$J_{\alpha}^{\prime}(J_{\alpha}^{\prime})^{-1}J_{\alpha}^{\prime}u=(J_{\alpha}^{\prime})^{-1}(J_{\alpha}^{\prime}J_{\alpha}^{\prime}u)$.
Setting $v:=J_{\alpha}^{\prime}J_{\alpha}^{\prime}u$, we have

$None$
$\partial_{t}^{\alpha}v+Av=J_{\alpha}^{\prime}J_{\alpha}^{\prime}F,\quad v\in
H_{\alpha}(0,T;L^{2}(\Omega)).$

We note that Proposition 2.9 (iii) yields
$J_{\alpha}^{\prime}(J_{\alpha}^{\prime}u)=J^{\alpha}(J_{\alpha}^{\prime}u)$
by $J_{\alpha}^{\prime}u\in L^{2}(0,T;L^{2}(\Omega))$ and that
$J_{\alpha}^{\prime}F\in L^{2}(0,T;L^{2}(\Omega))$ by
$F\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$ and Proposition 2.9 (ii). Moreover
$J_{\alpha}^{\prime}(J_{\alpha}^{\prime}F)\in H_{\alpha}(0,T;L^{2}(\Omega))$
by Propositions 2.9 (iii) and 2.2.

Applying Proposition 2.9 (ii) twice to
$F\in\,^{-2\alpha}H(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$, we obtain
$J_{\alpha}^{\prime}F\in{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$, so that $J_{\alpha}^{\prime}(J_{\alpha}^{\prime}F)\in
L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$. Therefore the application of
Corollary 6.1 to (6.33), yields

$A^{2}v=A^{2}J_{\alpha}^{\prime}J_{\alpha}^{\prime}u\in
L^{2}(0,T;L^{2}(\Omega)).$

By $u\in L^{2}(0,T;L^{2}(\Omega))$, we see that
$J_{\alpha}^{\prime}J_{\alpha}^{\prime}u\in
L^{2}(0,T;\mathcal{D}(A^{2}))\subset L^{2}(0,T;H^{4}(\Omega))$. In terms of
Proposition 2.9 (ii), it follows
$u\in\,^{-2\alpha}{H}(0,T;\mathcal{D}(A^{2}))$. Thus the proof of Proposition
6.2 is complete. $\blacksquare$

§6.5. Initial boundary value problem for multi-term time-fractional partial
differential equations

Let $N\in\mathbb{N}$, $0<\alpha_{1}<\cdots<\alpha_{N}<\alpha<1$, $q_{k}\in
C(\overline{\Omega})$ for $1\leq k\leq N$. We recall that $A$ is defined by
(1.1) and (6.1). We discuss an initial boundary value problem for a multi-term
time-fractional partial differential equation:

$None$
$\partial_{t}^{\alpha}(u-a)+\sum_{k=1}^{N}q_{k}(x)\partial_{t}^{\alpha_{k}}(u-a)+Au=F$

and

$None$ $u-a\in H_{\alpha}(0,T;L^{2}(\Omega)).$

Then we can will prove
Theorem 6.6.
For each $a\in H^{1}_{0}(\Omega)$ and $F\in L^{2}(0,T;L^{2}(\Omega))$, there
exists a unique solution $u\in L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$
to (6.34) - (6.35). Moreover there exists a constant $C>0$ such that

$None$ $\|u\|_{L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}+\|u-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}\leq
C(\|a\|_{H^{1}_{0}(\Omega)}+\|F\|_{L^{2}(0,T;L^{2}(\Omega))})$

for each $a\in H^{1}_{0}(\Omega)$ and $F\in L^{2}(0,T;L^{2}(\Omega))$.

Here the constant $C>0$ depends on $\Omega$, $T$, $A$, $\alpha_{k}$, $q_{k}$
for $1\leq k\leq N$.
In the same way as in Subsections 6.3 and 6.4, we can establish stronger and
weaker regular solutions, but we omit the details.

The arguments are direct within our framework based on
$\partial_{t}^{\alpha}:\,^{-\beta}H(0,T)\,\longrightarrow\,^{-\alpha-\beta}H(0,T)$
and
$\partial_{t}^{\alpha}:\,H_{\alpha+\gamma}(0,T)\,\longrightarrow\,H_{\gamma}(0,T)$
with $\gamma\geq 0$ and $\beta\geq 0$. For fractional ordinary differential
equations, we mentioned the corresponding initial value problem as (5.4) in
Section 5.

The multi-term time-fractional partial differential equations are considered
for example in Kian [15], Li, Imanuvilov and Yamamoto [20], Li, Liu and
Yamamoto [21] for symmetric $A$ which makes arguments simple. Non-symmetric
$A$ contains an advection term $\sum_{j=1}^{d}b_{j}\partial_{j}u$, and so it
is physically more feasible model. However, we do not know the existing
results on the corresponding well-posedness for non-symmetric $A$. In fact,
our method for initial boundary value problems is not restricted to symmetric
$A$. Moreover we can treat weaker and stronger solutions in the same way as in
Subsections 6.3 and 6.4, and we omit the details.

We can easily extend our method to the case where the coefficients $q_{k}$,
$1\leq k\leq N$ and $b_{j}$, $1\leq j\leq d$ and $c$ depend on time.

Proof.
First Step.
We prove the theorem under the assumption that $b_{j}=0$ for $1\leq j\leq d$
and $c\leq 0$ on $\Omega$. By Proposition 6.1 and Lemma 6.2, it suffices to
prove that there exists a unique $u-a\in H_{\alpha}(0,T;L^{2}(\Omega))$
satisfying

$None$
$u(t)-a=S(t)a-a+\int^{t}_{0}K(t-s)F(s)ds+\sum_{k=1}^{N}\int^{t}_{0}K(t-s)q_{k}\partial_{t}^{\alpha_{k}}(u(s)-a)ds,\quad
0<t<T.$

In this step, we prove
Lemma 6.3.
Let $k=1,...,N$.
(i)

$\|\partial_{t}^{\alpha_{k}}K(t)a\|_{L^{2}(\Omega)}\leq
Ct^{\alpha-\alpha_{k}-1}\|a\|_{L^{2}(\Omega)},\quad 0<t<T,\quad a\in
L^{2}(\Omega).$

(ii) For $G\in L^{2}(0,T;L^{2}(\Omega))$, we have $\int^{t}_{0}K(t-s)G(s)ds\in
H_{\alpha_{k}}(0,T;L^{2}(\Omega))$ and

$\partial_{t}^{\alpha_{k}}\int^{t}_{0}K(t-s)G(s)ds=\int^{t}_{0}\partial_{t}^{\alpha_{k}}K(t-s)G(s)ds,\quad
0<t<T.$

Proof of Lemma 6.3.
(i) We set

$K_{m}(t)a:=\sum_{n=1}^{m}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}\quad\mbox{for
$m\in\mathbb{N}$ and $a\in L^{2}(\Omega)$.}$

Then, by Lemma 4.1 (ii), we see that

$\displaystyle\partial_{t}^{\alpha_{k}}K_{m}(t)a=\sum_{n=1}^{m}\partial_{t}^{\alpha_{k}}(t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha}))(a,\varphi_{n})\varphi_{n}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{m}t^{\alpha-\alpha_{k}-1}E_{\alpha,\alpha-\alpha_{k}}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}.$

Therefore, applying (4.4), we obtain

$\|\partial_{t}^{\alpha_{k}}K_{m}(t)a\|_{L^{2}(\Omega)}^{2}=\sum_{n=1}^{m}t^{2(\alpha-\alpha_{k}-1)}|E_{\alpha,\alpha-\alpha_{k}}(-\lambda_{n}t^{\alpha})|^{2}|(a,\varphi_{n})|^{2}$
$None$ $\leq Ct^{2(\alpha-\alpha_{k}-1)}\sum_{n=1}^{m}|(a,\varphi_{n})|^{2}.$

Letting $m\to\infty$, we complete the proof of part (i).
(ii) By Proposition 4.2, we can readily prove $\int^{t}_{0}K(t-s)G(s)ds\in
H_{\alpha}(0,T;L^{2}(\Omega))\subset H_{\alpha_{k}}(0,T;L^{2}(\Omega))$.
Noting that $G(x,\cdot)\in L^{2}(0,T)$ for almost all $x\in\Omega$, in terms
of Lemma 4.1 (iii), we have

$\partial_{t}^{\alpha_{k}}\int^{t}_{0}K_{m}(x,t-s)G(x,s)ds=\int^{t}_{0}\partial_{t}^{\alpha_{k}}K_{m}(x,t-s)G(x,s)ds,\quad
x\in\Omega,\,0<t<T$

for $m\in\mathbb{N}$.

Similarly to (6.38), we esimate

$\displaystyle\left\|\int^{t}_{0}(\partial_{t}^{\alpha_{k}}K_{m}-\partial_{t}^{\alpha_{k}}K)(t-s)G(\cdot,s)ds\right\|_{L^{2}(\Omega)}\leq\int^{t}_{0}\|(\partial_{t}^{\alpha_{k}}K_{m}-\partial_{t}^{\alpha_{k}}K)(t-s)G(\cdot,s)\|_{L^{2}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}(t-s)^{\alpha-\alpha_{k}-1}\left(\sum_{n=m+1}^{\infty}|(G(s),\varphi_{n})_{L^{2}(\Omega)}|^{2}\right)^{\frac{1}{2}}ds.$

The Young inequality on the convolution yields

$\displaystyle\left\|\int^{t}_{0}(\partial_{t}^{\alpha_{k}}K_{m}-\partial_{t}^{\alpha_{k}}K)(t-s)G(\cdot,s)ds\right\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\|s^{\alpha-\alpha_{k}-1}\|_{L^{1}(0,T)}\left\|\left(\sum_{n=m+1}^{\infty}|(G(s),\varphi_{n})_{L^{2}(\Omega)}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{2}(0,T)}$
$\displaystyle\leq$ $\displaystyle
CT^{\alpha-\alpha_{k}}\left(\int^{T}_{0}\sum_{n=m+1}^{\infty}|(G(s),\varphi_{n})_{L^{2}(\Omega)}|^{2}ds\right)^{\frac{1}{2}}\,\longrightarrow\,0$

as $m\to\infty$ by $G\in L^{2}(0,T;L^{2}(\Omega))$. Therefore,

$None$
$\lim_{m\to\infty}\int^{t}_{0}\partial_{t}^{\alpha_{k}}K_{m}(t-s)G(s)ds=\int^{t}_{0}\partial_{t}^{\alpha_{k}}K(t-s)G(s)ds\quad\mbox{in
$L^{1}(0,T;L^{2}(\Omega))$}.$

Similarly, using (4.4), we can verify

$None$
$\lim_{m\to\infty}\int^{t}_{0}K_{m}(t-s)G(s)ds=\int^{t}_{0}K(t-s)G(s)ds\quad\mbox{in
$L^{1}(0,T;L^{2}(\Omega))$}.$

Choosing $\gamma>\frac{1}{2}$, we see that
$L^{1}(0,T)\subset\,^{-\gamma}H(0,T)$ and the series in (6.40) converges also
in $\,{}^{-\gamma}H(0,T;L^{2}(\Omega))$. Since
$\partial_{t}^{\alpha_{k}}:\,^{-\gamma}H(0,T)\,\longrightarrow\,^{-\gamma-\alpha_{k}}H(0,T)$
is an isomorphism by Theorem 2.1, we obtain

$\lim_{m\to\infty}\partial_{t}^{\alpha_{k}}\int^{t}_{0}K_{m}(t-s)G(s)ds=\partial_{t}^{\alpha_{k}}\int^{t}_{0}K(t-s)G(s)ds\quad\mbox{in
$\,{}^{-\gamma-\alpha_{k}}H(0,T;L^{2}(\Omega))$}.$

Combining this with (6.39), we finish the proof of Lemma 6.3. $\blacksquare$

Second Step.
We will prove the unique existence of $u$ to (6.37) by the contraction mapping
theorem. We set

$Qu(t):=S(t)a+\int^{t}_{0}K(t-s)F(s)ds+\sum_{k=1}^{N}\int^{t}_{0}K(t-s)q_{k}\partial_{s}^{\alpha_{k}}(u(s)-a)ds,\quad
0<t<T.$

By Lemma 6.2, we see

$None$ $S(t)a-a,\,\int^{t}_{0}K(t-s)F(s)ds\in
H_{\alpha}(0,T;L^{2}(\Omega))\quad\mbox{for $a\in H^{1}_{0}(\Omega)$ and $F\in
L^{2}(0,T;L^{2}(\Omega))$}.$

We estimate $\sum_{k=1}^{N}\|Qu-Qv\|_{H_{\alpha_{k}}(0,T:L^{2}(\Omega))}$ for
$u-a,\,v-a\in H_{\alpha_{k}}(0,T;L^{2}(\Omega))$. To this end, we show

$None$
$\left\|\partial_{t}^{\alpha_{j}}\int^{t}_{0}K(t-s)w(s)ds\right\|_{L^{2}(\Omega)}\leq
C\int^{t}_{0}(t-s)^{\alpha-\alpha_{j}-1}\|w(s)\|_{L^{2}(\Omega)}ds,\quad 1\leq
j\leq N,\,0<t<T.$

Verification of (6.42).
The application of Lemma 6.3 (ii) and (i) yields

$\displaystyle\left\|\partial_{t}^{\alpha_{j}}\int^{t}_{0}K(t-s)w(s)ds\right\|_{L^{2}(\Omega)}=\left\|\int^{t}_{0}\partial_{t}^{\alpha_{j}}K(t-s)w(s)ds\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\int^{t}_{0}\|\partial_{t}^{\alpha_{j}}K(t-s)w(s)\|_{L^{2}(\Omega)}ds\leq
C\int^{t}_{0}(t-s)^{\alpha-\alpha_{j}-1}|w(s)|_{L^{2}(\Omega)}ds.$

Thus (6.42) follows directly. $\blacksquare$

We note that if $u-a,v-a\in H_{\alpha_{N}}(0,T;L^{2}(\Omega))\subset
H_{\alpha_{k}}(0,T;L^{2}(\Omega))$ with $1\leq k\leq N$, then
$(u-a)-(v-a)=u-v\in H_{\alpha_{k}}(0,T;L^{2}(\Omega))$ for $1\leq k\leq N$.
Therefore, using $q_{k}\in L^{\infty}(\Omega)$ for $1\leq k\leq N$, we have

$\displaystyle\|\partial_{t}^{\alpha_{j}}(Qu(t)-Qv(t))\|_{L^{2}(\Omega)}=\left\|\sum_{k=1}^{N}\partial_{t}^{\alpha_{j}}\int^{t}_{0}K(t-s)q_{k}\partial_{s}^{\alpha_{k}}(u(s)-v(s))ds\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{N}\left\|\partial_{t}^{\alpha_{j}}\int^{t}_{0}K(t-s)q_{k}\partial_{s}^{\alpha_{k}}(u(s)-v(s))ds\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{N}\int^{t}_{0}(t-s)^{\alpha-\alpha_{j}-1}\|\partial_{s}^{\alpha_{k}}(u(s)-v(s))\|_{L^{2}(\Omega)}ds.$

Summing up over $j=1,...,N$ and applying $(t-s)^{\alpha-\alpha_{j}-1}\leq
C(t-s)^{\alpha-\alpha_{N}-1}$ for $1\leq j\leq N$, we reach

$\sum_{j=1}^{N}\|\partial_{t}^{\alpha_{j}}(Qu(t)-Qv(t))\|_{L^{2}(\Omega)}$
$None$ $\leq
C\int^{t}_{0}(t-s)^{\alpha-\alpha_{N}-1}\sum_{k=1}^{N}\|\partial_{s}^{\alpha_{k}}(u(s)-v(s))\|_{L^{2}(\Omega)}ds.$

Hence, applying (6.42) for estimating
$\sum_{j=1}^{N}\|\partial_{t}^{\alpha_{j}}(Q^{2}u(t)-Q^{2}v(t))\|_{L^{2}(\Omega)}$,
we obtain

$\displaystyle\sum_{j=1}^{N}\|\partial_{t}^{\alpha_{j}}(Q^{2}u(t)-Q^{2}v(t))\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}(t-s)^{\alpha-\alpha_{N}-1}\sum_{k=1}^{N}\|\partial_{s}^{\alpha_{k}}(Qu(s)-Qv(s))\|_{L^{2}(\Omega)}ds.$

Substituting (6.43) and exchanging the order of the integral, we obtain

$\displaystyle\sum_{k=1}^{N}\|\partial_{t}^{\alpha_{k}}(Q^{2}u(t)-Q^{2}v(t))\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C^{2}\int^{t}_{0}(t-s)^{\alpha-\alpha_{N}-1}\left(\int^{s}_{0}(t-\xi)^{\alpha-\alpha_{N}-1}\sum_{k=1}^{N}\|\partial^{\alpha_{k}}_{\xi}(u-v)(\xi)\|_{L^{2}(\Omega)}d\xi\right)ds$
$\displaystyle=$ $\displaystyle
C^{2}\int^{t}_{0}\left(\int^{t}_{\xi}(t-s)^{\alpha-\alpha_{N}-1}(t-\xi)^{\alpha-\alpha_{N}-1}ds\right)\sum_{k=1}^{N}\|\partial^{\alpha_{k}}_{\xi}(u-v)(\xi)\|_{L^{2}(\Omega)}d\xi$
$\displaystyle=$
$\displaystyle\frac{C^{2}\Gamma(\alpha-\alpha_{N})^{2}}{\Gamma(2(\alpha-\alpha_{N}))}\int^{t}_{0}(t-\xi)^{2\alpha-2\alpha_{N}-1}\sum_{k=1}^{N}\|\partial^{\alpha_{k}}_{\xi}(u-v)(\xi)\|_{L^{2}(\Omega)}d\xi.$

Continuing this estimation, we can reach

$\displaystyle\sum_{k=1}^{N}\|\partial_{t}^{\alpha_{k}}(Q^{m}u(t)-Q^{m}v(t))\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{(C\Gamma(\alpha-\alpha_{N}))^{m}}{\Gamma(m(\alpha-\alpha_{N}))}\int^{t}_{0}(t-s)^{m(\alpha-\alpha_{N})-1}\sum_{k=1}^{N}\|\partial_{s}^{\alpha_{k}}(u-v)(s)\|_{L^{2}(\Omega)}ds,\quad
0<t<T,\,m\in\mathbb{N}.$

Consequently, the Young inequality yields

$\displaystyle\left(\int^{T}_{0}\left(\sum_{k=1}^{N}\|\partial_{t}^{\alpha_{k}}(Q^{m}u(t)-Q^{m}v(t))\|_{L^{2}(\Omega)}\right)^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{(C\Gamma(\alpha-\alpha_{N}))^{m}}{\Gamma(m(\alpha-\alpha_{N}))}\left\|s^{m(\alpha-\alpha_{N})-1}\,*\,\sum_{k=1}^{N}\|\partial_{s}^{\alpha_{k}}(u-v)(s)\|_{L^{2}(\Omega)}\right\|_{L^{2}(0,T)}$
$\displaystyle\leq$
$\displaystyle\frac{(C\Gamma(\alpha-\alpha_{N}))^{m}}{\Gamma(m(\alpha-\alpha_{N}))}\frac{T^{m(\alpha-\alpha_{N})}}{m(\alpha-\alpha_{N})}\left(\int^{T}_{0}\left(\sum_{k=1}^{N}\|\partial_{s}^{\alpha_{k}}(u-v)(s)\|_{L^{2}(\Omega)}\right)^{2}ds\right)^{\frac{1}{2}}.$

For estimating the right-hand side of the above inequality, we find a constant
$C_{N}>0$ such that

$C_{N}^{-1}\sum_{k=1}^{N}|\xi_{k}|^{2}\leq\left(\sum_{k=1}^{N}|\xi_{k}|\right)^{2}\leq
C_{N}\sum_{k=1}^{N}|\xi_{k}|^{2}$

for all $\xi_{1},...,\xi_{N}\in\mathbb{R}$, we obtain

$\displaystyle\sum_{k=1}^{N}\|\partial_{t}^{\alpha_{k}}(Q^{m}u-Q^{m}v)\|_{L^{2}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\frac{(C\Gamma(\alpha-\alpha_{N})T^{\alpha-\alpha_{N}})^{m}}{\Gamma(m(\alpha-\alpha_{N})+1)}\sum_{k=1}^{N}\|\partial_{t}^{\alpha_{k}}(u-v)\|_{L^{2}(0,T;L^{2}(\Omega))},$

that is,

$\|Q^{m}u-Q^{m}v\|_{H_{\alpha_{N}}(0,T;L^{2}(\Omega))}\leq\frac{(C\Gamma(\alpha-\alpha_{N})T^{\alpha-\alpha_{N}})^{m}}{\Gamma(m(\alpha-\alpha_{N})+1)}\|u-v\|_{H_{\alpha_{N}}(0,T;L^{2}(\Omega))}$

for all $u-a,v-a\in H_{\alpha_{N}}(0,T;L^{2}(\Omega))$ and $m\in\mathbb{N}$.

By the asymptotic behavior of the gamma function, we know

$\lim_{m\to\infty}\frac{(C\Gamma(\alpha-\alpha_{N})T^{\alpha-\alpha_{N}})^{m}}{\Gamma(m(\alpha-\alpha_{N})+1)}=0,$

and choosing $m\in N$ sufficiently large, we see that
$Q^{m}:\,H_{\alpha_{N}}(0,T;L^{2}(\Omega))\,\longrightarrow\,H_{\alpha_{N}}(0,T;L^{2}(\Omega))$
is a contraction. Thus we have proved the unique existence of a solution $u$
to (6.37) such that $u-a\in H_{\alpha_{N}}(0,T;L^{2}(\Omega))$, which
completes the proof of Theorem 6.6, provided that $b_{j}=0$ for $1\leq j\leq
d$ and $c\leq 0$ in $\Omega$.

For general $b_{j},c$, setting

$-Lv(x):=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v)(x)$

in place of (6.6), we replace (6.37) by

$u(t)-a=S(t)a-a+\int^{t}_{0}K(t-s)\sum_{j=1}^{d}b_{j}\partial_{j}u(s)ds+\int^{t}_{0}K(t-s)cu(s)ds$
$None$
$+\int^{t}_{0}K(t-s)F(s)ds+\sum_{k=1}^{N}\int^{t}_{0}K(t-s)\partial_{s}^{\alpha_{k}}(u(s)-a)ds,\quad
0<t<T$

and argue similarly. For the estimation of the second term on the right-hand
side of (6.44), in view of Theorem 3.4, we see

$\displaystyle\partial_{t}^{\alpha_{k}}\int^{t}_{0}K(t-s)b_{j}\partial_{j}u(s)=\partial_{t}^{\alpha_{k}}\int^{t}_{0}K(s)b_{j}\partial_{j}u(t-s)ds$
$\displaystyle=$
$\displaystyle\int^{t}_{0}K(s)b_{j}\partial_{t}^{\alpha_{k}}\partial_{j}u(t-s)ds=\int^{t}_{0}L^{\frac{1}{2}}K(s)L^{-\frac{1}{2}}(b_{j}\partial_{j}\partial_{t}^{\alpha_{k}}u(t-s))ds,$

and so we can apply Lemma 6.1 (iii) to argue similarly. Here we omit the
details. Thus the proof of Theorem 6.6 is complete. $\blacksquare$

##  7\. Application to an inverse source problem: illustrating example

We construct our framework for fractional derivatives. In this article, within
the framework we study initial value problems for fractional ordiary
differential equations and initial boundary value problems for fractional
partial differential equations which are classified into so called ”direct
problems”. On the other hand, ”inverse problems” are important where we are
required to determine some of initial values, boundary values, order $\alpha$,
and coefficients in the fractional differential equations by observation data
of solution $u$.

The inverse problem is indispensable for accurate modelling for analyzing
phenomena such as anomalous diffusion. After relevant studies of inverse
problems, we can identify coefficicients, etc. to determine equations
themselves and can proceed to initial value problems and initial boundary
value problems. Thus the inverse problem is the premise for the study of the
forward problem.

Actually, the main purpose of the current article is to not only establish a
theory for direct problems concerning time-fractional differential equations,
but also apply the theory to inverse problems. The inverse problems are very
various and here we discuss only one inverse problem in order to illustrate
how our framework for fractional derivatives works.

Let $0<\alpha<1$ and $A$ be the same elliptic operator as the previous
sections, that is, defined by (6.1) and (6.4). We consider

$None$
$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}u+Au=\mu(t)f(x)\quad\mbox{in
${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$},\\\ &u\in
L^{2}(0,T;L^{2}(\Omega)),\end{array}\right.$

where $\mu\in{{}^{-\alpha}{H}}(0,T)$, $\not\equiv 0$ and $f\in L^{2}(\Omega)$.
Inverse source problem.
Let $\theta\in L^{2}(\Omega)$, $\not\equiv 0$ be arbitrarily chosen, and let
$f\in L^{2}(\Omega)$ be given. Then determine
$\mu=\mu(t)\in{{}^{-\alpha}{H}}(0,T)$ by data

$None$ $\int_{\Omega}u(x,t)\theta(x)dx,\quad 0<t<T.$

By Theorem 6.5, we know that the solution $u$ exists uniquely and $u\in
L^{2}(0,T;L^{2}(\Omega))\cap{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$. Therefore the data (7.2) are well-defined in
$L^{2}(0,T)$. As other choice of data in the case where the spatial dimensions
$d\leq 3$, we can choose: $u(x_{0},t)$ for $0<t<T$ with fixed point
$x_{0}\in\Omega$. By $1\leq d\leq 3$, the Sobolev embedding implies
$u\in{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega))\subset{{}^{-\alpha}{H}}(0,T;C(\overline{\Omega}))$
and so $u(x_{0},\cdot)\in{{}^{-\alpha}{H}}(0,T)$, which implies that data are
well-defined in ${{}^{-\alpha}{H}}(0,T)$. However we are restricted to data
(7.2).

Data (7.2) are spatial average values of $u(x,t)$ with the weight function
$\theta(x)$. We can choose $\theta(x)$ in (7.2) such that supp $\theta$ is
concentrating near one fixed point $x_{0}\in\Omega$, which means that data are
mean values of $u(\cdot,t)$ in a neighborhood of $x_{0}$.

If $\alpha>\frac{1}{2}$, then ${{}^{\alpha}{H}}(0,T)\subset C[0,T]$ by the
Sobolev embedding, and so the space ${{}^{-\alpha}{H}}(0,T)$ can contain a
linear combination of Dirac delta functions:

$\sum_{k=1}^{N}q_{k}\delta_{t_{k}}(t),$

where $q_{k}\in\mathbb{R},\neq 0$, $t_{j}\in(0,T)$ for $1\leq k\leq N$ and
${}_{H^{-1}(\Omega)}<\delta_{t_{k}},\,\varphi>_{H^{1}_{0}(\Omega)}=\varphi(t_{k})$
for all $\varphi\in C^{\infty}_{0}(0,T)$. Then our inverse source problem is
concerned with determination of $N$, $q_{k}$, $t_{k}$ for $1\leq k\leq N$.

Now we are ready to state
Theorem 7.1 (uniqueness).
We assume that $f$ satisfies

$None$ $\int_{\Omega}\theta(x)f(x)dx\neq 0.$

If

$\int_{\Omega}\theta(x)u(x,t)dx=0,\quad 0<t<T,$

then $\mu=0$ in ${{}^{-\alpha}{H}}(0,T)$.
Proof.
The following uniqueness is known: Let $v\in H_{\alpha}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$ satisfy

$None$ $\partial_{t}^{\alpha}v+Av=g(t)f(x),\quad 0<t<T,$

where $g\in L^{2}(0,T)$ and $f\in L^{2}(\Omega)$. Under assumption (7.3),
condition

$None$ $\int_{\Omega}\theta(x)v(x,t)dx=0,\quad 0<t<T,$

yields $g=0$ in $L^{2}(0,T)$.

This uniqueness for $g\in L^{2}(0,T)$ can be proved by combining Duhamel’s
principle and the uniqueness result for initial boundary value problem for
fractional partial differential equation with $f=0$. We can refer also to pp.
411-464 by Li, Liu and Yamamoto in a handbook [17], Sakamoto and Yamamoto
[32]. In particular, Jiang, Li, Liu and Yamamoto [13], which establishes the
uniqueness for the case of $f=0$ with non-symmetric $A$. Thus the uniqueness
in the inverse source problem of determining $g$ is classical within the
caregory of $L^{2}(0,T)$, and so we omit the proof.

Now we can readily reduce the proof of Theorem 7.1 to the case of $g\in
L^{2}(0,T)$. We set $v:=J_{\alpha}^{\prime}u$. Then, by Propositions 2.5 and
2.9 (iii), we see that $v\in H_{\alpha}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$ satisfy

$\partial_{t}^{\alpha}v+Av=(J_{\alpha}^{\prime}\mu)(t)f(x)\quad\mbox{in
$L^{2}(0,T;L^{2}(\Omega))$},$

where $J_{\alpha}^{\prime}\mu\in L^{2}(0,T)$.

Noting by Proposition 2.9 (ii) that $\int_{\Omega}\theta(x)u(x,\cdot)dx\in
L^{2}(0,T)$, by (7.2) we obtain

$\displaystyle
0=J_{\alpha}^{\prime}\left(\int_{\Omega}\theta(x)u(x,\cdot)dx\right)=J^{\alpha}\left(\int_{\Omega}\theta(x)u(x,\cdot)dx\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}\left(\int_{\Omega}\theta(x)u(x,s)dx\right)ds=\int_{\Omega}\theta(x)(J^{\alpha}u)(x,t)dx=0,\quad
0<t<T.$

That is, we reach (7.5), which yields $J_{\alpha}^{\prime}\mu=0$ in $(0,T)$ by
the uniqueness in the inverse problem in the case of
$J_{\alpha}^{\prime}\mu\in L^{2}(0,T)$. Since
$J_{\alpha}^{\prime}:{{}^{-\alpha}{H}}(0,T)\longrightarrow L^{2}(0,T)$ is
injective by Proposition 2.9 (ii), we obtain that $\mu=0$ in
${{}^{-\alpha}{H}}(0,T)$. Thus the proof of Theorem 7.1 is complete.
$\blacksquare$

##  8\. Concluding remarks

8.1. Main messages

(A) We establish fractional derivatives as operators in subspaces of the
Sobolev-Slobodecki spaces, whose orders are not necessarily integer nor
positive, and consider fractional derivatives in spaces of distribution.
Accordingly we extend classes of functions of which we take fractional
derivatives.

In such distribution spaces, we can justify the fractional calculus, and we
can argue as if all the functions under consideration would be in
$L^{1}(0,T)$. Such a typical example is a fractional derivative of a Heaviside
function in Example 2.4.

(B) More importantly, with our framework of fractional calculus, we can
construct a fundamental theory for linear fractional differential equations,
which is easily adjusted to weak solutions and strong solutions, and slso
classical solutions.

We intend this article to be introductory accounts aiming at opertator
theoretical treatments of time fractional partial differential equations.
Comprehensive descriptions should require more works and so this artcile can
provide the essence of foundations. Next we give some summary and prospects.

8.2. What we have done

(i) In Section 2, we introduced Sobolev spaces $H_{\alpha}(0,T)$ as subspace
of the Sobolev-Slobodecki spaces $H^{\alpha}(0,T)$, and we defined a
fractional derivative $\partial_{t}^{\alpha}$ as an isomorphism from
$H_{\alpha}(0,T)$ to $L^{2}(0,T)$, and finally as isomorphisms from
$H_{\alpha+\beta}(0,T)\,\longrightarrow\,H_{\beta}(0,T)$ and
$\,{}^{-\beta}H(0,T)\,\longrightarrow\,^{-\alpha-\beta}H(0,T)$ for $\alpha>0$
and $\beta\geq 0$, where $\,{}^{-\beta}H(0,T)$ and
$\,{}^{-\alpha-\beta}H(0,T)$ are defined by the duality and are subspace of
the distribution space. The key for the definition of $\partial_{t}^{\alpha}$
is an operator theory, and we always attach the fractional derivatives with
their domains. The extension procedure is governed by the Riemann-Liouville
fractional integral operator $J^{\alpha}$, and $\partial_{t}^{\alpha}$ is
defined by the inverse to $J^{\alpha}$ with suitable domain.

(ii) In Section 3, we established several basic properties of
$\partial_{t}^{\alpha}$, which are naturally expected to hold as formulae of
derivatives. We apply some of them to fractional differential equations.

(iii) In Sections 5 and 6, we formulate initial value problems for linear
fractional ordinary differential equations and initial boundary value problems
for linear fractional partial differential equations to prove the well-
posedness and the regularity of solutions within two categories: weak solution
and strong solutoin. Our defined $\partial_{t}^{\alpha}$ enables us to treat
fractional differential equations in a feasible manner.

(iv) Our researches are strongly motivated by inverse problems for fractional
differential equations. The studies of inverse problems can be well based on
the formulation proposed in this article. Taking into consideration the great
variety of inverse problems, in Section 7, we are obliged to be limited to one
simple inverse problem in order to illustrate the effectiveness of our
framework.

8.3. What we will do

Naturally we had to skip many important and intersting issues. Moreover many
researches are going on and here we give up comprehensive prospects for future
research topics. We write some of them, related to our framework.

(I) As fractional partial differential equations, we exclusively consider
evolution equations with elliptic operators of second order for spatial part,
which is classified into fractional diffusion-wave equation. The mathematical
researches should not be limited to such evolution equations. For exmple, the
fractional transport equation is also significant:

$\partial_{t}^{\alpha}u(x,t)+H(x,t)\cdot\nabla u(x,t)=F(x,t).$

One can refer to Luchko, Suzuki and Yamamoto [25] for a maximum principle, and
Namba [28] for viscosity solution.

(II) In this article, we did not apply our framework to nonlinear fractional
differential equations, although necessary steps are prepared. We can refer to
Lucko and Yamamoto [27] as one recent work.

(III) For fractional partial differetial equations, we did not consider non-
homogeneous boundary values in spite of the necessity and the importance. We
need more delicate treatments and see e.g., Yamamoto [33].

(IV) We should study variants of fractional derivatives according to physical
backgrounds. We mention a distributed derivative just as one example:

$\int^{1}_{0}\partial_{t}^{\alpha}u(x,t)d\alpha.$

The work Yamamoto [34] studies similar treatments for a generalized fractional
derivative

$\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}g(t-s)\frac{dv}{ds}(s)ds$

with $0<\alpha<1$ and suitable function $g$, and discusses corresponding
initial value problems for time-fractional ordinary differential equations. It
is another future issue about how much our framework works for such
derivatives.

(V) Our approach is based essentially on the $L^{2}$-space, and our theory is
consistent within $L^{2}$-based Sobolev spaces of any orders
$\alpha\in\mathbb{R}$. Therefore, for example, for treating $L^{1}$-functions
in time as source term $F(x,t)$, we have to embed such functions to an
$L^{2}$-based Sobolev space of negative order. This is not the best possible
way for gaining $L^{p}$-regularity in time with $p\neq 2$, and we should
construct the corresponding $L^{p}$-theory for $\partial_{t}^{\alpha}$. We can
refer to Yamamoto [35] as for a similar work discussing some fundamental
properties in the $L^{p}$-case with $1\leq p<\infty$.

(VI) With the aid of our framework for the direct problem, results on inverse
problems can be expected to be sharpened in view of required regularity for
instance. This should be main future topics after this article.

##  9\. Appendix: Sketches of proofs of Lemmata 6.1 and 6.2

Proof of Lemma 6.1.
(i) By (6.7) and (6.8), noting that

$L^{-\frac{1}{2}}v=\sum_{n=1}^{\infty}\lambda_{n}^{-\frac{1}{2}}(v,\varphi_{n})_{L^{2}(\Omega)}\varphi_{n}$

for $v\in L^{2}(\Omega)$, we obtain

$None$ $L^{\frac{1}{2}}L^{-\frac{1}{2}}v=v\quad\mbox{for $v\in
L^{2}(\Omega)$}$

and

$None$
$(L^{\frac{1}{2}}y,\,z)_{L^{2}(\Omega)}=(y,\,L^{\frac{1}{2}}z)_{L^{2}(\Omega)}\quad\mbox{for
$y,z\in\mathcal{D}(L^{\frac{1}{2}})=H^{1}_{0}(\Omega)$}.$

Next we will prove that there exists a constant $C>0$ such that

$None$
$C^{-1}\|v\|_{H^{-1}(\Omega)}\leq\|L^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}\leq
C\|v\|_{H^{-1}(\Omega)}\quad\mbox{for $v\in L^{2}(\Omega)$}.$

Verification of (9.3).
Let $v\in L^{2}(\Omega)$. Then we have $L^{-\frac{1}{2}}v\in
H^{1}_{0}(\Omega)$ by (6.7) and $L^{\frac{1}{2}}L^{-\frac{1}{2}}v=v$. Hence,
by applying (6.7) again, equalities (9.1) and (9.2) yield

${}_{H^{-1}(\Omega)}<v,\,\psi>_{H^{1}_{0}(\Omega)}=\,_{H^{-1}(\Omega)}<L^{\frac{1}{2}}L^{-\frac{1}{2}}v,\,\psi>_{H^{1}_{0}(\Omega)}$
$\displaystyle=$
$\displaystyle(L^{\frac{1}{2}}L^{-\frac{1}{2}}v,\,\psi)_{L^{2}(\Omega)}=(L^{-\frac{1}{2}}v\,,L^{\frac{1}{2}}\psi)_{L^{2}(\Omega)}\quad\mbox{for
$v\in L^{2}(\Omega)$ and $\psi\in H^{1}_{0}(\Omega)$}.$

Consequently, applying (6.8), we obtain

$\displaystyle\|v\|_{H^{-1}(\Omega)}=\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}\,|_{H^{-1}(\Omega)}<v,\,\psi>_{H^{1}_{0}(\Omega)}|\,=\,\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}|(L^{-\frac{1}{2}}v,\,L^{\frac{1}{2}}\psi)_{L^{2}(\Omega)}|$
$\displaystyle\leq$
$\displaystyle\|L^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}\|L^{\frac{1}{2}}\psi\|_{L^{2}(\Omega)}\leq
C\|L^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}.$

Therefore, the first inequality of (9.3) is proved.

Next we have

$\|L^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}=\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}|(L^{-\frac{1}{2}}v,\,\psi)_{L^{2}(\Omega)}|=\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}|(v,\,L^{-\frac{1}{2}}\psi)_{L^{2}(\Omega)}|.$

Since

$\|L^{-\frac{1}{2}}\psi\|_{H^{1}_{0}(\Omega)}\leq
C\|L^{\frac{1}{2}}(L^{-\frac{1}{2}}\psi)\|_{L^{2}(\Omega)}=C\|\psi\|_{L^{2}(\Omega)},$

by (9.1) we obtain

$\displaystyle\sup_{\|\psi\|_{L^{2}(\Omega)}=1}|(v,\,L^{-\frac{1}{2}}\psi)_{L^{2}(\Omega)}|\leq\sup_{\|\widetilde{\psi}\|_{H^{1}_{0}(\Omega)}\leq
C}|(v,\,\widetilde{\psi})_{L^{2}(\Omega)}|$ $\displaystyle\leq$ $\displaystyle
C\sup_{\|\mu\|_{H^{1}_{0}(\Omega)}=1}|_{H^{-1}(\Omega)}<v,\,\mu\,>_{H^{1}_{0}(\Omega)}|\leq
C\|v\|_{H^{-1}(\Omega)}.$

Hence, the second inequality of (9.3) is proved, and the verification of (9.3)
is complete. $\blacksquare$

Since $L^{2}(\Omega)$ is dense in $H^{-1}(\Omega)$, in view of (9.3), we can
extend $L^{-\frac{1}{2}}:L^{2}(\Omega)\,\longrightarrow\,L^{2}(\Omega)$ to
$L^{-\frac{1}{2}}:H^{-1}(\Omega)\,\longrightarrow\,L^{2}(\Omega)$. More
precisely, for any $v\in H^{-1}(\Omega)$, we choose $v_{n}\in L^{2}(\Omega)$,
$n\in\mathbb{N}$ such that $\lim_{n\to\infty}v_{n}=v$ in $H^{-1}(\Omega)$.
Then (9.3) implies
$\lim_{m,n\to\infty}\|L^{-\frac{1}{2}}(v_{m}-v_{n})\|_{L^{2}(\Omega)}=0$, that
is, a sequence $\\{L^{-\frac{1}{2}}v_{n}\\}_{n\in\mathbb{N}}\subset
L^{2}(\Omega)$ is a Cauchy sequence and we can define

$L^{-\frac{1}{2}}v:=\lim_{n\to\infty}L^{-\frac{1}{2}}v_{n}\quad\mbox{in
$L^{2}(\Omega)$}.$

Then, by (9.3), the limit is independent of choices of
$\\{v_{n}\\}_{n\in\mathbb{N}}$, and we see that (9.3) holds for $v\in
H^{-1}(\Omega)$. Thus the proof of part (i) of Lemma 6.1 is complete.
$\blacksquare$

(ii) By (6.7), we can directly verify

$K(t)v=L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}\quad\mbox{for $t>0$ and $v\in
L^{2}(\Omega)$}.$

Let $v\in H^{-1}(\Omega)$ be arbitrarily given. Since $L^{2}(\Omega)$ is dense
in $H^{-1}(\Omega)$, there exists a sequence $v_{n}\in L^{2}(\Omega)$,
$n\in\mathbb{N}$ such that $v_{n}\longrightarrow v$ in $H^{-1}(\Omega)$ as
$n\to\infty$. By part (i) and the extended $L^{-\frac{1}{2}}$, estimate (9.3)
holds for $v\in H^{-1}(\Omega)$. Hence,
$L^{-\frac{1}{2}}v_{n}\,\longrightarrow\,L^{-\frac{1}{2}}v$ in
$L^{2}(\Omega)$.

In terms of (6.9), we see that

$L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v_{n}\,\longrightarrow\,L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v\quad\mbox{in
$L^{2}(\Omega)$}$

as $n\to\infty$ for $t>0$. Since
$L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v_{n}=v_{n}$ by $v_{n}\in L^{2}(\Omega)$,
we obtain $v_{n}\,\longrightarrow\,L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v$ in
$L^{2}(\Omega)$ as $n\to\infty$ for $t>0$. Using that $v_{n}\longrightarrow v$
in $H^{-1}(\Omega)$, we reach $L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v=v$ for
$t>0$. Thus the proof of part (ii) is complete. $\blacksquare$

(iii) Let $b_{j}\in C^{1}(\overline{\Omega})$, $1\leq j\leq d$. Then we will
verify

$None$ $\|b_{j}\partial_{j}u\|_{H^{-1}(\Omega)}\leq
C\|u\|_{L^{2}(\Omega)},\quad 1\leq j\leq d\quad\mbox{for all $u\in
L^{2}(\Omega)$}.$

Here and henceforth constants $C>0$ depend on $b_{j}$.

If (9.4) is verified, then we can complete the proof of part (iii) of the
lemma as follows: by combining (9.4) with part (i) of the lemma, we obtain

$\|L^{-\frac{1}{2}}(b_{j}\partial_{j}u)\|_{L^{2}(\Omega)}\leq
C\|b_{j}\partial_{j}u\|_{H^{-1}(\Omega)}\leq C\|u\|_{L^{2}(\Omega)}.$

$\blacksquare$
Verification of (9.4).
By the definition of $b_{j}\partial_{j}u$ as element in $H^{-1}(\Omega)$, in
terms of $b_{j}\in C^{1}(\overline{\Omega})$, we have

$\displaystyle\|b_{j}\partial_{j}u\|_{H^{-1}(\Omega)}=\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}\left|{}_{H^{-1}(\Omega)}<b_{j}\partial_{j}u,\,\psi>_{H^{1}_{0}(\Omega)}\right|$
$\displaystyle=$
$\displaystyle\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}|-(u,\,\partial_{j}(b_{j}\psi))_{L^{2}(\Omega)}|\leq\sup_{\|\psi\|_{H^{1}_{0}(\Omega)}=1}\|u\|_{L^{2}(\Omega)}\|\partial_{j}(b_{j}\psi)\|_{L^{2}(\Omega)}\leq
C\|u\|_{L^{2}(\Omega)}.$

Thus the verification of (9.4), and accordingly the proof of Lemma 6.1 are
complete. $\blacksquare$

Proof of Lemma 6.2.
(i) We set

$S_{m}(t)a:=\sum_{n=1}^{m}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}\quad\mbox{for
$m\in\mathbb{N}$.}$

Proposition 4.1 (i) yields

$\partial_{t}^{\alpha}(S_{m}(t)a-a)=\sum_{n=1}^{m}-\lambda_{n}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}=-L\left(\sum_{n=1}^{m}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}\right),$

that is,

$None$ $\partial_{t}^{\alpha}(S_{m}(t)a-a)+LS_{m}(t)a=0,\quad
0<t<T,\,m\in\mathbb{N}.$

Since

$LS_{m}(t)a=t^{-\alpha}\sum_{n=1}^{m}-\lambda_{n}t^{\alpha}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n}),\quad
0<t<T,$

by (4.4) we estimate

$\displaystyle\|LS_{m}(t)a\|^{2}_{L^{2}(\Omega)}=t^{-2\alpha}\sum_{n=1}^{m}|-\lambda_{n}t^{\alpha}E_{\alpha,1}(-\lambda_{n}t^{\alpha})|^{2}|(a,\varphi_{n})|^{2}$
$\displaystyle\leq$ $\displaystyle
Ct^{-2\alpha}\sum_{n=1}^{m}|(a,\varphi_{n})|^{2},\quad 0<t<T.$

Therefore, $\lim_{m\to\infty}LS_{m}(t)a=LS(t)a$ in $L^{2}(\Omega)$ for fixed
$t>0$. Hence, letting $m\to\infty$ in (9.5), we obtain
$\partial_{t}^{\alpha}(S(t)a-a)+LS(t)a=0$ for $0<t<T$.

Next let $a\in H^{1}_{0}(\Omega)$. Since

$\lambda_{n}^{\frac{1}{2}}(a,\varphi_{n})=(a,\,L^{\frac{1}{2}}\varphi_{n})=(L^{\frac{1}{2}}a,\,\varphi_{n})\quad\mbox{for
$a\in H^{1}_{0}(\Omega)$},$

we have

$\displaystyle
LS(t)a=\sum_{n=1}^{\infty}E_{\alpha,1}(-\lambda_{n}t^{\alpha})\lambda_{n}(a,\varphi_{n})\varphi_{n}=\sum_{n=1}^{\infty}\lambda_{n}^{\frac{1}{2}}E_{\alpha,1}(-\lambda_{n}t^{\alpha})\lambda_{n}^{\frac{1}{2}}(a,\varphi_{n})\varphi_{n}$
$\displaystyle=$ $\displaystyle
t^{-\frac{\alpha}{2}}\sum_{n=1}^{\infty}(\lambda_{n}t^{\alpha})^{\frac{1}{2}}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(L^{\frac{1}{2}}a,\,\varphi_{n})\varphi_{n},$

so that

$\displaystyle\|LS(t)a\|_{L^{2}(\Omega)}\leq
Ct^{-\frac{\alpha}{2}}\left(\sum_{n=1}^{\infty}|\lambda_{n}t^{\alpha}||E_{\alpha,1}(-\lambda_{n}t^{\alpha})|^{2}|(L^{\frac{1}{2}}a,\,\varphi_{n})|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
Ct^{-\frac{\alpha}{2}}\|L^{\frac{1}{2}}a\|_{L^{2}(\Omega)}\leq
Ct^{-\frac{\alpha}{2}}\|a\|_{H^{1}_{0}(\Omega)}$

by (4.4). Therefore, this means that $LS(t)a\in L^{2}(0,T;L^{2}(\Omega))$ for
$a\in H^{1}_{0}(\Omega)$ and

$None$ $\|LS(t)a\|_{L^{2}(0,T;L^{2}(\Omega))}\leq C\|a\|_{H^{1}_{0}(\Omega)}.$

Since $\partial_{t}^{\alpha}(S(t)a-a)+LS(t)a=0$ for $0<t<T$, by (9.6), the
proof of part (i) is complete. $\blacksquare$
(ii) We set

$\left\\{\begin{array}[]{rl}&R(t):=\int^{t}_{0}K(t-s)F(s)ds,\\\
&R_{m}(t):=\int^{t}_{0}K_{m}(t-s)F(s)ds\\\
=&\sum_{n=1}^{m}\left(\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})(F(s),\,\varphi_{n})ds\right)\varphi_{n}(x),\\\
&F_{m}:=\sum_{n=1}^{m}(F,\,\varphi_{n})\varphi_{n}\end{array}\right.$

for $0<t<T$ and $m\in\mathbb{N}$. Then by Proposition 4.2, we can readily
verify

$None$ $\partial_{t}^{\alpha}R_{m}+LR_{m}=F_{m}\quad\mbox{in
$L^{2}(0,T;L^{2}(\Omega))$ for all $m\in\mathbb{N}$.}$

Moreover

$\|LR_{m}(t)\|^{2}_{L^{2}(\Omega)}=\sum_{n=1}^{m}\left|\int^{t}_{0}\lambda_{n}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})(F(s),\,\varphi_{n})ds\right|^{2}.$

Hence,

$\displaystyle\|LR_{m}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}=\sum_{n=1}^{m}\int^{T}_{0}\left|\int^{t}_{0}\lambda_{n}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})(F(s),\,\varphi_{n})ds\right|^{2}dt$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{m}\|(\lambda_{n}s^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}s^{\alpha})\,*\,(F(s),\,\varphi_{n})\|^{2}_{L^{2}(0,T)}$
$\displaystyle\leq$
$\displaystyle\sum_{n=1}^{m}\left(\int^{T}_{0}|\lambda_{n}s^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}s^{\alpha})|ds\right)^{2}\int^{T}_{0}|(F(s),\,\varphi_{n})|^{2}ds$
$\displaystyle\leq$
$\displaystyle\int^{T}_{0}\sum_{n=1}^{m}|(F(s),\,\varphi_{n})|^{2}ds=\|F_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}.$

For the last inequality and the second to last inequality, we applied the
Young inequality on the convolution and (4.10) respectively.

Consequently,

$None$ $\|LR_{m}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}\leq
C\|F_{m}\|^{2}_{L^{2}(0,T;L^{2}(\Omega))},\quad m\in\mathbb{N}.$

Since $F\in L^{2}(0,T;L^{2}(\Omega))$, we see that $LR_{m}\longrightarrow LR$
in $L^{2}(0,T;L^{2}(\Omega))$ as $m\to\infty$. By (9.7) we see

$None$
$\lim_{m\to\infty}\partial_{t}^{\alpha}R_{m}=\lim_{m\to\infty}(F_{m}-LR_{m})=F-LR\quad\mbox{in
$L^{2}(0,T;L^{2}(\Omega))$}.$

Since $\lim_{m\to\infty}R_{m}=R$ in $L^{2}(0,T;L^{2}(\Omega))$, it follows
from (6.25) that
$\lim_{m\to\infty}\partial_{t}^{\alpha}R_{m}=\partial_{t}^{\alpha}R$ in
${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$. Hence (9.9) yields
$\partial_{t}^{\alpha}R=F-LR$ in ${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$.

In view of (9.7) and (9.8), we reach

$\displaystyle\|\partial_{t}^{\alpha}R_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}=\|-LR_{m}+F_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|-LR_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}+\|F_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}\leq
C\|F_{m}\|_{L^{2}(0,T;L^{2}(\Omega))}\quad\mbox{for each $m\in\mathbb{N}$.}$

Letting $m\to\infty$, we reach

$\|\partial_{t}^{\alpha}R\|_{L^{2}(0,T;L^{2}(\Omega))}\leq
C\|F\|_{L^{2}(0,T;L^{2}(\Omega))}.$

Thus the proof of Lemma 6.2 is complete. $\blacksquare$

Acknowledgements.
The author is supported by Grant-in-Aid for Scientific Research (A) 20H00117,
JSPS and by the National Natural Science Foundation of China (Nos. 11771270,
91730303). This paper has been supported by the RUDN University Strategic
Academic Leadership Program. The author thanks Professor Bangti Jin
(University College London) for the careful reading and comments.

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