1 Introduction

The fractional calculus has already become a powerful tool which describes many nonlinear complex phenomena arising in fluid mechanics, thermodynamics, plasma dynamics, continuum mechanics, quantum mechanics, electrodynamics and biological systems [1, 2]. In particular, the fractional diffusion equations capture well the anomalous diffusion process with continuous time random walks [3, 4].

In this paper, we consider the following initial value problem for the multi-term time-space Caputo-Riesz fractional diffusion equation:

$$\begin{aligned}& \sum_{j=0}^{n-1}a_{j} D^{\alpha_{j}}u(t,x)=-b(-\bigtriangleup)^{\beta }u(t,x), \end{aligned}$$
(1.1)
$$\begin{aligned}& u(0,x)=g(x), \end{aligned}$$
(1.2)

where \(n \geq1 \), \(a_{0}=1 ,a_{i}>0, \alpha_{i}>0, \alpha_{n-1}<\cdots <\alpha_{0} \leq1, b>0, 0<\beta\leq1, x \in R=(-\infty, \infty)\), \(t \geq 0\), the symbol \(D^{\alpha}\) denotes the Caputo-type fractional derivative defined by [5]

$$D^{\alpha}u(t)=\frac{1}{\Gamma(\lceil\alpha\rceil-\alpha) } \int ^{t}_{0}{(t-s)}^{\lceil\alpha\rceil-\alpha-1} u^{(\lceil\alpha\rceil )}(s) \,ds $$

and the symbol \((-\bigtriangleup)^{\beta} \) denotes the fractional Laplacian defined by [6]

$$ (-\bigtriangleup)^{\beta}u(t)=F^{-1}\bigl\{ \vert s \vert ^{2\beta} Fu(s)\bigr\} (t), $$
(1.3)

where F means the Fourier transform.

In fractional calculus the most popular fractional derivatives are Caputo derivative and Riemann-Liouville derivative. Because of the convenience in handling initial conditions, the Caputo fractional derivative has been more widely used in practice [7]. However, the Caputo fractional derivative is usually defined for the continuously differentiable functions [5, 7]. In [8] the authors gave a new definition of the Caputo fractional derivative on a bounded interval in the fractional Sobolev space and proved the maximal regularity of solutions of time fractional diffusion equations. The fractional Laplacian is also a well-known nonlocal operator which plays an important role in the potential theory [9]. The authors of [10] considered the relation between fractional Laplacian and fractional Sovolev space. The fractional Laplacian operator on a bounded interval is defined in terms of the eigenvalues and eigenfunctions of the Laplacian operator [8, 11]. The fractional Laplacian on a unbounded interval is usually defined in the Schwartz space which is too narrow for many important applications. Thus in [12] the solution space of analytical solutions of fractional time-space Caputo-Riesz diffusion equations on an infinite domain was not illustrated and the authors [13] established mild solutions by deriving an equivalent integral equation.

Since multi-term fractional diffusion equations are more flexible than single-term fractional diffusion equations in modeling the anomalous diffusion phenomena, they have often appeared in recent publications [11, 14–17]. By establishing the maximum principle for multi-term time fractional diffusion equations with Caputo derivatives and proving some properties of multivariate Mittag-Leffler functions, the authors [14, 15] studied the well-posedness and the long-time asymptotic behavior. In [17] the authors proved the maximum principle for multi-term time-space Caputo-Riesz fractional diffusion equations and derived the uniqueness and continuous dependence of the solution. The authors of [11] used the Luchko theorem to obtain the analytical solutions for multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a bounded interval. However, to the best of our knowledge, multi-term time-space Caputo-Riesz fractional diffusion equations on an infinite domain have not been considered in the literature yet.

In the present paper, by extending the domain of the fractional Laplacian to a Banach space and using the multivariate Mittag-Leffler function, the analytical solutions of the multi-term fractional diffusion equation (1.1)-(1.2) are obtained. Especially the meaning of the analytical solutions is found.

2 Extension of domain of fractional Laplacian

In this section the domain of the fractional Laplacian operator (1.3) is extended to a Banach space. Firstly we recall the concepts of Lebesgue space and Schwartz space.

Definition 2.1

[18], p.110

The space \(L^{2} \) means the set of all measurable functions \(u:R \rightarrow R \) such that \(\|u\|_{L^{2}} < \infty\), where

$$\Vert u \Vert _{L^{2}}= \int_{R} \bigl\vert u(x) \bigr\vert ^{2} \,dx. $$

Definition 2.2

[18], p.214

The space S means the set of all \(C^{\infty}\) functions \(u:R \rightarrow R \) such that \(\|u\|_{r,q} < \infty\) for all \(r,q=0,1,\ldots\) , where

$$\Vert u \Vert _{r,q}=\sup_{x\in R}\bigl(1+ \vert x \vert ^{r}\bigr)\sum_{m=0}^{q} \bigl\vert u^{(m)}(x) \bigr\vert . $$

Definition 2.3

By \(M_{\beta}\) we mean the completion of the Schwartz space S over R with the norm \(\|\cdot\|_{M_{\beta}} \) defined by

$$ \Vert f \Vert _{M_{\beta}}= \bigl\Vert \vert t \vert ^{2\beta}Ff(t) \bigr\Vert _{L^{2}},\quad f \in S. $$
(2.1)

For any \(f \in M_{\beta}\), there exists a sequence \(\{ f_{m} \in S \}\) such that \(\|f\|_{M_{\beta}}=\lim_{m \to\infty}\|f_{m}\|_{M_{\beta}} \) and \(\|f_{m}-f_{r}\|_{M_{\beta}} \rightarrow0 \) as \(m,r \rightarrow\infty\).

Theorem 2.4

The fractional Laplacian \((-\bigtriangleup)^{\beta} \) is extended to the Banach space \(M_{\beta}\).

Proof

By using the extension principle, we can easily prove the result. □

Theorem 2.5

\(H_{\beta}=\{ f\in L^{2}:|t|^{2\beta}Ff(t)\in L^{2}\} \subset M_{\beta}\).

Proof

Let suppose that \(f \in H_{\beta}\) and \(\epsilon>0 \). Then there exists a real number \(r_{\epsilon}>0 \) such that

$$\int_{|t|>r_{\epsilon}} \vert t \vert ^{4\beta}(Ff)^{2}(t) \,dt < \epsilon^{2}. $$

There exists a function \(g_{\epsilon}\in C_{0}^{\infty}([-r_{\epsilon},r_{\epsilon}]) \) such that

$$\int_{|t|< r_{\epsilon}} (Ff-g_{\epsilon})^{2}(t)\,dt < \frac{\epsilon ^{2}}{r_{\epsilon}^{4\beta}}. $$

Let

$$g^{*}_{\epsilon}(t):= \textstyle\begin{cases} g_{\epsilon}& \text{for $t \in[-r_{\epsilon},r_{\epsilon}]$,}\\ 0 & \text{else,} \end{cases} $$

and \(f_{\epsilon}:=F^{-1}(g^{*}_{\epsilon}) \). Then \(f_{\epsilon}\in S \). We have

$$\begin{aligned} \bigl\Vert \vert t \vert ^{2\beta}\bigl(Ff(t)-Ff_{\epsilon}(t) \bigr) \bigr\Vert ^{2}_{L^{2}} = & \int_{|t|>r_{\epsilon}} \vert t \vert ^{4\beta}(Ff)^{2}(t) \,dt+ \int_{|t|< r_{\epsilon}} \vert t \vert ^{4\beta} (Ff-Ff_{\epsilon})^{2}(t)\,dt \\ \leq & \epsilon^{2}+r_{\epsilon}^{4\beta} \int_{|t|< r_{\epsilon}} (Ff-Ff_{\epsilon})^{2}(t)\,dt\leq2 \epsilon^{2}. \end{aligned}$$

Then \(\|f-f_{\frac{1}{m}}\|_{M_{\beta}}=\||t|^{2\beta}(Ff(t)-Ff_{\frac {1}{m}}(t))\|_{L^{2}} \rightarrow0 \) as \(m \rightarrow\infty\), which implies that \(f \in M_{\beta}\). □

3 Solution of the multi-term fractional diffusion equation

In this section the analytical solution to the initial value problem (1.1)-(1.2) is obtained by using the Luchko theorem.

Definition 3.1

[19], p.3

A real- or complex-valued function \(f (x), x > 0 \), is said to be in the space \(C_{\alpha}, \alpha\in R \), if there exists a real number \(p > \alpha\) such that \(f (x) = x^{p} f_{1}(x) \), with a function \(f_{1}(x) \in C[0, \infty)\).

Definition 3.2

[19], p.4

A function \(f (x), x > 0\), is said to be in the space \(C_{\alpha}^{m}, m\in N \cup\{0\}\), if and only if \(f^{(m)} \in C_{\alpha}\).

Lemma 3.3

[19], p.6

Let \(u \in C_{-1}^{r}, r\in N \cup\{0\} \). Then the Caputo fractional derivative \(D^{\alpha}u, 0 \leq\alpha\leq r \), is well defined and the inclusion

$$D^{\alpha}u \in \textstyle\begin{cases} C_{-1}, & r-1< \alpha\leq r,\\ C^{r-1}[0,\infty)\subset C_{-1} , & r-k-1< \alpha\leq r-k,k=1,\ldots,r-1, \end{cases} $$

holds true.

The following is the well-known Luchko theorem (Theorem 4.1 in [19]).

Lemma 3.4

[19], p.15

Let \(\gamma_{0}>\cdots>\gamma_{p} \geq0 \) and \(c_{i} \in R \). The initial value problem

$$ \begin{aligned} & D^{\gamma_{0}}v(t)-\sum _{j=1}^{p}c_{j}D^{\gamma_{j}}v(t)=G(t), \\ & v^{(j)}(0)=d_{j}, \quad j=0,1,\ldots,\lceil \gamma_{0} \rceil-1, \end{aligned} $$
(3.1)

where the function G is assumed to lie in \(C_{-1} \) if \(\gamma_{0} \in N \), in \(C_{-1}^{1} \) if \(\gamma_{0} \notin N \), and the unknown function \(v(t) \) is to be determined in the space \(C_{-1}^{\lceil\gamma_{0} \rceil}\), and it has a solution, unique in the space \(C_{-1}^{\lceil\gamma_{0} \rceil}\), of the form

$$v(t)=v_{G}(t)+\sum_{j=0}^{\lceil\gamma_{0} \rceil-1} \,d_{j} v_{j}(t),\quad t\geq0. $$

Here

$$v_{G}(t)= \int_{0}^{t} s^{ \gamma_{0} -1}E_{(\cdot),\gamma_{0}}(s)G(t-s) \,ds $$

is a solution of the problem (3.1) with zero initial conditions, and the system of functions

$$v_{j}(t)=\frac{t^{j}}{j!}+\sum_{l=l_{j}+1}^{p} c_{l} t^{j+\gamma_{0}-\gamma _{l}}E_{(\cdot),j+1+\gamma_{0}-\gamma_{l}}(t),\quad j=0,\ldots,\lceil \gamma_{0} \rceil-1, $$

fulfills the initial conditions \(v_{j}^{(l)}=\delta_{jl}\), \(j,l= 0,\ldots,\lceil\gamma_{0} \rceil-1\). The function

$$E_{(\cdot),\beta}(t)=E_{(\gamma_{0}-\gamma_{1},\ldots,\gamma_{0}-\gamma_{p}),\beta }\bigl(c_{1} t^{\gamma_{0}-\gamma_{1}}, \ldots,c_{p} t^{\gamma_{0}-\gamma_{p}}\bigr) $$

is a particular case of the multivariate Mittag-Leffler function

$$ E_{(x_{1},\ldots, x_{p}),y}(z_{1},\ldots,z_{p})=\sum _{k=0}^{\infty} \mathop{\sum _{l_{1}+\cdots+l_{p}=k}}_{l_{1}\geq0,\ldots,l_{p}\geq0} \frac{k!}{l_{1}!\cdots l_{p}!}\frac{\prod_{j=1}^{p} z_{j}^{l_{j}}}{\Gamma (y+\sum_{j=1}^{p} x_{j} l_{j})}. $$
(3.2)

The natural numbers \(l_{j} \) are determined from the condition

$$\textstyle\begin{cases} \lceil\gamma_{l_{j}} \rceil\geq j+1,\\ \lceil\gamma_{l_{j}+1} \rceil\leq j. \end{cases} $$

In the case \(\lceil\gamma_{r} \rceil\leq j \) for any \(r=1,\ldots,p \), we set \(l_{j}=0 \) and, if \(\lceil\gamma_{r} \rceil\geq j+1 \) for any \(r=1,\ldots,p \), then \(l_{j}=p \).

The Mittag-Leffer type functions are very crucial in the theory of fractional differential equations [7, 20–22]. Now we prove a property of the multivariate Mittag-Leffer function which appears in the analytical solution of the initial value problem (1.1)-(1.2).

Lemma 3.5

Let \(0 \leq x_{p}<\cdots< x_{0}\leq1,c_{0},\ldots,c_{p}> 0 \). Then the function

$$\bigl\vert t^{x_{0}} E_{(x_{0}-x_{1},\ldots, x_{0}-x_{p},x_{0}),1+x_{0}}\bigl(-c_{1} t^{x_{0}-x_{1}},\ldots,-c_{p} t^{x_{0}-x_{p}},-c_{0} t^{x_{0}}\bigr) \bigr\vert $$

is bounded for all \(t \geq0 \).

Proof

The multivariate Mittag-Leffer function can be rewritten by using the Hankel integral representation of \(1/\Gamma(z) \) [5],

$$\frac{1}{\Gamma(z)}=\frac{1}{2\pi i} \int_{Ha(\epsilon+)} e^{s} s^{-z} \,ds, $$

where \(r>0, Ha(\epsilon+)=\{ z\in C:|z|=\epsilon, 0\leq|\operatorname{arg}(z)|\leq \pi\} \cup\{ z\in C:|z|>\epsilon, |\operatorname{arg}(z)|= \pi\} \). For any \(t>0\), there exists a \(r_{t}>0 \) such that

$$r_{t}>\max \Biggl\{ t,t \Biggl(\sum_{i=0}^{p} \vert c_{i} \vert \Biggr)^{1/(x_{0}-x_{1})} \Biggr\} . $$

Then we have, for \(r>r_{t}\),

$$\begin{aligned}& t^{x_{0}} E_{(x_{0}-x_{1},\ldots, x_{0}-x_{p},x_{0}),1+x_{0}}\bigl(-c_{1} t^{x_{0}-x_{1}}, \ldots,-c_{p} t^{x_{0}-x_{p}},-c_{0} t^{x_{0}}\bigr) \\& \quad=\frac{t^{x_{0}}}{2\pi i} \int_{Ha(r+)}\sum_{k=0}^{\infty} \mathop{\sum_{l_{0}+\cdots+l_{p}=k}}_{l_{0}\geq0,\ldots,l_{p}\geq0} \frac{(-1)^{k} k!}{l_{0}!\cdots l_{p}!}\prod_{j=0}^{p} c_{j}^{l_{j}}t^{x_{0}l_{0}+\sum_{j=1}^{p} (x_{0}-x_{j})l_{j}} \frac {e^{s}}{s^{1+x_{0}+x_{0}l_{0}+\sum_{j=1}^{p} (x_{0}-x_{j})l_{j}}} \,ds \\& \quad=\frac{t^{x_{0}}}{2\pi i} \int_{Ha(r+)}\sum_{k=0}^{\infty}(-1)^{k} \mathop{\sum_{l_{0}+\cdots+l_{p}=k}}_{l_{0}\geq0,\ldots,l_{p}\geq0} \frac{ k!}{l_{0}!\cdots l_{p}!}\prod_{j=0}^{p} c_{j}^{l_{j}} \biggl(\frac {t}{s} \biggr)^{x_{0}l_{0}+\sum_{j=1}^{p} (x_{0}-x_{j})l_{j}} \frac{e^{s}}{s^{1+x_{0}}}\,ds \\& \quad=\frac{1}{2\pi i} \int_{Ha(r/t+)}\sum_{k=0}^{\infty}(-1)^{k} \mathop{\sum_{l_{0}+\cdots+l_{p}=k}}_{l_{0}\geq0,\ldots,l_{p}\geq0} \frac{ k!}{l_{0}!\cdots l_{p}!}\prod_{j=0}^{p} c_{j}^{l_{j}}\xi ^{-x_{0}l_{0}-\sum_{j=1}^{p} (x_{0}-x_{j})l_{j}}\frac{e^{\xi t}}{\xi^{1+x_{0}}}\,d\xi \\& \quad=\frac{1}{2\pi i} \int_{Ha(r/t+)}\sum_{k=0}^{\infty}(-1)^{k} \Biggl(c_{0}\xi ^{-x_{0}}+\sum_{j=1}^{p}c_{j} \xi^{x_{j}-x_{0}} \Biggr)^{k}\frac{e^{\xi t}}{\xi ^{1+x_{0}}}\,d\xi \\& \quad=\frac{1}{2\pi i} \int_{Ha(r/t+)}\frac{1}{1+c_{0}\xi^{-x_{0}}+\sum_{j=1}^{p}c_{j}\xi^{x_{j}-x_{0}}}\frac{e^{\xi t}}{\xi^{1+x_{0}}}\,d\xi \\& \quad=\frac{1}{2\pi i} \int_{Ha(r/t+)}\frac{1}{\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi ^{x_{j}}+c_{0}}\frac{e^{\xi t}}{\xi}\,d\xi. \end{aligned}$$

Let \(r_{0}>r_{t} \) be a sufficiently large real number that satisfies the condition: all zeros of the function \(\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi ^{x_{j}}+c_{0} \) are contained in the circle \(O(r_{0})=\{z \in C:|s|=r_{0}, 0\leq|\operatorname{arg}(z)|\leq\pi\} \). Let \(L(r_{0},\phi)=\{z \in C:|z|>r_{0}, |\operatorname{arg}(z)|= \pi\} \). For simplicity, we denote

$$h(\xi):=\frac{1}{\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi^{x_{j}}+c_{0}}\frac{e^{\xi t}}{\xi}. $$

Then we have

$$\int_{Ha(r_{0}+)}h(\xi)\,d\xi= \int_{L(r_{0},\phi)+O(r_{0})}h(\xi)\,d\xi=K_{1}+K_{2}, $$

where

$$\begin{aligned}& K_{1}= \int_{L(r_{0},\phi)}h(\xi)\,d\xi, \qquad K_{2}= \int_{O(r_{0})}h(\xi)\,d\xi, \\& K_{1}= \int_{r_{0}}^{\infty} \biggl(\frac{e^{rt\cos\pi}e^{irt\sin\pi }}{r^{x_{0}}e^{i\pi x_{0}}+\sum_{j=1}^{p}c_{j}r^{x_{j}}e^{i\pi x_{j}}+c_{0}}- \frac{e^{rt\cos\pi}e^{-irt\sin\pi}}{r^{x_{0}}e^{-i\pi x_{0}}+\sum_{j=1}^{p}c_{j}r^{x_{j}}e^{-i\pi x_{j}}+c_{0}} \biggr), \\& \frac{dr}{r}\leq \int_{r_{0}}^{\infty}\frac{2e^{-rt }}{ |r^{x_{0}}-\sum_{j=1}^{p}|c_{j}|r^{x_{j}}-|c_{0}| |} \frac{dr}{r} \rightarrow0,\quad r_{0} \rightarrow\infty. \end{aligned}$$

If \(x_{0},\ldots,x_{p} \) are all rational numbers, then the function \(\xi (\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi^{x_{j}}+c_{0})\) has finitely many zeros. Then by Cauchy’s residue theorem, we have

$$K_{2}=2\pi i\sum_{i=1}^{k} \operatorname{Res}(h,z_{i}), $$

where \(z_{i}\) is a zero of the function \(\xi(\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi^{x_{j}}+c_{0})\) and \(\operatorname{Res}(h,z_{i}) \) is the residue of \(h(\xi )\) at \(z_{i}\). If \(z_{i} \) is a pole of order m, then the residue of \(h(\xi)\) at \(z_{i}\) is obtained by the formula

$$\operatorname{Res}(h,z_{i})=\frac{1}{(m-1)!}\lim _{z \rightarrow z_{i}}\frac {d^{m-1}}{dz^{m-1}}\bigl((z-z_{i})^{m}h(z) \bigr). $$

Then there exists a function \(h_{i} \) such that

$$\operatorname{Res}(h,z_{i})=h_{i}(z_{i})e^{z_{i}t}=h_{i}(z_{i})e^{|z_{i}|t\cos \operatorname{arg}(z_{i})}e^{it\sin \operatorname{arg}(z_{i})}. $$

It follows from \(c_{i} > 0 \) for any i that, if \(|\operatorname{arg}(\xi)|\leq\pi/2 \), then \(0<|\operatorname{arg}(\xi^{x_{0}}+\sum_{j=1}^{p}c_{j}\xi^{x_{j}}+c_{0})|\leq\pi/2 \). Therefore \(|\operatorname{arg}(z_{i})|>\pi/2 \) and \(|\operatorname{Res}(h,z_{i})| \leq|h_{i}(z_{i})| \). Thus we have

$$\vert K_{2} \vert \leq2\pi\sum_{i=1}^{k} \bigl\vert h_{i}(z_{i}) \bigr\vert , $$

which implies that

$$\biggl\vert \int_{Ha(r_{0}+)}h(\xi)\,d\xi \biggr\vert \leq \vert K_{1} \vert + \vert K_{2} \vert \leq2\pi\sum _{i=1}^{k} \bigl\vert h_{i}(z_{i}) \bigr\vert . $$

If \(x_{0},\ldots,x_{p} \) are all real numbers, then, since the set of rational numbers is everywhere dense in the set of real numbers and the function

$$t^{x_{0}} E_{(x_{0}-x_{1},\ldots, x_{0}-x_{p},x_{0}),1+x_{0}}\bigl(-c_{1} t^{x_{0}-x_{1}}, \ldots,-c_{p} t^{x_{0}-x_{p}},-c_{0} t^{x_{0}}\bigr) $$

is continuous with respect to \(x_{0},\ldots,x_{p} \), we can obtain the desired result. □

Lemma 3.6

Let \(0 < x_{p}<\cdots< x_{0} \leq1,y > 0\). Let \(z_{0},z_{1},\ldots,z_{p}\in C \) satisfy \(\mu\leq|\arg z_{0}|\leq\pi\) and \(-l \leq z_{j} \leq0 \) \((j=1,\ldots,p)\) for some fixed \(\mu\in(x_{0}\pi/2,x_{0}\pi) \) and \(l>0\). Then there exists a \(K>0 \) depending only on \(\mu, l, x_{j} \) \(( j=0,\ldots,p) \) and y such that

$$\bigl\vert E_{(x_{0}-x_{1},\ldots, x_{0}-x_{p},x_{0}),y}(z_{1},\ldots,z_{p},z_{0}) \bigr\vert < \frac{K}{1+|z_{0}|}. $$

Proof

By (3.2), it is obvious that

$$E_{(x_{0}-x_{1},\ldots, x_{0}-x_{p},x_{0}),y}(z_{1},\ldots,z_{p},z_{0})= E_{(x_{0},x_{0}-x_{1},\ldots, x_{0}-x_{p}),y}(z_{0},z_{1},\ldots,z_{p}). $$

Then, using Lemma 3.2 in [14], we can prove the result. □

Theorem 3.7

Let \(g \in H_{\beta}\). Then the Cauchy problem (1.1)-(1.2) has a unique solution in \(C^{1}_{-1}([0,\infty), M_{\beta}) \). In particular, the solution is in \(C^{1}_{-1}([0,\infty), H_{\beta}) \) and is given by

$$\begin{aligned} u(t,x) = & \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{g}(\xi) \bigl[ 1-|\xi |^{2\beta}t^{\alpha_{0}} E_{(\alpha_{0}-\alpha_{1}, \ldots,\alpha_{0}-\alpha _{n-1},\alpha_{0}),1+\alpha_{0}} \\ &{} \bigl(-a_{1} t^{\alpha_{0}-\alpha_{1}},\ldots,-a_{n-1} t^{\alpha_{0}-\alpha _{n-1}},-|\xi|^{2\beta}t^{\alpha_{0}} \bigr) \bigr]\cos(x\xi) \,d\xi, \end{aligned}$$

where ĝ means the Fourier transform of g. The solution \(u(t,x)\) is bounded for all \(t \geq0 \) and \(x \in R \).

Proof

Applying the Fourier transform to equation (1.1) with respect to the space variable x, we have

$$\begin{aligned}& \sum_{j=0}^{n-1}a_{j} D^{\alpha_{j}}\hat{u}(t,\xi)+ \vert \xi \vert ^{2\beta} \hat{u}(t, \xi)=0, \\& \hat{u}(0,\xi)=\hat{g}(\xi). \end{aligned}$$

By Lemma 3.4, we have

$$\begin{aligned} \hat{u}(t,\xi) =& \hat{g}(\xi) \bigl[ 1- \vert \xi \vert ^{2\beta}t^{\alpha_{0}} E_{(\alpha_{0}-\alpha_{1}, \ldots,\alpha_{0}-\alpha_{n-1},\alpha_{0}),1+\alpha _{0}} \bigl(-a_{1} t^{\alpha_{0}-\alpha_{1}}, \\ &{}\ldots,-a_{n-1} t^{\alpha_{0}-\alpha_{n-1}},- \vert \xi \vert ^{2\beta}t^{\alpha_{0}} \bigr) \bigr]. \end{aligned}$$

By Lemma 3.6, for any \(t>0 \), there exists a \(M_{t}>0 \) such that

$$\bigl\vert \vert \xi \vert ^{2\beta}t^{\alpha_{0}} E_{(\alpha_{0}-\alpha_{1}, \ldots,\alpha_{0}-\alpha_{n-1},\alpha_{0}),1+\alpha _{0}} \bigl(-a_{1} t^{\alpha_{0}-\alpha_{1}},\ldots,-a_{n-1} t^{\alpha_{0}-\alpha _{n-1}},-|\xi|^{2\beta}t^{\alpha_{0}} \bigr) \bigr\vert < M_{t} $$

for any \(\xi\in R \) and \(|\hat{u}(t,\xi)| \leq(M_{t}+1)|\hat{g}(\xi)|\). Then \(u(t,\cdot) \in H_{\beta}\). Using the inverse Fourier transform with respect to ξ, we obtain

$$\begin{aligned} u(t,x) =& \frac{1}{2\pi} \int_{-\infty}^{\infty}\hat{g}(\xi) \bigl[ 1- \vert \xi \vert ^{2\beta}t^{\alpha_{0}} E_{(\alpha_{0}-\alpha_{1}, \ldots,\alpha_{0}-\alpha _{n-1},\alpha_{0}),1+\alpha_{0}} \\ &{} \bigl(-a_{1} t^{\alpha_{0}-\alpha_{1}},\ldots,-a_{n-1} t^{\alpha_{0}-\alpha _{n-1}},- \vert \xi \vert ^{2\beta}t^{\alpha_{0}} \bigr) \bigr] \cos(x\xi) \,d\xi. \end{aligned}$$

Then we have

$$\bigl\vert u(t,x) \bigr\vert \leq\frac{(M_{t}+1)}{2\pi} \int_{-\infty}^{\infty} \bigl\vert \hat{g}(\xi) \bigr\vert \,d\xi. $$

Meanwhile, by Lemma 3.5, we obtain

$$\bigl\vert u(t,x) \bigr\vert \leq\frac{1}{2\pi} \int_{-\infty}^{\infty} \bigl\vert \hat{g}(\xi) \bigr\vert \bigl(1+K_{\xi} \vert \xi \vert ^{2\beta}\bigr) \,d\xi, $$

where

$$K_{\xi}=\sup_{t>0} \bigl\vert t^{\alpha_{0}}E_{(\alpha_{0}-\alpha_{1}, \ldots,\alpha _{0}-\alpha_{n-1},\alpha_{0}),1+\alpha_{0}} \bigl(-a_{1} t^{\alpha_{0}-\alpha_{1}}, \ldots,-a_{n-1} t^{\alpha_{0}-\alpha _{n-1}},- \vert \xi \vert ^{2\beta}t^{\alpha_{0}} \bigr) \bigr\vert . $$

From Lemma 3.6, there exists a \(K>0 \) such that \(K_{\xi}< K \) for any \(\xi\in R \). Then we have

$$\bigl\vert u(t,x) \bigr\vert \leq\frac{1}{2\pi} \int_{-\infty}^{\infty} \bigl\vert \hat{g}(\xi) \bigr\vert \bigl(1+K \vert \xi \vert ^{2\beta}\bigr) \,d\xi, $$

which implies that \(u(t,x)\) is bounded. □