R E S E A R C H
Open Access
Existence and uniqueness of the solution
to a coupled fractional diffusion system
Lang Li, Lingyu Jin and Shaomei Fang
**Correspondence:
[email protected] Department of Applied Mathematics, South China Agricultural University, Guangzhou, China
Abstract
In this paper, we consider the initial boundary value problem for a coupled fractional diffusion system. By using eigenfunction expansions and a priori estimates, we establish the existence and uniqueness of the weak solution and then the regularity of the solution.
Keywords: fractional diffusion system; initial boundary value problem; weak solution
1 Introduction
The traditional diffusion equation has commonly been used to describe the phenomenon of Brownian motion. However, it may not be adequate for the motion of the particles if they do not obey the law of Markov diffusion. An example is the diffusion that takes place in a highly heterogeneous aquifer, the probability density may have a heavier tail than the Gaussia density. This phenomenon, called anomalous diffusion, can be well modeled by fractional diffusion equations, including time-fractional diffusion equations, space frac-tional diffusion equations, and time-space fracfrac-tional diffusion equations. The main rea-son is that the fractional derivative is nonlocal, which provides an excellent instrument to model the processes with fractal geometry, hereditary, and non-Markovian properties.
Recently, the fractional diffusion equation has attracted a great deal of interest and more applications of it have been investigated to represent many natural processes in physics chemistry, biology, medicine, food processing, aerodynamics,etc.[–]. By using different methods, many authors have discussed the mathematical properties of the solution to the fractional differential equation and its numerical result. For example, by using a finite sine transform technique, Agarwal [] obtained the general solution to a fractional diffusion equation and presented some numerical results to show the influence of the fractional orderαon the solution. Meanwhile, Mainardi solved a fractional diffusion equation by using the Laplace transform method []. In [], Eidelman and Kochubei considered the Cauchy problem for fractional differential equations, they constructed the fundamental solution and proved the maximum principle. Furthermore, Luchko established the maxi-mum principle for the generalized fractional differential equations []. Then Luchko ex-tended the result to the generalized multi-term time-fractional diffusion equation []. In [], Sakamoto and Yamamoto established the existence of the weak solution and asymp-totic behavior for fractional diffusion-wave equation and then considered its application to some inverse problems.
In this paper, we extend the existing result to a coupled system and consider the follow-ing coupled fractional diffusion equations
∂tαu(x,t) =Lu(x,t) +F(u,v), () ∂tαv(x,t) =Lv(x,t) +F(u,v), ()
u=v= onx∈∂,t∈(,T], ()
u|t==a(x), v|t==a(x), x∈, ()
whereis a bounded domain inRdwith sufficiently smooth boundary∂,Lis a sym-metric uniformly elliptic operator,F(u,v) andF(u,v) are the forcing terms depending on
uandv, the initial dataa(x) anda(x) are given functions on, andT> is a fixed value. Here∂α
t is the Caputo fractional derivative of orderαand is defined by
∂tαg(t) = (n–α)
t
(t–τ)n–α– d
n
dτng(τ)dτ, n– <α<n,n∈N,
where(·) is the Gamma function.
Since we address fractional diffusion equations, we restrict ourselves to the two cases <α< and <α < . If <α < , this system describes a relaxation process (sub-diffusion). If <α< , an oscillation may occur (super-diffusion). Ifα= andL=, it corresponds to a standard reaction-diffusion system. So we can regard the above system as the generalization of reaction-diffusion system to fractional order.
IfL=,F(u,v) =f(u), andF(u,v) =g(v), this system is decoupled and becomes two fractional nonlinear diffusion equations. For the one nonlinear fractional diffusion equa-tion of type () with unknown funcequa-tion, many authors have considered the initial boundary value problem. For example, Gafiychuk and Datsko investigated possible scenarios of pat-tern formations in fractional reaction-diffusion systems with initial boundary (Neumann) conditions []. In [], Luchko established the existence and uniqueness of the solution for the initial boundary (Dirichlet) value problem. In this paper, motivated by the above discussion, we extend it to a more involved and complex system and study the existence of the solution.
To our knowledge, there have not been many papers published concerning the existence and uniqueness of the solution to equations ()-(). The goal of this paper is to prove the existence result as regards the weak solution and establish some results on the regularity of the solution.
2 Preliminaries
LetL(),H(),H() be the usual Sobolev spaces.L˙() is a homogeneous Hilbert space defined as follows
˙
L() =u∈L(),u= onx∈∂.
LetH=L˙()× ˙L() be the product Hilbert space with the norm
φH=
u
L+vL,
for anyφ= (u,v)∈H. DefineV=V×V, where
V=
u∈H()∩H(),u= onx∈∂,
V=
v∈H()∩H(),v= onx∈∂.
Then the function spaceVis a product Hilbert space with the induced norm
φV=
uH+vH,
whereφ= (u,v)∈V.
LetVbe the dual space ofV, we have
V →H=H →V,
where the embeddings above are dense and continuous. We denote byφ,ψthe duality betweenφ∈Vandψ∈V.
Under the assumption thatLis an uniformly elliptic operator and D(–L) =H()∩
H
(), we define
–Lϕn=λnϕn, n= , , . . . ,
where{λn}∞n=is the eigenvalues of –Land{ϕn}∞n=is the corresponding orthonormal eigen-functions which belong toH()∩H
(). It follows that the sequence{ϕn}∞n= is an or-thonormal basis inL(). For anyn∈N,λ
n> . Without loss of generality, we set
<λ≤λ≤ · · ·.
Then we define
D(–L)γ=
ψ∈L();
∞
n=
λnγ(ψ,ϕn)
<∞
for anyγ ∈R. It follows thatD((–L)γ) is a Hilbert space with the norm
ψD((–L)γ)= ∞
n=
λnγ(ψ,ϕn)
.
SinceD((–L)γ)⊂L() forγ> and (L())=L(), we have
D(–L)γ⊂L() =L()⊂D(–L)γ.
Set (D((–L)γ))=D((–L)–γ). Then it is easy to show thatD((–L)–γ) is a Hilbert space with
the norm
fD((–L)–γ)= ∞
n=
λ–nγ–γf,ϕnγ
.
According to Riesz representation theorem, we have
–γf,ϕnγ = (f,ϕn),
iff ∈L() andϕ
n∈D((–L)γ).
The forcing terms are assumed to satisfy the following assumptions:
(A.)
F(u,v)L()≤K
+uL()+vL()
,
F(u,v)L()≤K
+uL()+vL()
,
(A.)
F(u,v)H()≤K
+uH()+vH()
,
F(u,v)H()≤K
+uH()+vH()
,
(A.)
F(u,v) –F
u,vL()≤Lu–uL()+v–vL()
,
F(u,v) –F
u,vL()≤Lu–uL()+v–vL()
,
for anyt∈[,T],(u,v)∈H,(u,v)∈H.
Now we give some examples to show the assumptions (A.)-(A.) can be satisfied.
Example Suppose thatF(u,v) =u–vandF(u,v) =u–v+A, whereAis an external parameter, it is easy to show the assumptions (A.)-(A.) are satisfied.
Example Suppose thatF(u,v) =u–g(|u|)u–vandF(u,v) =v–u, whereg(r) = , if
r>N;g(r) =r–N, ifr≤Nand ≤g(r)≤C, ≤g(r)≤C. We can also verifyFandF satisfy the assumptions; for more details, see pp. of [].
When no confusion arises, we simply denote Lp() by Lp and Hp() by Hp for
Moreover, the Mittag-Leffler function is defined as follows
Eα,β(z) :=
∞
k=
zk
(αk+β), z∈C,
whereα> andβ∈Rare arbitrary constants.
Now, we give three lemmas concerning the Mittag-Leffler function which will be used later.
Lemma [] Let <α< andβ∈Rbe arbitrary.We suppose thatμis such thatπ α/ < μ<min{π,π α}.Then there exists a constant C=C(α,β,μ) > such that
Eα,β(z)≤
C
+|z|, μ≤arg(z)≤π.
Lemma [] Forλ> ,α> ,and positive integer m∈N,we have
dm dtmEα,
–λtα= –λtα–mEα,α–m+
–λtα, t> , and
d dt
tα–Eα,α
–λtα=tα–Eα,α–
–λtα, t≥. Lemma [] For <α< ,we have
Eα,α(–x)≥, x≥.
3 Existence result
In this paper, we are interested in studying the existence of weak solutions. So we first give the definition of weak solutions as follows.
Definition (u,v) is called a weak solution to equations ()-() if equations ()-() hold in H and (u,v)∈V for almost all t∈(,T). Moreover, (u,v)∈C([,T];D((–L)–γ))×
C([,T];D((–L)–γ)) and satisfy
lim
t→u(·,t) –aD((–L)–γ)=limt→v(·,t) –aD((–L)–γ)=
with someγ > .
Now we state our first main result as follows.
Theorem (i)Let
≤α< ,and the initial data(a,a)∈H.Then there exists a unique
weak solution(u,v)to equations()-()with
(u,v)∈C(,T];V∩C[,T];H, ∂tαu,∂tαv∈C(,T];H, and the following inequalities hold
uH+vH≤C, t∈(,T], and the following inequalities hold
uH+vH≤C, t∈[,T],
∂tαuL+∂tαvL≤C, t∈[,T].
Proof (i) Using the Fourier method, we have eigenfunction expansions of the solution to equations ()-() (p. of []),
Thanks to Lemma , Hölder’s inequality and the assumption (A.), we can deduce
Similarly, we can also obtain
together with equation (), we arrive at
u(·,t)L+v(·,t)
where we used the fact ≤α< . By using Gronwall’s inequality, we have
u(·,t)L+v(·,t)
Under the assumption (A.), it follows that
F(u,v,t)L∞(,T;L)≤C, F(u,v,t)L∞(,T;L)≤C.
From equation (), it is easy to see thatu∈C([,T];L) for∞
n=(a,ϕn)Eα,(–λntα)ϕn(x)
and∞n=(t(F(u,v,τ),ϕn)(t–τ)α–Eα,α(–λn(t–τ)α)dτ)ϕn(x) are convergent inL
uni-formly int∈[,T]. Similarly, we havev∈C([,T];L) from equation (). From equation () and equation (), we then obtain
∂tαu(x,t) = –
Therefore, by Lemma , Lemma and Lemma , we have
+
∞
n=
sup
≤τ≤T
F(u,v,τ),ϕn
–
t
d dτEα,
–λnτα
dτ
≤ aLt–α+F(u,v)
L
+F(u,v,t)
L∞(,T;L)
–Eα,
–λntα
≤ aLt–α+C
K,K,T,α,aL,aL
, t∈(,T],
where we have used the fact
t
τα–Eα,α
–λnταdτ= t
τα–Eα,α
–λnτα
dτ= – λn
t
d dτEα,
–λnτα
dτ.
Similarly, we can also get
∂tαv(·,t)L≤ aLt–α+C
K,K,T,α,aL,aL
, t∈(,T].
From equations ()-(), we also know that
u(·,t)H+v(·,t)H ≤CLu(·,t)L+Lv(·,t)L
≤C∂tαu(·,t)L+∂tαv(·,t)L+F(u,v)L +F(u,v)L
≤Ct–α+C, t∈(,T],
whereCdepends onaL,aL,T,K,K,α.
Owing to ∞n=λn(a,ϕn)Eα,(–λntα)ϕn(x) and ∞n=λn(a,ϕn)Eα,(–λntα)ϕn(x) being
convergent in t∈[δ,T] with any given δ> ,n∞=(F(u,v,t),ϕn)ϕn(x),∞n=(F(u,v,t), ϕn)ϕn(x), ∞n=λn(
t
(F(u,v,τ),ϕn)(t – τ)
α–E
α,α(–λn(t –τ)α)dτ)ϕn(x), and ∞n=λn×
(t(F(u,v,τ),ϕn)(t–τ)α–Eα,α(–λn(t–τ)α)dτ)ϕn(x) being convergent in t∈[,T], we
see that (∂α
t u,∂tαv)∈C((,T];H) and (u,v)∈C((,T];V).
Next we prove the uniqueness.
Taking the duality pairing –γ·,·γ of equations ()-() with ϕn and setting un(t) =
–γu(x,t),ϕnγ,vn(t) =–γv(x,t),ϕnγ, we obtain
∂tαun(t) = –λnun(t) +Fn(t), ()
∂tαvn(t) = –λnvn(t) +Fn(t), ()
un(t) =vn(t) = , t∈(,T], ()
un|t== (a,ϕn), vn|t== (a,ϕn). ()
Due to the existence of the ordinary fractional differential equations and the Lipschitz assumption (A.), we obtain the existence and uniqueness of the solution to equations ()-(). Since{ϕn}∞n=is a complete orthonormal basis inL, we get the uniqueness of the weak solution (u,v) to equations ()-().
Finally, we have to prove
lim
t→
u(·,t) –aL= , lim
t→
In fact,
by using the Lebesgue dominated convergence theorem, we have
lim
Thus we complete the proof of (i).
≤CaH+CF(u,v)
From equations ()-(), it follows that
∂tαu(·,t)L+∂
Next we have to prove
lim
for eachn∈N, by using the Lebesgue dominated convergence theorem, we have
Sinceλn≥Cn
d (e.g.[]), we obtain
S(t)≤C
∞
n=
nγd+ <∞
forγ>d– . Hence, the Lebesgue dominated convergence theorem yields
lim
t→S(t) = .
Thus we complete the proof of (ii).
Remark In the proof of Theorem , we have used the fact≤α< . But we do not know whether it still holds for the case <α<
. The energy estimation for the case <α< is technically more complicated and will be done in the future.
Theorem Let <α< ,the initial data(a,a)∈V and∂tu|t== ,∂tv|t== ,x∈.
Then there exists a unique solution(u,v)to equations()-()with
(u,v)∈C[,T],V, ∂tαu,∂tαv∈C[,T];H, and the following estimates hold
u(·,t)H+v(·,t)H≤C, t∈[,T],
∂tαu(·,t)L+∂tαv(·,t)L≤C, t∈[,T].
Proof By assumption (A.) and equations ()-(), we have
u(·,t)H≤CLu(·,t)
L
≤C
∞
n=
λn(a,ϕn)Eα,
–λntα
+C
∞
n= λn
t
F(u,v,τ),ϕn
(t–τ)α–Eα,α
–λn(t–τ)α
dτ
≤CaH+C
t
F(u,v)
Hdτ
≤C+C
t
u(·,τ)H+v(·,τ)
H
dτ.
Similarly,
v(·,t)H≤C+C
t
u(·,t)
H+v(·,t)
H
dτ.
Combining the above two inequalities, and using Gronwall’s inequality, we have
u(·,t)H+v(·,t)
H≤C
aH,aH,T,K,K,α
Furthermore,
u(·,t)
L∞(,T;H)≤C, v(·,t)L∞(,T;H)≤C.
From equations ()-(), we can also get the estimate
∂tαu(·,t)L+∂
α
tv(·,t)
L≤C.
The proof of uniqueness is similar to the proof of Theorem .
Corollary Let <α< and the initial data a=a= ,then we have
u(·,t)
D((–L)αα–)+v(·,t)D((–L)αα–)≤C, t∈[,T].
Proof From equation (), we have
(–L)αα–u(·,t) L
=
∞
n= λ
α–
α n
tF(u,v,τ),ϕn
(t–τ)α–Eα,α
–λn(t–τ)α
dτ
≤C
∞
n=
sup
≤t≤T
F(u,v,t),ϕn
t (λnτα) α–
α
+λnτα
dτ
.
Since
sup
τ≥ (λnτα)
α–
α
+λnτα
=(α– )
α–
α
α ,
we deduce that
(–L)αα–u(·,t)
L≤CTF(u,v)L∞(,T;L)≤C, t∈[,T].
Similarly,
(–L)αα–v(·,t)
L≤C, t∈[,T].
Corollary Let <α< and the initial data a=a= ,then we have
∂tu(·,t)D((–L)–γ)+∂tv(·,t)D((–L)–γ)≤C, t∈[,T],
forγ >d
.
Proof By Lemma , we have
∂tu(·,t) =
∞
n=
t
F(u,v,τ),ϕn
(t–τ)α–E
α,α–
–λn(t–τ)α
dτ
∂tv(·,t) =
∞
n=
t
F(u,v,τ),ϕn
(t–τ)α–Eα,α–
–λn(t–τ)α
dτ
ϕn(x).
Therefore, by Lemma andλn≥Cn
d, we have
∂tu(·,t)D((–L)–γ)
=
∞
n= λnγ
tF(u,v,τ),ϕn
(t–τ)α–Eα,α–
–λn(t–τ)α
dτ
≤
∞
n= λnγ
sup
≤t≤T
F(u,v,t),ϕn
tτα–Eα,α–
–λnτα
dτ
≤CF(u,v)
L∞(,T;L)
∞
n=
λnγ
tα–
≤CF(u,v)L∞(,T;L)T
α–
∞
n=
ndγ <∞
forγ>d. In the same way, we can also obtain the estimate
∂tu(·,t)
D((–L)–γ)<∞.
Thus we complete the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
Lang Li was partially supported by the NSFC under grant No. 11426069 and No. 61375006. Lingyu Jin was supported by the NSFC under grant No. 11101160. Shaomei Fang was supported by the NSFC under grant No. 11271141. The authors would like to thank the anonymous referee for many helpful comments and suggestions.
Received: 2 October 2015 Accepted: 19 November 2015 References
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