CHAPTER 2. Well-posedness 31 Hence,
t(α+n−1)kADtnv(t)k ≤Cku0k+C Z t
0
τ(α+n−1)kADnτv(τ)k dτ.
Invite Gronwall’s lemma to obtain
(2.50) t(α+n−1)kADtnv(t)k ≤Cku0k. Now, in view of Lemma 2.3.3 and (2.50), we obtain for i= 1,
kADtnu(t)k ≤Ct−(α+n)ku0k+Ct−(α+n−1)ku0k ≤Ct−(α+n)ku0k. Thus, the relation (2.40) holds for m=n.
Remark 2.3.1. Theorem 2.3.1 shows the regularity property in time of the homogeneous SEM with the initial data u0 ∈L2(Ω). Observe that when α →1−, the presented result is similar to the standard PIDE [75, Theorem 2.3.4].
CHAPTER 2. Well-posedness 32 Also, we have from (2.11)
(2.55) ku(t)k2 ≤Cku0k2.
For 0 ≤ p ≤ 2, q = 0, the estimate (2.51) follows from the interpolation of estimates (2.53) and (2.54). Again, interpolation of (2.54) and (2.55) gives (2.51), for p= 2,0 ≤ q ≤2. Finally, we invite interpolation of (2.53) and (2.55) to obtain (2.51) for 0≤q = p≤2.
Next, to estimatek∂tαu(t)kp for different real numberp, 0≤p≤2, we first show that k∂tαu(t)kp makes sense forp= 2. The estimates of k∂tαu(t)kp for p= 0 are contained in Theorems 2.2.1 and 2.2.3 when the data u0 belongs to ˙H2(Ω) and L2(Ω), respectively.
Then, we apply interpolation of estimates to obtain (2.52).
Putting f = 0 in (2.8) and taking the Caputo fractional derivative of order α, we have
∂tαu(t) =
∞
X
j=1
−λj(u0, φj)Eα,1(−λjtα)φj(x) + ˜Bu(t)
−
∞
X
j=1
Z t 0
λj(t−τ)α−1Eα,α(−λj(t−τ)α)( ˜Bu(τ), φj)dτ
φj(x), (2.56)
where ∂tαE(t)u0 =
∞
X
j=1
−λj(u0, φj)Eα,1(−λjtα)φj(x). Setting
Fj = Z t
0
λj(t−τ)α−1Eα,α(−λj(t−τ)α)( ˜Bu(τ), φj)dτ, using (2.20) and integrating by parts, we first note that
Fj = Z t
0
d
dτEα,1(−λj(t−τ)α)( ˜Bu(τ), φj)dτ
= ( ˜Bu(t), φj)− Z t
0
Eα,1(−λj(t−τ)α)( ˜Bτu(τ), φj)dτ
− Z t
0
Eα,1(−λj(t−τ)α)(B(τ, τ), φj)dτ.
The last term of the right-hand side of (2.56) can be expressed as
∞
X
j=1
Fjφj(x) = ˜Bu(t)−
∞
X
j=1
Z t 0
Eα,1(−λj(t−τ)α)( ˜Bτu(τ) +B(τ, τ)u(τ), φj)dτ
φj(x).
CHAPTER 2. Well-posedness 33 From (2.56), we can write
∂tαu(t) =
∞
X
j=1
−λj(u0, φj)Eα,1(−λjtα)φj(x) + Z t
0
E(t−τ)( ˜Bτu(τ) +B(τ, τ)u(τ))dτ (2.57)
:=S1+S2.
The quantity kAS1k is meaningful only when the data u0 ∈ H˙2(Ω). Following Jin et al. [25, Theorem 2.2, ` = 1], we have kAS1k ≤ Ct−αku0k2. Now we show thatkAS2k makes sense for u0 ∈H˙2(Ω). Since Bt and B are dominated by A(cf. (1.9)), we obtain using (2.11)
kAS2k ≤ Z t
0
kAE(t−τ)( ˜Bτu(τ) +B(τ, τ)u(τ))k dτ
≤C Z t
0
(t−τ)−α
kB˜τu(τ)k+kB(τ, τ)u(τ)k dτ
≤Cku0k2 ≤Ct−αku0k2. It follows from (2.57) that
kA∂tαu(t)k ≤Ct−αku0k2. (2.58)
Further, the L2(Ω)-estimates of ∂tαu(t) for nonsmooth and smooth data u0, are respec- tively (see Theorems 2.2.3 and 2.2.1),
k∂tαu(t)k ≤Ct−αku0k, t >0, (2.59)
and
k∂tαu(t)k ≤Cku0k2. (2.60)
Interpolation of the estimates (2.59) and (2.60) gives (2.52) forp= 0 and 0≤q ≤2. The estimates for 0 ≤p= q ≤2 follow from (2.59) and (2.58). Finally, again interpolation of the estimates (2.60) and (2.58) yields (2.52), for 0 < p < q ≤2. This completes the proof.
Remark 2.4.1. (i) By Theorem 2.4.1, for the homogeneous problem with u0 ∈H01(Ω), we get ku(t)k2+k∂tαu(t)k ≤Ct−α/2ku0k1, t >0, which is an immediate consequence of the interpolation of estimates for q= 0 and q= 2.
(ii) The estimate (2.52) invites the restriction p ≤ q. Such restriction do present for SE (see [25, Theorem 2.2]). This shows that like SE, SEM also has a narrow smoothing property.
CHAPTER 2. Well-posedness 34 In the following, we prove some stability results for the solution of the nonhomoge- neous problem. The first theorem presents stability estimates for the Caputo derivative while the second one describes those for the usual partial derivative of the continuous solution. These results will be useful in the semidiscrete error analysis.
Theorem 2.4.2. Assume that u0 ∈ H˙2(Ω) and f ∈ C1(L2(Ω)). With the assumption (A), we have
(2.61) ku(t)k2 ≤C ku0k2+kfkC1(L2(Ω))
, 0≤t ≤T, and
(2.62) k∂tαu(t)kp ≤C t−αp/2[ku0k2+kf(0)k] +kfkC1(L2(Ω))
, p= 1,2, t >0.
Proof. The proof of (2.61) is given by Theorem 2.2.1 and the proof of (2.62) follows from Theorem 2.4.1.
Remark 2.4.2. We remark that the a priori bound (2.61) remains valid for H¨older continuous source functions f. However, the differentiability of f is needed only to estimate (2.62). The singularity t−αp/2, p= 1,2, onf(0) in (2.62) can be removed if we consider f(0) is an element of the space H˙1(Ω) for p= 1 and f(0) in H˙2(Ω) for p= 2.
Theorem 2.4.3. Assume that u0 ∈ H˙2(Ω) and f ∈ C1( ˙H1(Ω)). With the assumption (A) and p= 1,2, we have for positive time t,
ku0(t)kp ≤C t−1+(2−p)α/2[ku0k2+t(p−1)α/2kf(0)k(p−1)] +tα/2kfkC1( ˙H(p−1)(Ω))
. Proof. To prove the assertion, we decompose the problem (2.1) in two parts as
(2.63)
∂tαv(t) +Av(t) = ˜Bv(t) in Ω, 0< t≤T, 0< α <1, v(t) = 0 on ∂Ω, 0< t≤T,
v(0) =u0(x) in Ω, where ˜Bv(t) = Rt
0 B(t, s)v(s)ds and
(2.64)
∂tαw(t) +Aw(t) = f(t) in Ω, 0< t≤T, 0< α <1, w(t) = 0 on ∂Ω, 0< t≤T,
w(0) = 0 in Ω.
CHAPTER 2. Well-posedness 35 Following Theorem 2.3.1, for smooth initial data we can obtain kv0(t)k ≤Ct−1+αku0k2 and kv0(t)k2 ≤Ct−1ku0k2 for t >0. By interpolation we get
kv0(t)k1 ≤Ct−1+α/2ku0k2, t >0.
(2.65)
Now, the solution for problem (2.64) (cf. [24]) is given by w(t) =
Z t 0
E(tb −s)f(s)ds= Z t
0
E(s)fb (t−s)ds, (2.66)
where the operator E(t) is defined byb E(t)χb =
∞
X
j=1
tα−1Eα,α(−λjtα)(χ, φj)φj(x), t >0, χ∈L2(Ω),
with Eα,α(−λjtα) =P∞ j=0
(−λjtα)j
Γ(jα+α). Differentiating (2.66) with respect to time to have w0(t) =
Z t 0
E(s)fb 0(t−s)ds+E(t)fb (0) = Z t
0
E(tb −s)f0(s)ds+E(t)fb (0).
Invoking Lemma 2.2 [24, p. 565] with p= 1, q= 0, we obtain for t >0 kw0(t)k1 ≤
Z t 0
kE(tb −s)f0(s)k1 ds+kE(t)fb (0)k1
≤ Z t
0
(t−s)−1+α/2kf0(s)k ds+Ct−1+α/2kf(0)k
≤Ctα/2kfkC1(L2)+Ct−1+α/2kf(0)k. (2.67)
In view of (2.65) and (2.67), we get for t >0
ku0(t)k1 ≤ kv0(t)k1+kw0(t)k1 ≤Ct−1+α/2 ku0k2+kf(0)k) +Ctα/2kfkC1(L2). (2.68)
Similarly, putting p= 2, q= 1, in Lemma 2.2 [24, p. 565], we obtain kw0(t)k2 ≤Ctα/2kfkC1( ˙H1)+Ct−1+α/2kf(0)k1, t >0.
Thus, for positive time t,
ku0(t)k2 ≤ kv0(t)k2+kw0(t)k2 ≤Ct−1 ku0k2+tα/2kf(0)k1) +Ctα/2kfkC1( ˙H1). (2.69)
The estimates (2.68)-(2.69) complete the proof of the assertion.
In the following, as applications to Theorems 2.2.1 and 2.2.2, we provide two exam- ples of SEM in one and two space dimensions.
CHAPTER 2. Well-posedness 36 Example 2.1. Let Ω = (0,1) and T > 0 be a fixed value. Consider the SEM (cf. [75])
∂tαu(t)− ∂2u(t)
∂x2 =f(t) + Z t
0
e(t−s)u(s)ds in Ω, 0< t≤T, 0< α <1, u(0, t) = u(1, t) = 0, 0< t≤T,
u(x,0) = sin(πx) in Ω.
Here A = −∂x∂22 and B(t, s) = e(t−s)I. In addition, B(t, s) is dominated by A. Choose the source function f as
f(x, t) = etγ(1−α, t)
Γ(1−α) sin(πx) +π2etsin(πx)−tetsin(πx),
where γ(1−α, t) is an incomplete gamma function [23, 44]. Thenf is a H¨older contin- uous function. The exact solution u is given by u(x, t) =etsin(πx) and Theorems 2.2.1 and 2.2.2 hold.
Example 2.2. Let Ω = (0,1)2. Forα∈(0,1)andT <∞, consider the following IBVP in u(x, y, t):
(2.70)
∂tαu(t)−∆u(t) = f(t) + Z t
0
e(t−s)∆u(s)ds in Ω, 0< t≤T, u(x,0, t) = u(x,1, t) = 0 for 0≤x≤1, 0< t≤T, u(0, y, t) = u(1, y, t) = 0 for 0≤y≤1, 0< t≤T, u(x, y,0) = x(1−x)y(1−y) in Ω,
where ∆ = ∂x∂22 + ∂y∂22. Here we see that A = −∆ and B(t, s) = e(t−s)∆. Consider the source function f as
f(x, y, t) =etγ(1−α, t)
Γ(1−α) x(1−x)y(1−y)
+ 2et(x(1−x) +y(1−y)) + 2tet(x(1−x) +y(1−y)).
Then, it is not so hard to verify that the functionu(x, y, t) = etx(1−x)y(1−y) satisfies problem (2.70) and the estimates of Theorems 2.2.1 and 2.2.2 hold for the solution u.
At the end, we remind our readers that our analysis of (2.1) is still valid when the operator A is a more general elliptic operator of the form:
A=−
d
X
i,j=1
∂
∂xj
cij(x) ∂
∂xi
+c0(x)I.
Here, the coefficients cij = cji and c0 ≥ 0 are smooth functions of the space variable only and cij form a uniformly positive definite matrix on Ω.
CHAPTER 2. Well-posedness 37