt−τGθ2τdτHpM≤C¯1,α,β∫0tt−τ−βGθ1τ−Gθ2τHp−2Mdτ+C¯2,α,β∫0tt−τ−βGθ1τ−Gθ2τHp−2−2βMdτ.
Sincep≤2, we know the Sobolev embedding L2M°Hp−2M, and so we get(18)Gθ1τ−Gθ2τHp−2M≤C1Gθ1τ−Gθ2τL2M≤C1BGθ1τ−θ2τL2M,Gθ1τ−Gθ2τHp−2−2βM≤C1,βGθ1τ−Gθ2τL2M≤C1,βBKθ1τ−θ2τL2M.
From two above observations, we deduce that(19)tϑF1θ1−F1θ2HpM≤C¯1,α,βC1BG+C¯2,α,βC1BKtϑ∫0tt−τ−βθ1τ−θ2τL2Mdτ≤C¯3,p,α,βtϑ∫0tt−τ−βτ−ϑτϑθ1τ−θ2τHpMdτ≤C¯3,p,α,βtϑ∫0tt−τ−βτ−ϑdτθ1−θ2Lϑ∞0,T;HpM=C¯3,p,α,βt1−βB1−β,1−ϑθ1−θ2Lϑ∞0,T;HpM≤C¯3,p,α,βT1−βB1−β,1−ϑθ1−θ2Lϑ∞0,T;HpM.
Because the right side of the above expression does not depend ont, we have the following assertion:(20)F1θ1−F1θ2Lϑ∞0,T;HpM≤C¯3,p,α,βT1−βB1−β,1−ϑθ1−θ2Lϑ∞0,T;HpM.
Step 2. EstimateF2θ1−F2θ2HpM for any θ1,θ2 that belongs to the space HpM.
From the definition ofF1 as in (16), we find that(21)F2θ1−F2θ2HpM=∫0tPαt−τ∫0τψτ−ξKθ1ξdξdτ−∫0tPαt−τ∫0τψτ−ξKθ2ξdξdτHpM≤C¯1,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHp−2Mdτ+C¯2,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHp−2−2βMdτ.
It is easy to see that(22)C¯1,α,β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHp−2M≤C¯1,α,βC1BGD∫0ττ−ξ−δθ1ξ−θ2ξL2Mdξ≤C¯1,α,βC1BGD∫0ττ−ξ−δξ−ϑξϑθ1ξ−θ2ξL2Mdξ≤C¯1,α,βC1BGD∫0ττ−ξ−δξ−ϑdξθ1−θ2Lϑ∞0,T;HpM=C¯1,α,βC1BGDτ1−δ−ϑB1−δ,1−ϑθ1−θ2Lϑ∞0,T;HpM.
By a similar way as above, we also obtain that(23)C¯2,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHp−2−2βMdτ≤C¯2,α,βC1,βBKDτ1−δ−ϑB1−δ,1−ϑθ1−θ2Lϑ∞0,T;HpM.
From two recent observations and noting thatβ+δ<2, we find that(24)tϑF2θ1−F2θ2HpM=C¯3B1−δ,1−ϑB1−β,2−δ−ϑt2−β−δθ1−θ2Lϑ∞0,T;HpM≤C¯3B1−δ,1−ϑB1−β,2−δ−ϑT2−β−δθ1−θ2Lϑ∞0,T;HpM,where(25)C¯3=C¯1,α,βC1BGD+C¯2,α,βC1,βBKD.
Due to the right hand side of (24) being independent of t, we can deduce that(26)F2θ1−F2θ2Lϑ∞0,T;HpM≤C¯3B1−δ,1−ϑB1−β,2−δ−ϑT2−β−δθ1−θ2Lϑ∞0,T;HpM.
Combining (20) and (26), we derive that(27)Fθ1−Fθ2Lϑ∞0,T;HpM≤F1θ1−F1θ2Lϑ∞0,T;HpM+F2θ1−F2θ2Lϑ∞0,T;HpM≤C¯3,p,α,βT1−βB1−β,1−ϑθ1−θ2Lϑ∞0,T;HpM+C¯3B1−δ,1−ϑB1−β,2−δ−ϑT2−β−δθ1−θ2Lϑ∞0,T;HpM.
Moreover, by applying Lemma1 and noting that ϑ≥β, we can confirm the following results:(28)tϑPαtu0HpM≤tϑC¯1,α,βt−βu0Hp−2M+C¯2,α,βt−βu0Hp−2−2βM≤C¯1,α,βtϑ−βu0Hp−2M+C¯2,α,βtϑ−βu0Hp−2−2βM≤C¯1,α,βTϑ−βu0Hp−2M+C¯2,α,βTϑ−βu0Hp−2−2βM.☐


## 4. Global Existence Results under a Global Lipschitz Case

In this section, we derive the global results under the assumption of the nonlinear source functionF, a global Lipschitz. Let F and g satisfy that(29)Gu−FvHsM≤Lgu−vHqM,1≤s≤q,(30)Ku−KvHsM≤Lku−vHqM,1≤s≤q,where Kf,Kg are postive constants. Our results in this section are to present the well-posedness of the problem. Let ν>0 and q≥1. In order to establish the existence of the mild solution, we need to define the following space:(31)Xd,m0,T;HqM=f:M×0,T⟶ℝ:tde−mt∥w·.,tHqM<∞,t∈0,T,associated with the following norm:(32)fXd,m0,T;HqM=sup0≤t≤Ttde−mtw·.,tHqM.Let us provide the following results that will be valuable in justifying our key results. We can find and view it in Lemma 8 of [33] (page 9).Lemma 2.
Letc>−1, d>−1 such that c+d≥−1, h>0 and t∈0,T. For ε>0, the following limit holds:(33)limε⟶∞supt∈0,Tth∫01τc1−τde−εt1−τdτ=0.
Now, we are in the position to introduce the main contributions of this work. Our main results address the global existence of the mild solution.Theorem 2.
Let0<α<1. Let us assume that(34)θt≤Cθt−δ.
Letu0∈Hq−2−2βM∩Hq−2M. Then, there exists a positive number m0 such that problem (1) has a unique solution in Xd,m00,T;HqM. Here, β,d,δ satisfy that(35)0<β≤d<1,β+d<1,β+δ<2,δ<1.Proof.
Let the functionF:Xd,m0,T;HqM⟶Xd,m0,T;HqM be as follows:(36)Fθt=Pαtu0+∫0tPαt−τGθτdτ+∫0tPαt−τ∫0τψτ−ξKθξdξdτ=Pαtu0+F1θt+F2θt.
First, we have the following observation:(37)Pαtu0HqM≤C¯1,α,βt−βu0Hq−2M+C¯2,α,βt−βu0Hq−2−2βM.
By multiplying the two sides of the above inequality bytde−mt and noting that e−mt≤1, one has(38)tde−mtPαtu0HqM≤C¯1,α,βtd−βu0Hq−2M+C¯2,α,βtd−βu0Hq−2−2βM.
Noting thatd≥β, we deduce that if u0∈Hq−2−2βM∩Hq−2M, then the following holds:(39)Pαtu0∈Xd,m0,T;HqM.
Take two functionsθ1,θ2∈Xd,m0,T;HqM. First, we need to derive the estimation for the term:(40)I=∫0tPαt−τGθ1τdτ−∫0tPαt−τGθ2τdτHqM.
Using Lemma1 and Sobolev embedding HsM°Hq−2M and HsM°Hq−2−2βM (since s≥q−2), we arrive at(41)I=∫0tPαt−τGθ1τdτ−∫0tPαt−τGθ2τdτHqM≤C¯1,α,β∫0tt−τ−βGθ1τ−Gθ2τHq−2Mdτ+C¯2,α,β∫0tt−τ−βGθ1τ−Gθ2τHq−2−2βMdτ≤C~1,α,β+C~2,α,β∫0tt−τ−βGθ1τ−Gθ2τHsMdτ.
Since the assumption (29), we know that(42)∫0tt−τ−βGθ1τ−Gθ2τHsMdτ≤Lg∫0tt−τ−βθ1τ−θ2τHqMdτ=Lg∫0tt−τ−βτ−demτθ1−θ2Xd,m0,T;HqM.
Combining (40) and (41), we find that(43)tde−mt∫0tPαt−τGθ1τdτ−∫0tPαt−τGθ2τdτHqM≤C~1,α,β+C~2,α,βLgtd∫0tt−τ−βτ−de−mt−τdτθ1−θ2Xd,m0,T;HqM=C~1,α,β+C~2,α,βLgt1−β∫011−ξ−βξ−de−mt1−ξdξθ1−θ2Xd,m0,T;HqM,where we have used the fact that(44)td∫0tt−τ−βτ−de−mt−τdτ=t1−β∫011−ξ−βξ−de−mt1−ξdξ.
Let us continue to treat the term. First, we need to derive the estimation for the term:(45)II=∫0tPαt−τ∫0τψτ−ξKθ1ξdξdτ−∫0tPαt−τ∫0τψτ−ξKθ2ξdξdτHqM.
Using Lemma1 and Sobolev embedding HsM°Hq−2M and HsM°Hq−2−2βM (since s≥q−2), we get the following bound:(46)II=∫0tPαt−τ∫0τψτ−ξKθ1ξdξdτ−∫0tPαt−τ∫0τψτ−ξKθ2ξdξdτHqM≤C¯1,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHq−2Mdτ+C¯2,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHq−2−2βMdτ≤C~1,α,β+C~2,α,β∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHsMdτ.
Using the Lipschitz property ofK, we know that(47)∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHqM≤CθLk∫0ττ−ξ−δθ1ξ−θ2ξHqMdξ=CθLk∫0ττ−ξ−δξ−demξξde−mξθ1ξ−θ2ξHqMdξ≤CθLk∫0ττ−ξ−δξ−demξdξθ1−θ2Xd,m0,T;HqM.
It is obvious to see that(48)∫0ττ−ξ−δξ−demξdξ≤emτ∫0ττ−ξ−δξ−ddξ=emττ1−δ−dB1−δ,1−d.
This implies that the following estimation is for the integral term on the right hand side of (45):(49)∫0tt−τ−β∫0τψτ−ξKθ1ξdξ−∫0τψτ−ξKθ2ξdξHqMdτ≤CθLkB1−δ,1−d∫0tt−τ−βτ1−δ−demτdτθ1−θ2Xd,m0,T;HqM.
This implies that(50)tde−mt∫0tPαt−τ∫0τψτ−ξKθ1ξdξdτ−∫0tPαt−τ∫0τψτ−ξKθ2ξdξdτHqM≤C3td∫0tt−τ−βτ1−δ−de−mt−τdτθ1−θ2Xd,m0,T;HqM=C3t2−β−δ∫011−ξ−βξ1−d−δe−mt1−ξdξθ1−θ2Xd,m0,T;HqM,where we note that(51)C3=C~1,α,β+C~2,α,βCθLkB1−δ,1−d,td∫0tt−τ−βτ1−δ−de−mt−τdτ=t2−β−δ∫011−ξ−βξ1−d−δe−mt1−ξdξ.
Combining (42) and (49), we obtain the following bound:(52)tde−mtFθ1t−Fθ2tHqM≤tde−mt∫0tPαt−τGθ1τdτ−∫0tPαt−τGθ2τdτHqM+tde−mt∫0tPαt−τ∫0τψτ−ξKθ1ξdξdτ−∫0tPαt−τ∫0τψτ−ξKθ2ξdξdτHqM≤C~1,α,β+C~2,α,βLgt1−β∫011−ξ−βξ−de−mt1−ξdξθ1−θ2Xd,m0,T;HqM+C3t2−β−δ∫011−ξ−βξ1−d−δe−mt1−ξdξθ1−θ2Xd,m0,T;HqM.
From the condition (34), we can verify the following condition:(53)1−β>0,−β>−1,−d>−1,−β−d>−1,β+δ<2,1−d−δ>−1,`−β+1−d−δ>−1.
By using Lemma2, we have two statements immediately:(54)limm⟶∞supt∈0,Tt1−β∫011−ξ−βξ−de−mt1−ξdξ=0,limm⟶∞supt∈0,Tt2−β−δ∫011−ξ−βξ1−d−δe−mt1−ξdξ=0.
From the last two observations, we can find that the positive numberm0 such that(55)C~1,α,β+C~2,α,βLgt1−β∫011−ξ−βξ−de−mt1−ξdξ+C3t2−β−δ∫011−ξ−βξ1−d−δe−mt1−ξdξis less than 1. By applying the Banach fixed point theorem, we know that problem (1) has a unique solution in Xd,m00,T;HqM.☐


## 5. Conclusion

The result of the paper is one of the first works on the topic of memory for equations with Caputo-Fabrizio derivatives. We obtain the following results: first, prove the existence of local solutions. The second is a survey of the global solution. The main technique is to use the Banach fixed point theorem in combination with Sobolev embeddings.



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