# Classical Solutions to the Initial-Boundary Value Problems for Nonautonomous Fractional Diffusion Equations

Authors: Jia Mu; Yang Liu; Huanhuan Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.


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## Body

## 1. Introduction

Due to the nonlocal kernel of fractional differential operators, the time fractional diffusion equations (FDEs) of order 0 to 1 can describe irregular diffusion phenomena with long tails. In real life, the regular diffusion phenomenon (integer order case) only occurs in a few special cases. Therefore, FDE has attracted the attention of many scholars.Qualitative analysis on FDE is the premise of practical application. At present, the research in this area mainly includes the existence, regularity, and stability of solutions. El-Sayed and Herzallah [1] discussed the maximal regularity of strong solutions of the autonomous fractional nonhomogeneous evolution equations under the condition of Hölder continuity. The existence and uniqueness of the mild solution of the autonomous fractional evolution equations (AFEE) involving almost sectorial operators and the existence of classical solutions under the condition of Hölder continuity were researched by Wang et al. [2]. Other studies on the maximal regularity of classical solutions in the autonomous case in the function space of Hölder continuous functions can refer to [3–5]. It is known that Hölder continuity is a special case of weighted Hölder continuity. Mu et al. [6] studied the existence, maximum regularity, and spatial regularity of classical solutions to the autonomous fractional diffusion equations (AFDE) under the condition of weighted Hölder continuity and extended some results in existing research. Later, the Mittag-Leffler function and eigenfunction expansion were employed by Zhou et al. [7] to study the existence, uniqueness, and regularity of mild solutions to the backward problem of the AFDE in the function space of weighted Hölder continuous functions. For other relevant results, please refer to [8–15].The FDE are often nonautonomous in practical problems, which makes it necessary to research the NFDE. The diffusion coefficient of the NFDE is related not only to the spatial variable but also to the time variable, which brings great difficulties to the research. For example, the diffusion term generates a continuous semigroup in the autonomous case, rather than a two-parameter family of evolution operators in the nonautonomous case. Nevertheless, El-Borai [16] obtained the existence of classical solutions of non-autonomous fractional evolution equations (NFEE) under the condition of Hölder continuity. A new resolvent family concept and a fixed point theorem were used by Debbouche and Baleanu [17] to establish some control results for nonlocal impulsive quasilinear delay integrodifferential systems. Chalishajar et al. [18] used Sadovskii’s fixed point theorem and Banach’s fixed point theorem to study the existence of mild solutions to nonlocal problems of NFEE. In [19], the fractional resolvent family and the fixed point theorem are applied to investigate the global existence of mild solutions to NFEE. Chen et al. [20] applied noncompactness measure and Sadovskii’s fixed point theorem to study the local existence and blow up of mild solutions to the Volterra-type NFEE. For other studies, see [21, 22]. On the basis of the above analysis, it can be found that the regularity of solutions to the NFDE and NFEE need to further study.In this paper, we consider NFDE(1)∂tαux,t+∑p≤2mbpx,tDpux,t=fx,t,x,t∈Ω×I,Dpux,t=0,p≤m,x,t∈∂Ω×I,ux,0=u0x,x∈Ω,where ∂tα is the Caputo fractional partial derivative with respect to t, Ω⊂Rn is a bounded open domain, whose boundary ∂Ω is sufficiently smooth, I=0,TT>0, u0 is the initial data for u. Ax,t=∑p≤2mbpx,tDpux,t, m is a positive integer, multi-index x=x1,x2,…,xn, p=p1,p2,…,pn, p=∑i=1npi, Dp=(∂p1/∂x1p1∂p2/∂x1p2⋯∂pn/∂x1pn). Additionally, the following hypotheses are satisfied:H0The operators Ax,t are uniformly elliptic operator in Ω¯×I¯. That is, there exists a constant C such that(2)Cξ2m≤−1mRe∑p=2mbpx,tξp,x,t∈Ω¯×I¯,where ξ=ξ1,ξ2,…,ξn∈ℝn, ξ2=∑i=1nξi2;H1For t∈I¯, the coefficients bqx,t are smooth functions with respect tox∈Ω¯, and there exists 0<σ≤1 such that(3)bpx,t−bpx,s≤Ct−sσ,forx∈Ω¯,andt,s∈I¯.Firstly, when the inhomogeneous term satisfied the weighted Hölder continuityf and the initial value u0 belonged to DA0, the recursive method is applied to determine the representation of the solution to (1). The existence of a unique classical solution to (1) is proved by virtue of the properties of the probability density function and the evolution operator family. In addition, some estimates of the classical solution, directly connected with the regularity of u0 and f, are carried out. Finally, the continuity of the classical solution to (1) in some fractional power norm is proved and a reasonable estimate is obtained. Theorem 1 extends Theorem 2.2 in [16], where f is Höler continuous.The structure of this paper is as follows. The second section expounds the basic knowledge used later. In the third section, the existence and uniqueness of the classical solution to (1) and its continuity in the sense of fractional power are described, and the corresponding estimates are presented.


## 2. Preliminaries

Throughout this paper, the notationC represents a constant in a particular situation, Γ⋅ denotes the Gamma function, and B⋅,⋅ denotes the Beta function. Let X be a Banach space with the norm ⋅X and γ, β are two constants satisfying 0<γ<β≤1. We define the space of weighted Hölder continuous functions, which is Hölder continuous with the exponent γ and the weight s1−β+γ, by Cβ,γI,X={h:I¯⟶X|h is continuous on I for β∈0,1(or I¯ for β=1), limt⟶0t1−βft exsits for β∈0,1,(4)sup0≤s<t≤Ts1−β+γht−hsXt−sγ<∞,limt⟶0sup0≤s<ts1−β+γht−hsXt−sγ⟶.The norm is(5)hCβ,γ=sup0≤t≤Tt1−βhtX+sup0≤s<t≤Ts1−β+γht−hsXt−sγ.Remark 1.
Ifh is Hölder continuous with exponent γγ∈0,1 on I¯, then h∈C1,γI,X [23]. Because of this, our results could generalize some existing conclusions which need Hölder continuity.
Setutx=ux,t, ftx=fx,t. Define At:D⊂L2Ω⟶L2Ω by(6)Atutx=Ax,tux,t,DAt=D=H2mΩ∩H0mΩ.
Then we turn (1) into the abstract fractional equations(7)Dtαut+Atut=ft,t∈I,u0=u0,in the Hilbert space L2Ω, where Dtα is the Caputo fractional derivative. We say that u:I¯⟶L2Ω is a classical solution of (7), if ut is continuous on I¯, Atut and Dtαu exist and are continuous on I, and (7) is satisfied on I¯.
It is well known that each−Ass∈I¯ generates an analytic evolution family Tt,st≥0. Under the assumptions H0 and H1, there exists a constant k≥0 such that Bt≔At+kI satisfies the following properties [24]:
H2 For all λ satisfying Reλ≤0, the resolvent Rλ;Bt of Bt exists and(8)Rλ;BtL2Ω≤Cλ+1,for each t∈I¯.
H3(9)Bt−BsB−1τL2Ω≤Ct−sσ,for all t,s,τ∈I¯.
Without losing generality, we supposeAt satisfies H2 and H3 in the following sections.
SetUt,s=α∫0∞θζαθTtαθ,sdθ, φ1t,s=t−sα−1At−AsUt−s,s, φn+1t,s=∫stφnt,τφ1τ,sdτ, n∈ℤ+, φt,s=∑n=1∞φnt,s, Vt=−AtA−10−∫0tφt,sAsA−10ds, where t,s∈I¯, ζα is probability density function defined on 0,∞:(10)ζαθ=1αθ−1−1/αραθ−1/α,ραθ=1π∑n=0∞−1n−1θ−αn−1Γnα+1n!sinnπα,θ∈0,∞,and ∫0∞θλζαθdθ=Γλ+1/Γαλ+1 for λ∈−1,∞ [25]. In order to obtain the main results, we need to recall the fractional powers of At [24]. Let us denote by(11)A−qt=1Γq∫0∞sq−1Ts,tds,forq>0,t≥0.
Then we define the fractional power ofAt by Aqt=(A−qt)−1 for q>0, and A0t=I. The following conclusions follow from some results of [16, 23, 24, 26].Lemma 1.
(i)
t−sα−1Ut−s,s and t−sα−1AtUt−s,s are uniformly continuous, where t,s∈I¯, t−s≥ε, and ϵ is an arbitrary positive number;(ii)
AtUt−s,s−Ut−s,tL2Ω≤Ct−sσ−αfor0≤s<t≤T;(iii)
AqsUt,sL2Ω≤C/tαqforq≥0,t>0,s∈I¯;(iv)
AqsTt,sL2Ω≤C/tqforq≥0,t>0,s∈I¯;(v)
φnt,sL2Ω≤Cnt−sσn−1/Γσn, φt,sL2Ω≤Ct−sσ−1, VtL2Ω≤C+Ctσ, φt,s and Vt are uniformly continuous in t,s, and(12)φt,τ−φs,τL2Ω≤Ct−sσ−νs−τν−1,for0≤τ<s<t≤T,Vt−VsL2Ω≤Ct−sσ+Ct−sσ−νsν,for0≤s<t≤T,whereν∈0,σ,(vi)
Aq1tA−q2s≤C,wheret,s∈I¯,0≤q1<q2;(vii)
AtA−1s≤C,wheret,s∈I¯;(viii)
Aqtt≥0 is a closed operator, whose domain DAqt is a Banach space;(ix)
Aq1+q2t=Aq1tAq2t,forq1,q2∈ℝ;(x)
DAq1t⊂DAq2t,for0<q2≤q1.


## 3. Classical Solutions

In the following, we state the main results.Theorem 1.
Iff∈Cβ,γI,L2Ω, u0∈DA0, 0<γ<β≤1, and α+β>1, then (7) has a unique classical solution:(13)ut=u0+∫0tt−sα−1Ut−s,sVsA0u0ds+∫0tt−sα−1Ut−s,sfsds+∫0t∫0τt−sα−1Ut−s,sφs,τfτdτds,fort∈I¯,(14)utL2Ω≤Cmax1,tα+σu0L2Ω+Cmaxtα+β−1,tα+σ+β−1fCβ,γ,fort∈I¯,(15)AtutL2Ω+DtαutL2Ω≤CA0u0L2Ω+Cmaxtβ−1,t2σ+β−1fCβ,γ,fort∈I.Proof.
We set(16)ut=u0+∫0tt−sα−1Ut−s,sVsA0u0ds+∫0tt−sα−1Ut−s,swsds.
Substituting it into (7), by Theorem 2.2 in [16], we get(17)Dαu0+∫0tt−sα−1Ut−s,sVsA0u0ds+Atu0+∫0tt−sα−1Ut−s,sVsA0u0ds=0.
Then formally using Lemma1 of [16], we obtain that(18)ft=Dα∫0tt−sα−1Ut−s,swsds+At∫0tt−sα−1Ut−s,swsds=wt−∫0tt−sα−1AsUt−s,swsds+At∫0tt−sα−1Ut−s,swsds=wt+∫0tφ1t−s,swsds.
Writew0t=ft, wn+1=−∫0tφ1t,swnsds, and wt=∑n=0∞wnt. According to the character of φn and Fubini’s theorem, we have(19)wnt=∫0tφnt,sfsds.
In view of Lemma1, we obtain that(20)∑n=0∞wntL2Ω≤∑n=0∞∫0tCnt−sσn−1Γσnsβ−1dsfCβ,γ≤fCβ,γ∑n=0∞CnBσn,βΓσnTnσ+β−1≤fCβ,γTβ−1Γβ∑n=0∞CTσnΓσn+β=fCβ,γTβ−1ΓβEσ,βCTσ,which implies that ∑n=0∞wnt uniformly converges on I¯ and(21)wt=ft+∫0tφt,sfsds.
Besides,(22)wt−wsL2Ω≤ft−fsL2Ω+∫0sφt,τ−φs,τL2ΩfτL2Ωdτ+∫stφt,τfτL2Ωdτ≤s−1+β−γt−sγfCβ,γ+CfCβ,γ∫0st−sσ−νs−τν−1τβ−1dτ+CfCβ,γ∫stt−τσ−1τβ−1dτ≤s−1+β−γt−sγfCβ,γ+CBν,βt−sσ−νsν+β−1fCβ,γ+Cσsβ−1t−sσfCβ,γ,provided 0<s<t≤T. Using Lemma 1, we deduce that u is continuous on I. In addition, we conclude from Lemma 1 that(23)ut−u0L2Ω≤C∫0tt−sα−11+sσdsu0L2Ω+C∫0tt−sα−1sβ−1ds+∫0t∫0st−sα−1s−τσ−1τβ−1dτdsfCβ,γ≤Ctαα+CBα,σ+1tα+σu0L2Ω+CBα,βtα+β−1+Bσ,βBα,σ+βtα+σ+β−1fCβ,γ,implies that u is continuous at t=0. Next, we show that ut∈D for t∈I. In view of(24)∫0tt−sα−1AtUt−s,sVsA0u0ds=∫0tt−sα−1AtUt−s,tVs−VtA0u0ds+∫0tt−sα−1AtUt−s,s−Ut−s,tVsA0u0ds−∫0∞ζαθTtαθ,tVtA0u0dθ+VtA0u0,we conclude from Lemma 1 and (22) that(25)Atu0+∫0tt−sα−1Ut−s,sVsA0u0dsL2Ω≤AtA−10A0u0L2Ω+C∫0tt−s−1t−sσ+t−sσ−νsνdsA0u0L2Ω+C∫0tt−sσ−11+sσdsA0u0L2Ω+C1+tσA0u0L2Ω≤CA0u0L2Ω+CtσA0u0L2Ω+Ct2σA0u0L2Ω.
That is,u0+∫0tt−sα−1Ut−s,sVsA0u0ds∈D for t∈I. We also know that Atu0+∫0tt−sα−1Ut−s,sVsA0u0ds is continuous for t∈I ([16]). Next, we write(26)∫0tt−sα−1AtUt−s,swsds=∫0tt−sα−1AtUt−s,tws−wtds+∫0tt−sα−1AtUt−s,s−Ut−s,twsds−∫0∞ζαθTtαθ,twtdθ+wt≔I1t+I2t+I3t+I4t.
Thanks to (22) and(27)wtL2Ω≤tβ−1fCβ,γ+∫0tt−sσ−1sβ−1dsfCβ,γ≤tβ−1+Bβ,σtσ+β−1fCβ,γ,drawn from Lemma 1, we come to the conclusion that(28)∫0tt−sα−1AtUt−s,swsdsL2Ω≤C∫0tt−s−1s−1+β−γt−sγ+t−sσ−νsν+β−1+sβ−1t−sσdsfCβ,γ+C∫0tt−sσ−1sβ−1+sσ+β−1dsfCβ,γ+Ctβ−1+tσ+β−1fCβ,γ≤CBγ,β−γtβ−1+Bσ−ν,ν+βtσ+β−1+Bσ,βtσ+β−1+tβ−1+tσ+β−1+Bσ,σ+βt2σ+β−1fCβ,γ.
That is∫0tt−sα−1Ut−s,swsds∈D for t∈ε,T∀ε∈I). Next, we show that A∫0tt−sα−1Ut−s,swsds∈CI,L2Ω. Moreover,(29)I1t−I1s=∫stt−τα−1AtUt−τ,twτ−wtdτ+∫0st−τα−1AtUt−τ,t−s−τα−1AsUs−τ,s⋅wτ−wsdτ+∫0st−τα−1AtUt−τ,tws−wtdτ≔J1t+J2t+J3t,and for t>s, (22) and Lemma 1 imply that(30)J1tL2Ω≤C∫stτ−1+β−γt−τγ−1+t−τσ−ν−1τν+β−1+τβ−1t−τσ−1dτ≤Cs−1+β−γγt−sγ+maxsν+β−1,tν+β−1σ−νt−sσ−ν+sβ−1σt−sσ.
Thenlimt⟶s+J1t=0s∈ε,T,∀ε∈I holds. Furthermore, using Lemma 1 we have(31)t−τα−1AtUt−τ,twτ−wsL2Ω≤Ct−τ−1wτ−wsL2Ω≤Cτ−1+β−γs−τγ−1+s−τσ−ν−1τν+β−1+τβ−1s−τσ−1∈L1I,ℝ.
Thuslimt⟶s+J2t=0s∈ε,T,∀ε∈I could be immediately gotten by Lebesgue’s dominated convergence theorem. Applying Lemma 1, we also find that(32)J3tL2Ω≤∫0∞ζαθTt−sαθ,tdθ−∫0∞ζαθTtαθ,tdθwt−wsL2Ω≤Cwt−wsL2Ω.
Thenlimt⟶s+J3t=0s∈ε,T follows from (22). Next, we write(33)I2t−I2s=∫stt−τα−1AtUt−τ,τ−Ut−τ,twτdτ+∫0st−τα−1AtUt−τ,τ−Ut−τ,t−s−τα−1AsUs−τ,τ−Us−τ,swτdτ≕K1+K2.
Then(34)K1tL2Ω≤C∫stt−τσ−1τβ−1+τσ+β−1dτ≤Cσsβ−1+maxsσ+β−1,tσ+β−1t−sσ,derived by Lemma 1 and (27). That is, limt⟶s+K1t=0s∈ε,T,∀ε∈I. Owing to(35)s−τα−1AsUs−τ,τ−Us−τ,twτL2Ω≤Cs−τσ−1τβ−1+τσ+β−1∈L1I,ℝ,limt⟶s+K2t=0 (s∈ε,T, ε is arbitrary and ε∈I) could be obtained by Lebesgue’s dominated convergence theorem. In addition, Lemma 1 and the formula (22) imply that limt⟶s+I3t=limt⟶s+I4t=0s∈ε,T. When t<s, the above limits are similar. Then by (16) and the properties of wt, using arguments similar to the ones in Theorem 2.2 and Lemma 1 in [16] one can easily obtain that Dtαu exists and is continuous on I, and u satisfies (7). Therefore, one obtains u is a classical solution of (7). It is also easily seen that (14) and (15) hold, by (23), (25), and (28).Remark 2.
Theorem1 extends Theorem 2.2 in [16], where f is Hӧler continuous.Theorem 2.
Iff∈Cβ,γI,L2Ω, u0∈DA0, 0<γ<β≤1, α+β>1, and 0<β1<1−1−β/α, then the classical solution to (7) has the property: Aβ1tut is continuous on I¯, and(36)A1βtutL2Ω≤CDAβ0u0L2Ω+CfCβ,γ.Proof.
The existence of the classical solution could be gotten immediately from Theorem1. Since DA0⊂DAβ0, u0∈DAβ0. In view of Aβ1t=Aβ1−1tAt and Aβ1−1t is a bounded linear operator for t∈I¯, using Theorem 1 we may find Aβ1tut is continuous on I. If α+β>1, then 0<β1<1−1−β/α<β. Next we show that Aβ1tut is continuous at t=0. In fact, note that(37)Aβ1tut−Aβ10u0=Aβ1tA−β0−Aβ1−β0Aβ0u0+∫0tt−sα−1AμsUt−s,sVsA−μtAβ1tA1−β0Aβ0u0ds+∫0tt−sα−1Aβ1−1tAtUt−s,swsds≕P1t+P2t+P3t,where 1−β+β1<μ<1.
It is clear thatH1 implies that limt⟶0P1t=0. We now estimate, using Lemma 1 and (27),(38)P2tL2Ω≤C∫0tt−sα−αμ−11+sσdsAβ0u0L2Ω=Ctα1−μ+Bα1−μ,σ+1tα1−μ+σAβ0u0L2Ω,P3tL2Ω≤∫0tt−sα−1Aβ2sA−β2sAβ1tUt−s,swsds≤C∫0tt−sα−αβ2−1sβ−1+Bβ,σsσ+β−1fCβ,γds≤CBα−αβ2,βtα−αβ2+β−1+Bβ,σBα−αβ2,β+σtα−αβ2+β+σ−1fCβ,γ,where β1<β2<1−1−β/α. These prove limt⟶0P2t=limt⟶0P3t=0. From the above, we see that Aβ1tut is continuous on I¯ and (36) holds.



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