On asymptotics of solutions for superdiffusion and subdiffusion equations with the Riemann-Liouville fractional derivative

1.   Introduction

The aim of this paper is to consider asymptotic behaviors of solutions for the following time-space fractional superdiffusion equation with integral initial conditions and $\alpha\in (1, 2)$:

and subdiffusion equation with integral initial condition and $\alpha\in (0, 1)$:

where $_{RL}{\rm{D}}_{0, t}^{\alpha} u$ is the Riemann-Liouville derivative of $u$, $(-\Delta)^{s}$ denotes the integral fractional Laplace operator with $s\in (0, 1)$, and $\varphi(x)$, $\psi(x)$, $\phi(x)$, $f(x, t)$, and $g(x, t)$ are given functions. Moreover, the symbols $_{RL}{\rm{D}}_{0, t}^{\alpha-2}$ in Eq (1.1) and $_{RL}{\rm{D}}_{0, t}^{\alpha-1}$ in Eq (1.2) are Riemann-Liouville integral operators, and the symbol $_{RL}{\rm{D}}_{0, t}^{\alpha-1}$ in Eq (1.1) is Riemann-Liouville derivative operator.

It is well known ^[14,19,28] that the Riemann-Liouville fractional integral of a function $f(t)\in L^{1}[a, b]\, (-\infty < a < b < +\infty)$ can be defined by

and the Riemann-Liouville fractional derivative may be represented in the form

where $n-1 < \alpha < n\in \mathbb{N}$ and $f(t)\in AC^{n}[a, b]$, here $AC^{n}[a, b]$ denotes the set of functions with an absolutely continuous $(n-1)$st derivative.

For a function $v(x)\in H^{2s}(\mathbb{R}^{d}) = \{v\in \mathcal{S}(\mathbb{R}^{d})\, |\, (-\Delta)^{s}v\in L^{2}(\mathbb{R}^{d}), s\in (0, 1)\}$ with $\mathcal{S}(\mathbb{R}^{d})$ being the Schwartz space, the integral fractional Laplacian of the function $v(x)$ is given by ^[8]

where P.V. denotes the Cauchy principal value and $C(d, s)$ is a dimensional constant

Superdiffusion and subdiffusion equations in the forms of Eqs (1.1) and (1.2) have drawn much interest in developing existence, uniqueness, stability as well as asymptotics of the solutions, due to their excellent modelling capability for various applications such as theory of viscoelasticity ^[23], signal and image processing ^[31], anomalous diffusion ^[24], control theory ^[26], epidemic phenomena in biology ^[1], economics ^[4], etc. For more widespread applications on fractional differential equations we refer the reader to other works ^{[3,9,10,12,14,19,27,28,33]} and the references cited therein.

As far as the asymptotic behaviors of solutions of fractional partial differential equations is concerned, we review some results on this topic in the current literatures. For the fractional superdiffusion (or call diffusion-wave) equation, the authors in ^[25] first studied the asymptotics of solutions in the sense of $L^{\infty}$ norm, where time derivatives are the Riemann-Liouville and Caputo ones respectively and spatial derivative is the standard Laplace operator. After that, the article ^[7] investigated the asymptotic estimates of solution under $L^{p}$ norm with $1\leq p\leq \infty$, where the Riemann-Liouville derivative replaced by Caputo derivative in Eq (1.1) and the initial conditions are written as $u(x, 0) = u_{0}(x)$ and $u_{t}(x, 0) = u_{1}(x)$. Recently, Li and Li ^[21,22] discussed the same problem as above and derived similar asymptotic behaviors, in which the temporal derivative are taken as Caputo-Hadamard and $\psi$-Caputo fractional ones.

On the other hand, concerning the fractional subdiffusion equation, Ma et al. ^[25] considered the asymptotic properties of such equation in the cases of the Riemann-Liouville and Caputo derivatives for Eq (1.2) when the force term is equal to zero, where the spatial direction is the standard Laplacian and the initial value is $u(x, 0) = u_{0}(x)$. Subsequently, the paper ^[17] generalized these conclusions of ^[25] and they established the asymptotic analysis of solution in terms of $L^{p}$ norm in which the Caputo derivative is used as temporal one. Shortly after, the results in ^[17] are further extended to time-space fractional subdiffusion equation ^[18] with the Caputo derivative and integral fractional Laplacian. Very recently, Li et al. ^[20,22] devoted to asymptotic properties of solution of Eq (1.2), where the Caputo-Hadamard and $\psi$-Caputo derivatives substituted for the Riemann-Liouville one. For other related studies we refer the reader to ^[16,32]. However, to the best of our knowledge, the asymptotic behaviors of solutions for Eqs (1.1) and (1.2) with the Riemann-Liouville derivative have been less studied and the literature ^[25] only considered very special cases for which the results obtained there can also be further improved.

Based on the above reasons and existing research works, the goal of this paper is to study the asymptotic behaviors of solutions of Eqs (1.1) and (1.2) in the sense of more general $L^{p}$ or weak $L^{p}$ norms. Specifically, we first investigate asymptotic estimates of the solution to Eq (1.1). Using the technique of integral transforms the solution of convolutional form of Eq (1.1) is constructed and the fundamental solutions are also explicitly expressed by the Fox $H$-function. Then we estimate the fundamental solutions by means of asymptotic expansions of the $H$-function and further obtain the asymptotics of solution with the help of Young's inequality for convolution. By applying similar argument we can derive gradient estimates and large time behaviors of solution to Eq (1.1). For the subdiffusion Eq (1.2), we likewise discuss the asymptotic properties, gradient estimates and large time behaviors of solution. In particular we obtain the optimal decay rate in the sense of $L^{2}$ norm. We find that these results with the Riemann-Liouville derivative in time are different from the Caputo case, for example, see Theorem 3.1 of this paper and Proposition 5.7 in ^[18].

The remaining part of this article is organized as follows. In Section 2, the asymptotic behaviors of solution of the fractional superdiffusion Eq (1.1) are studied by means of Young's inequality for convolution. Further, the gradient estimates and large time behaviors of the solution are also presented. By using the almost same methods, Section 3 discusses decay estimates of the solution for the fractional subdiffusion Eq (1.2) and the optimal $L^{2}$-decay rate is particularly derived. Some conclusions and remarks are presented in Section 4. At last, the Appendix recalls several integral transforms and concept of the Fox $H$-function. Throughout the paper we denote by $C$ a generic positive constant whose value may vary from line to line.

2.   Asymptotic estimates of solution for Eq (1.1)

In this section, we shall study asymptotic analysis of the solution to Eq (1.1). First, the solution of convolutional form for Eq (1.1) is constructed in terms of Fourier and Laplace transforms, where the fundamental solutions are written via the Fox $H$-functions. We subsequently investigate asymptotic behaviors and estimations of $L^{p}$-norm for the fundamental solutions. Then, the asymptotic estimates of solution of Eq (1.1) are established by means of Young's inequality for convolution. Finally, we present gradient estimates and large time behaviors of the solution to Eq (1.1) by using the almost same argument.

2.1.   Asymptotic behaviors of the solution

We first deduce the fundamental solutions and solution of Eq (1.1) by using integral transforms. Making use of the standard Laplace transform for temporal variable $t$ and the Fourier transform for spatial variable $x$, and taking the formulas (A2) and (A4) into account, it follows that

Furthermore,

where $\overline{\widehat{G}}_{\varphi}(\omega, \lambda) = \frac{\lambda}{\lambda^{\alpha}+|\omega|^{2s}}$ and $\overline{\widehat{G}}_{\psi}(\omega, \lambda) = \overline{\widehat{G}}_{f}(\omega, \lambda) = \frac{1}{\lambda^{\alpha}+|\omega|^{2s}}$.

Applying the inverse Fourier transform and inverse Laplace transform to the identity (2.2) we obtain

where the character $\ast$ denotes the standard convolution with respect to spatial variable, and the symbol $\star$ is used as a convolution in time and space directions.

In the following part, we present the explicit expressions of fundamental solutions $G_{\varphi}(x, t)$, $G_{\psi}(x, t)$, and $G_{f}(x, t)$ in (2.3). In terms of the relation (A3) for Laplace and Mellin transforms, one has

The inverse Fourier transform of the above equality yields

where $J_{\frac{d}{2}-1}(\rho|x|)$ is the first kind of Bessel function, see ^[14] for related definition and property. Observing that the formula (2.6.4) in ^[15], one has

Therefore, we arrive at

Finally, it follows from the inverse Mellin transform that

that is,

Similarly, we can derive the expression of fundamental solutions $G_{\psi}(x, t) = G_{f}(x, t)$ in the form

By using asymptotic expansions of the Fox $H$-function at infinity and zero ^[15], we can prove the following lemma on the fundamental solutions $G_{\varphi}(x, t)$ in (2.4) and $G_{\psi}(x, t) = G_{f}(x, t)$ in (2.5).

Lemma 2.1. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Suppose $R = t^{-\alpha}|x|^{2s}$. For the fundamental solutions $G_{\varphi}(x, t)$ in (2.4) and $G_{\psi}(x, t) = G_{f}(x, t)$ in (2.5), the following asymptotic behaviors hold.

(1) If $R > 1$, then

and if $R\leq 1$, then

$\rm (2)$ If $R > 1$, then

and if $R\leq 1$, then

Proof. $\rm (1)$ For the fundamental solution $G_{\varphi}(x, t)$ given by (2.4), we need to estimate asymptotic expansions of the $H$-function $H_{23}^{21}\left(\frac{|x|^{2s}}{2^{2s}t^{\alpha}}\right)$ which is the most important step. Noting that

then we find that $a^{*} = 2-\alpha > 0$.

We first prove (2.6) for $R > 1$. Using Theorems 1.4 and 1.7 in ^[15] gives

where

Hence, one get

Furthermore, there holds

This illustrates that there is a positive constant $M$ satisfying

In light of analyticity of the $H$-function $H_{23}^{21} \left(\frac{|x|^{2s}}{2^{2s}t^{\alpha}}\right)$, we find that it is bounded for $1 < R\leq M$. Hence,

Combining (2.10) and (2.11) we obtain

Next, we show (2.7) with $R\leq 1$. If $d > 4s$, then $b_{1\sigma} = -\frac{1+\sigma}{1} = -(\sigma+1)$ and $b_{2k} = -\frac{d/2+k}{s}$ for $\sigma, k = 0, 1, 2, \ldots$. Therefore, $b_{10}$ is a simple pole and Theorems 1.3 and 1.11 in ^[15] implies

Since

then it follows that

which indicates

Consequently, there exists a positive constant $\delta$ such that

Exploiting again analyticity of the $H$-function $H_{23}^{21}\left(\frac{|x |^{2s}}{2^{2s}t^{\alpha}}\right)$ we get

Using (2.12) and (2.13) yields

for $d > 4s$, and which is the first inequality in (2.7).

If $d = 4s$, then the poles $b_{10}$ is simple and the poles $b_{11} = b_{20} = -2$ are coincided. In view of Theorems 1.5 and 1.12 in ^[15] we have

where $H^{*}_{201} = \frac{\Gamma(2)}{s\Gamma(-\alpha-1)\Gamma(2s)}\neq0$. As a result,

That is to say that there exists a positive constant $\delta_{1}$ such that

We further derive

for $d = 4s$, and the second inequality in (2.7) is proved.

Finally, we show that the third inequality in (2.7) holds when $d < 4s$. To do this, we consider three cases respectively. If $d = 2s$, then the poles $b_{10} = -1$ and $b_{20} = -\frac{d}{2s}$ are coincide, but the coefficients $H_{100}^{*} = H_{101}^{*} = 0$ by a direct calculation in terms of Theorems 1.5 and 1.12 in ^[15]. If $d > 2s$, then $b_{10} = -1$ is a simple pole, but we find $h_{10}^{*} = 0$ in this case. If $d < 2s$, then $b_{20} = -\frac{d}{2s}$ is a simple pole, by using Theorem 1.11 in ^[15] one has

In either case, we can obtain

Consequently,

Furthermore it holds that

for $d < 4s$, and the third inequality holds.

Similarly, we can prove (2.8) and (2.9) by using the same technique as the above (1) and omit them. The proof is now completed. □

In our further consideration, $||\cdot||_{p}$ and $||\cdot||_{p, \infty}$ are used to simplify $||\cdot||_{L^{p}(\mathbb{R}^{d})}$ and $||\cdot||_{L^{p, \infty}(\mathbb{R}^{d})}$ respectively, where $L^{p, \infty}(\mathbb{R}^{d})$ means weak $L^{p}(\mathbb{R}^{d})$ space on $\mathbb{R}^{d}$, for example, see ^[11]. We can also introduce

and

The following estimates of the fundamental solution $G_{\varphi}(x, t)$ and $G_{\psi}(x, t) = G_{f}(x, t)$ in $L^{p}(\mathbb{R}^{d})$ and $L^{p, \infty}(\mathbb{R}^{d})$ norms are crucial in proving the asymptotic behaviors of the solution of Eq (1.1).

Lemma 2.2. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Then for any $t > 0$, it holds that $G_{\varphi}(x, t)\in L^{p}(\mathbb{R}^{d})$ and

for every $1\leq p < \kappa(d, s)$. Moreover, if $p = \frac{d}{d-4s}$ for $d > 4s$, we have $G_{\varphi}(x, t)\in L^{\frac{d}{d-4s}, \infty}(\mathbb{R}^{d})$ and

Proof. Firstly, we prove (2.14). Note that

From (2.6) one can get

namely,

On the other hand, when $d > 4s$ and $1\leq p < \kappa(d, s)$, it follows from the first inequality in (2.7) that

i.e.,

If $d = 4s$, applying the second inequality in (2.7) we obtain

for $1\leq p < \infty$. Consequently,

For $d < 4s$, we use the third inequality in (2.7) to derive

for $1\leq p < \infty$, which leads to

Collecting the above estimates (2.16)–(2.19), it follows that

for $1\leq p < \kappa(d, s)$ with $d\geq 1$ and $0 < s < 1$.

We next show (2.15). Let $R = t^{-\alpha}|x|^{2s}$ and $p = \frac{d}{d-4s}$ for $d > 4s$. Due to the fact

where $\chi_{\{E\}}(t)$ means the characteristic function of the set $E$. In terms of (2.16), there holds

To estimate $||G_{\varphi}(\texttt{{\rm{x}}}, t) \chi_{\{R\leq 1\}}(t)||_{p, \infty}$, we may use the first inequality in (2.7) to obtain

where $\varrho$ stands for the measure on $\mathbb{R}^{d}$. Thus we have

That is

Therefore the required result follows by using (2.20) and (2.21) and the proof is thus completed. □

Remark 2.1. If $d < 4s$, we infer from the third inequality of (2.7) in Lemma 2.1 that $G_{\varphi}(\cdot, t)\in L^{\infty}(\mathbb{R}^{d})$ and $||G_{\varphi}(x, t)||_{\infty}\leq Ct ^{\alpha-2-\frac{\alpha d}{2s}}$ for all $t > 0$.

Lemma 2.3. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. If $1\leq p < \kappa(d, s)$, then $G_{\psi}(x, t) = G_{f}(x, t) \in L^{p}(\mathbb{R}^{d})$ for any $t > 0$ and

Moreover, if $p = \frac{d}{d-4s}$ and $d > 4s$, then $G_{\psi}(x, t) = G_{f}(x, t)\in L^{\frac{d}{d-4s}, \infty}(\mathbb{R}^{d})$ for any $t > 0$ and

Proof. The proof is similar to that of Lemma 2.2 above. □

Remark 2.2. For the case $d < 4s$, it follows from the third inequality of (2.9) in Lemma 2.1 that $G_{\psi}(\cdot, t) = G_{f}(\cdot, t)\in L^{\infty}(\mathbb{R}^{d})$ and $||G_{\psi}(x, t)||_{\infty} = ||G_{f}(x, t)||_{\infty}\leq Ct^{\alpha-1-\frac{\alpha d}{2s}}$ for any $t > 0$.

Let us now turn our attention to the asymptotic estimates of the solution to Eq (1.1) when the force term $f\equiv 0$ and the initial values $\varphi = \psi = 0$, respectively.

Theorem 2.1. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Suppose $f\equiv 0$. Then the solution $u(x, t) = G_{\varphi}(x, t)\ast \varphi(x) +G_{\psi}(x, t)\ast \psi(x)$ to Eq (1.1), where $\varphi, \psi\in L^{q}(\mathbb{R}^{d})$ for $1\leq q\leq \infty$, has the following asymptotic estimates:

$\rm (1)$ If $q = \infty$, then

$\rm (2)$ If $1\leq q < \infty$, then

for any

Moreover, it holds that

if $d > 4sq$.

Proof. Let $1\leq p, q, r\leq \infty$ satisfy the relation

In view of Young's inequality for convolution, see (57) in ^[21], we get

(1) If $q = \infty$, then $r = \infty$ and $p = 1$ by (2.27). Observe that (2.14) in Lemma 2.2 and (2.22) in Lemma 2.3 for $p = 1$, one has

and

which together with (2.28) yields

(2) If $1\leq q < \infty$, then for $r\in \left[q, \frac{dq}{d-4sq}\right)$ when $d > 4sq$, we have $1\leq p < \frac{d}{d-4s}$ when $d > 4s$. Therefore, substituting (2.14) and (2.22) into (2.28) there holds

as required (2.25). Similarly, we can prove (2.25) for $r\in [q, \infty)$ if $d = 4sq$ and $r\in [q, \infty]$ if $d < 4sq$.

Finally we show (2.26) for $d > 4sq$. Recalling that Young's inequality for weak $L^{p}$-norm, see (58) in ^[21], combining (2.15), (2.23) and (2.27), it follows that

which is expected inequality (2.26) and the proof is thus complete. □

Theorem 2.2. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. And let $1\leq q < \infty$. Assume that $f(\cdot, t)\in L^{q}(\mathbb{R}^{d})$ for any $t > 0$ and there exists some $\gamma > 0$ such that

Then for every

the solution $u(x, t) = G_{f}(x, t)\star f(x, t)$ has the following estimates:

if $\gamma\neq 1$, and

if $\gamma = 1$.

Proof. The proof of this theorem can be referred to that of Proposition 5.15 in ^[18] or Theorem 3.9 in ^[20]. □

2.2.   Gradient estimates and large time behaviors

This subsection will develop gradient estimates and large time behaviors of the solution for Eq (1.1). Let us start with estimates of the derivatives for the fundamental solutions.

Lemma 2.4. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Suppose $R = t^{-\alpha}|x|^{2s}$. Then the spatial derivatives of $G_{\varphi}(x, t)$ and $G_{\psi}(x, t)$ and the temporal derivatives of $G_{f}(x, t)$ have the following estimates:

$\rm (1)$ If $R > 1$, then

and if $R\leq 1$, then

$\rm (2)$ If $R > 1$, then

and if $R\leq 1$, then

$\rm (3)$ If $R > 1$, then

and if $R\leq 1$, then

Proof. We only give the proof of (1), while (2) and (3) can be handled by similar method. Recalling that Property 2.8 in ^[15] results in

Thus the modulus of $\nabla G_{\varphi}(x, t)$ is

First of all, we prove (2.30). By means of Theorems 1.4 and 1.7 in ^[15] we find that

where

Hence, it follows that

from which one has

Using this inequality and the analyticity of the $H$-function $H_{34}^{31}\left(\frac{|x|^{2s}}{2^{2s}t^{\alpha}}\right)$, we conclude that there exists a positive constant $C$ such that

for $R > 1$ and which gives (2.30).

Next we show (2.31). According to Theorems 1.3 and 1.11 in ^[15], we find $h_{10}^{*} = h_{20}^{*} = h_{30}^{*} = 0$. Hence, $b_{11} = -2$ is a simple pole when $d+2 > 4s$ and

where

Thus there is a positive constant $C$ such that

We further obtain

for $R\leq 1$ and $d+2 > 4s$.

If $d+2 = 4s$, we see that the poles $b_{11} = b_{21} = b_{31} = -2$ are coincided, then

where $H^{*}_{111}\neq0$. Then one gets

Furthermore, it follows that

for $R\leq 1$ and $d+2 = 4s$.

It remains to show the case $d+2 < 4s$. Since the poles $b_{21} = b_{31} = -\frac{d+2}{2s}$ are coincided, then

with $H_{210}^{*}\neq 0$, which leads to

So we have

for $R\leq 1$ and $d+2 < 4s$.

Based on (2.37)–(2.39), the desired assertion (2.31) is obtained and the proof is now completed. □

According to Lemma 2.4, we can establish estimates of $||\nabla G_{\varphi}(x, t)||_{p}$, $||\nabla G_{\psi}(x, t)||_{p}$, and $||\nabla G_{f}(x, t)||_{p}$ and further derive asymptotic properties of $\nabla u(x, t)$ to Eq (1.1), whose proofs are very similar to those of counterparts in the previous subsection and omitted.

Lemma 2.5. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Assume that $1\leq p < \kappa^{*}(d, s)$. Then it holds that $\nabla G_{\varphi}(x, t)\in L^{p}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for any $t > 0$ and

Moreover, if $p = \frac{d}{d+1-4s}$ for $d+2 > 4s$, then $\nabla G_{\varphi}(x, t)\in L^{\frac{d}{d+1-4s}, \infty}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for all $t > 0$ and

Lemma 2.6. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Assume that $1\leq p < \kappa^{*}(d, s)$. Then it holds that $\nabla G_{\psi}(x, t) = \nabla G_{f}(x, t) \in L^{p}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for all $t > 0$ and

Moreover, if $p = \frac{d}{d+1-4s}$ for $d+2 > 4s$, then $\nabla G_{\psi}(x, t) = \nabla G_{f}(x, t) \in L^{\frac{d}{d+1-4s}, \infty}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for all $t > 0$ and

Theorem 2.3. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Suppose that $1\leq q\leq \infty$ and $f\equiv 0$. Then the following estimates hold on the gradient of solution $\nabla u(x, t) = \nabla G_{\varphi}(x, t)\ast \varphi(x)+\nabla G_{\psi}(x, t)\ast \psi(x)$ with $\varphi, \psi\in L^{q}(\mathbb{R}^{d})$.

$\rm (1)$ If $q = \infty$, then

$\rm (2)$ If $1\leq q < \infty$, then for any

one has

Moreover, if $d > (4s-1)q$, then

Theorem 2.4. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Let $1\leq q < \infty$ and the condition (2.29) be satisfied. Then for every

the gradient of solution $\nabla u(x, t) = \nabla G_{f}(x, t)\star f(x, t)$ has the following relations:

if $\gamma\neq 1$, and

if $\gamma = 1$.

The last two theorems present the large time behavior of the solution $u(x, t)$ for Eq (1.1).

Theorem 2.5. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Denote $M_{\varphi} = \int_{\mathbb{R}^{d}}\varphi(x){\rm{d}}x$ and $M_{\psi} = \int_{\mathbb{R}^{d}}\psi(x){\rm{d}}x$ with $\varphi, \psi\in L^{1}(\mathbb{R}^{d})$. Assume that $f\equiv 0$ and $1\leq p < \kappa^{*}(d, s)$. Then we have the following results.

$\rm (1)$ If $||\, |x|\varphi(x)||_{1} < \infty$ and $||\, |x|\psi(x)||_{1} < \infty$, then

for any $t > 0$. Moreover, when $p = \frac{d}{d+1-4s}$, one gets

for any $t > 0$.

$\rm (2)$ It follows that

as $t\rightarrow \infty$.

Proof. (1) Note that the conditions $\varphi, \psi\in L^{1}(\mathbb{R}^{d})$ and $||\, |x|\varphi(x)||_{1} < \infty$ and $||\, |x|\psi(x)||_{1} < \infty$. It follows from the decomposition lemma (see Lemma 8.4 ^[18]) that there exists functions $\Phi, \Psi\in L^{1}(\mathbb{R}^{d}; \mathbb{R}^{d})$ such that

where $||\Phi||_{1}\leq C||\, |x|\varphi(x)||_{1}$ and $||\Psi||_{1}\leq C||\, |x|\psi(x)||_{1}$. Therefore we find that

where $_{\bullet}\ast$ means the convolution between two vector functions. Further there holds

By using Young's inequality for convolution (57) in ^[21], and taking (2.40) and (2.42) into account, we obtain

for $1\leq p < \kappa^{*}(d, s)$, which gives

and the claim (2.46) holds. For the limit case $p = \frac{d}{d+1-4s}$, applying Young's inequality for convolution (58) in ^[21], (2.41) and (2.43) to (2.49) we immediately know that (2.47) is true.

(2) Let a sequence $\{\eta_{m}(x)\}\subseteq C_{0}^{\infty}(\mathbb{R}^{d})$ satisfy $\int_{\mathbb{R}^{d}}\eta_{m}(x){\rm{d}}x = M_{\varphi}$ for all $m$ and $\eta_{m}(x)\rightarrow \varphi(x)$ as $m\rightarrow \infty$ in $L^{1}(\mathbb{R}^{d})$. Likewise, set a sequence $\{\zeta_{n}(x)\}\subseteq C_{0}^{\infty}(\mathbb{R}^{d})$ satisfy $\int_{\mathbb{R}^{d}}\zeta_{n}(x){\rm{d}}x = M_{\psi}$ for all $n$ and $\zeta_{n}(x)\rightarrow \psi(x)$ as $n\rightarrow \infty$ in $L^{1}(\mathbb{R}^{d})$. Now we use Young's inequality for convolution (57) in ^[21], (2.14), (2.22) and the conclusion of (1) to derive for any $m, n$

Consequently,

from which we have

and the claimed result (2.48) can be achieved by letting $n\rightarrow \infty$. The proof of this theorem is now ended. □

Theorem 2.6. Let $d\in \mathbb{N}$, $1 < \alpha < 2$ and $0 < s < 1$. Let $\varphi\equiv\psi\equiv 0$ and denote $M_{f} = \int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(x, t){\rm{d}}x{\rm{d}}t$. Moreover, let us assume that $f(x, t)\in L^{1}\left(\mathbb{R}^{d}\times (0, \infty)\right)$ and there exists some $\gamma > 1$ such that

Then it holds that

as $t\rightarrow \infty$ for any

Proof. The proof can be completed by using the methods provided by Theorem 2.21 in ^[18] or Theorem 3.16 in ^[20] and the details are omitted here. □

3.   Decay estimates of solution for Eq (1.2)

In this section we shall deal with decay behaviors of the solution for Eq (1.2). Similar to the previous section, we present the asymptotics of the fundamental solution, decay estimates of the solution, gradient estimates and large time behaviors. In particular the optimal $L^{2}$-decay estimates are provided by virtue of Plancherel's theorem and the boundedness of Mittag-Leffler function. For the most of theorems and lemmas in the section, we directly give results without proofs since their proof techniques are very similar to ones of corresponding conclusions in the previous section.

3.1.   Decay behaviors of the solution

We first construct the solution of Eq (1.2) by integral transforms. Applying Fourier and Laplace transforms to Eq (1.2), and noticing that formula (2.248) in ^[28] and equality (A4), there holds

Then we get

Performing the inverse Fourier transform and inverse Laplace transform on both sides of (3.2), the solution of Eq (1.2) reads as

where the fundamental solution ^[18]

Lemma 3.1. ^[18] Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Suppose $R = t^{-\alpha}|x|^{2s}$. Then for the fundamental solution $\mathcal{G}(x, t)$ in (3.4) we have

for $R > 1$, and

for $R\leq 1$.

Lemma 3.2. ^[18] Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. If $1\leq p < \kappa(d, s)$, then we have $\mathcal{G}(x, t)\in L^{p}(\mathbb{R}^{d})$ for all $t > 0$ and

And if moreover $p = \frac{d}{d-4s}$ for $d > 4s$, then it follows that $\mathcal{G}(x, t)\in L^{\frac{d}{d-4s}, \infty}(\mathbb{R}^{d})$ for all $t > 0$ and

Remark 3.1. If $d < 4s$, by the third inequality of (3.6) in Lemma 3.1, it is clear that $\mathcal{G}(\cdot, t)\in L^{\infty}(\mathbb{R}^{d})$ and $||\mathcal{G}(x, t)||_{\infty}\leq Ct ^{\alpha-1-\frac{\alpha d}{2s}}$ hold for all $t > 0$.

Theorem 3.1. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $1\leq q\leq \infty$ and $g\equiv 0$. Then the solution $u(x, t) = \mathcal{G}(x, t)\ast \phi(x)$ to Eq (1.2) with $\phi\in L^{q}(\mathbb{R}^{d})$ has the following decay estimates:

$\rm (1)$ If $q = \infty$, then

$\rm (2)$ If $1\leq q < \infty$, then

holds for any

If moreover $d > 4sq$, then

Theorem 3.2. ^[18] Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $1\leq q < \infty$ and $\phi\equiv 0$. Assume that $g(\cdot, t)\in L^{q}(\mathbb{R}^{d})$ for all $t > 0$ and there is some $\gamma > 0$ such that

Then the solution $u(x, t) = \mathcal{G}(x, t)\star g(x, t)$ satisfies the following

and

for any $t > 0$ and

3.2.   Gradient estimates and large time behaviors

The gradient estimates and large time behaviors of the solution for Eq (1.2) are provided in the subsection.

Lemma 3.3. ^[18] Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Suppose $R = t^{-\alpha}|x|^{2s}$. Then the spatial and time derivatives of the fundamental solution $\mathcal{G}(x, t)$ in (3.4) have the following decay behaviors:

$\rm (1)$ If $R > 1$, then

and if $R\leq 1$, then

$\rm (2)$ If $R > 1$, then

and if $R\leq 1$, then

Lemma 3.4. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. For $1\leq p < \kappa^{*}(d, s)$, it holds that $\nabla \mathcal{G}(x, t)\in L^{p}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for any $t > 0$ and

Moreover, if $p = \frac{d}{d+1-4s}$ for $d+2 > 4s$, we have $\nabla \mathcal{G}(x, t)\in L^{\frac{d}{d+1-4s}, \infty}(\mathbb{R}^{d}; \mathbb{R}^{d})$ for any $t > 0$ and

Theorem 3.3. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $1\leq q\leq \infty$ and $g\equiv 0$. Then for $\nabla u(x, t) = \nabla \mathcal{G}(x, t)\ast \phi(x)$ with $\phi\in L^{q}(\mathbb{R}^{d})$, we have:

$\rm (1)$ If $q = \infty$, then

$\rm (2)$ If $1\leq q < \infty$, then

for any

Moreover if $d > (4s-1)q$, then

Theorem 3.4. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $1\leq q < \infty$ and $\phi\equiv 0$. Assume that $g(\cdot, t)\in L^{q}(\mathbb{R}^{d})$ for any $t > 0$ and the assumption (3.9) holds. Then $\nabla u(x, t) = \nabla \mathcal{G}(x, t)\star g(x, t)$ has the estimates

and

for all $t > 0$ and for every

Theorem 3.5. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $1\leq p < \kappa^{*}(d, s)$ and $g\equiv 0$. Assume $\phi\in L^{1}(\mathbb{R}^{d})$ and denote $M_{\phi} = \int_{\mathbb{R}^{d}}\phi(x){\rm{d}}x$. Then the following results hold.

$\rm (1)$ If $||\, |x|\phi(x)||_{1} < \infty$, then

When $p = \frac{d}{d+1-4s}$, one gets

$\rm (2)$ It follows that

as $t\rightarrow \infty$.

Theorem 3.6. Let $d\in \mathbb{N}$, $0 < \alpha < 1$ and $0 < s < 1$. Let $\phi\equiv 0$ and denote $M_{g} = \int_{0}^{\infty}\int_{\mathbb{R}^{d}}g(x, t){\rm{d}}x{\rm{d}}t$. Let us assume $g(x, t)\in L^{1}\left(\mathbb{R}^{d}\times (0, \infty)\right)$ and there exists some $\gamma > 1$ such that

Then for all

it holds that

as $t\rightarrow \infty$.

3.3.   The optimal $L^{2}$-decay estimate

In this subsection our attention will be restricted to decay estimate of the solution for Eq (1.2) in the sense of $L^{2}$-norm when $g\equiv 0$. To do this, let us consider the solution of Eq (1.2) under the Fourier transform. By using the inverse Laplace transform on both sides of (3.1) we obtain

where

To derive optimal $L^{2}$ decay rate, we need investigate some properties of the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$.

Lemma 3.5. ^[13] Let $0 < \alpha < 1$. Then the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta) > 0$ for $\eta\in (0, \infty)$.

Lemma 3.6. ^[13] Let $0\leq\alpha\leq1$ and $\beta\geq \alpha$. Then the Mittag-Leffler function $E_{\alpha, \beta}(-\eta)$ is completely monotone for $\eta\in (0, \infty)$.

Lemma 3.7. Let $0 < \alpha < 1$. Then there are positive constants $C_{1}$ and $C_{2}$ such that

Proof. First of all, we know that $E_{\alpha, \alpha}(-\eta) > 0$ for $0 < \alpha < 1$ and $\eta\geq 0$. In fact, one has $E_{\alpha, \alpha}(0) = 1/\Gamma(\alpha) > 0$. For $\eta > 0$, it is evident that $E_{\alpha, \alpha}(-\eta) > 0$ by Lemma 3.5.

In view of asymptotic expansions for the Mittag-Leffler function (see Theorem 1.4 in ^[28]) we obtain

This implies the function $E_{\alpha, \alpha}(-\eta)$ behaves as $C_{0}/\eta^{2}$ when $\eta\rightarrow \infty$ with some positive constant $C_{0}$, i.e., there exist a positive real number $X$ and two positive constants $C_{3}$ and $C_{4}$ such that

Since the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$ is analytic for any $\eta\in \mathbb{R}$, then it is bound on the interval $[0, X]$. Moreover, it follows from Lemma 3.6 that the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$ is monotonically decreasing for $\eta\in (0, \infty)$. Therefore, there are positive constants $C_{5}$ and $C_{6}$ satisfying $C_{5}\leq E_{\alpha, \alpha}(-\eta) \leq C_{6}$ with $0\leq \eta\leq X$. We further obtain

with positive constants $C_{7}$ and $C_{8}$.

The desired result follows from (3.16) and (3.17) and the proof is complete. □

We first present the estimates of lower bound for the solution $u(x, t) = \mathcal{G}(x, t)\ast \phi(x)$ to Eq (1.2) with $g\equiv 0$ in the sense of $L^{2}$-norm.

Theorem 3.7. Let $d\in \mathbb{N}$, $0 < \alpha < 1$, $0 < s < 1$, and $d\neq 8s$. Let $g\equiv 0$ in Eq (1.2). If $\phi\in L^{1}(\mathbb{R}^{d})\bigcap L^{2}(\mathbb{R}^{d})$ and $\int_{\mathbb{R}^{d}}\phi(x){\rm{d}}x\neq 0$, then the solution $u(x, t) = \mathcal{G}(x, t)\ast \phi(x)$ to Eq (1.2) has the lower bound estimate

Proof. Let $\rho = \rho(t)\in (0, \rho_{0}]$ with $t > 0$ and $\rho_{0} > 0$. According to Plancherel's theorem and the estimate for the Mittag-Leffler function in Lemma 3.7, it follows that

From the Plancherel's theorem and Riemann-Lebesgue lemma, it is easy to verify that $\widehat{\phi}\in C_{0}(\mathbb{R}^{d})\bigcap L^{2}(\mathbb{R}^{d})$ holds. By making use of Lebesgue differentiation theorem we have for a sufficient small $\rho_{0}$

Substituting this into (3.18) leads to

Now letting $\rho = \rho_{0}$ in (3.19) one gets

On the other hand, we may take $\rho = \frac{\rho_{0}}{(1+t^{2\alpha})^{1/4s}}$ in (3.19) to derive

Together with (3.20) and (3.21), the desired lower bound is established and the proof is thus completed. □

The upper bound for the solution $u(x, t) = \mathcal{G}(x, t)\ast \phi(x)$ to Eq (1.2) is estimated in the following theorem when $g\equiv 0$.

Theorem 3.8. Let $d\in \mathbb{N}$, $0 < \alpha < 1$, $0 < s < 1$, and $d\neq 8s$. Let $g\equiv 0$ in Eq (1.2). If $\phi\in L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ and $\int_{\mathbb{R}^{d}}\phi(x){\rm{d}}x\neq 0$, then the solution $u(x, t) = \mathcal{G}(x, t)\ast \phi(x)$ to Eq (1.2) satisfies the upper bound estimate

For $d = 8s$ one has

Proof. We divide into two cases to prove the assertion (3.22). When $d < 8s$, applying Plancherel's theorem and Lemma 3.7 one gets

i.e.,

If $d > 8s$, then $\phi\in L^{1}(\mathbb{R}^{d})\bigcap L^{2}(\mathbb{R}^{d})$ implies $\phi\in L^{\frac{2d}{d+8s}}(\mathbb{R}^{d})$ by interpolation. Furthermore, Theorem 8.5 in ^[18] gives

It follows from Plancherel's theorem and Lemma 3.7 that

Substituting (3.25) into (3.26) we have

Based on (3.24) with (3.27), the expected inequality (3.22) is proved.

If $d = 8s$, then by using Young's inequality for convolution and (2.15) in Lemma 2.2 it is evident that

which is the required result (3.23). This finishes the proof of theorem. □

Remark 3.2. For $1 < \alpha < 2$, we have not give the optimal $L^{2}$-norm estimates for the solution of Eq (1.1). The reason is as follows. We can apply the inverse Laplace transform to equality (2.1) to obtain the solution of Eq (1.1) in the Fourier domain

where

and

However, we can not derive positive lower bounds for the Mittag-Leffler functions $E_{\alpha, \alpha-1}(-\eta)$ and $E_{\alpha, \alpha}(-\eta)$ with $\eta\geq 0$ and $1 < \alpha < 2$ as Lemma 3.7 since these two functions have zeros on the real axis. In fact, using the result of Theorem 2 in ^[29] we immediately see that the Mittag-Leffler function $E_{\alpha, \alpha-1}(-\eta)$ exists real zeros. For the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$, it is obvious that $E_{\alpha, \alpha}(0) = 1/\Gamma(\alpha) > 0$. Additionally, In view of asymptotic expansion of the Mittag-Leffler function, see Theorem 1.4 in ^[28], we have

when $\eta\rightarrow \infty$. Since $1 < \alpha < 2$, then one gets $\Gamma(-\alpha) > 0$. Hence the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$ behaves like

which implies $E_{\alpha, \alpha}(-\eta) < 0$ for sufficiently large $\eta$. Combining the above analysis and taking the analyticity of the Mittag-Leffler function $E_{\alpha, \alpha}(-\eta)$ into account, we find that $E_{\alpha, \alpha}(-\eta)$ has one real zeros at least. In Figure 1, we depict graphs of the Mittag-Leffler functions $E_{\alpha, \alpha-1}(-\eta)$ and $E_{\alpha, \alpha}(-\eta)$ with $\eta\in [0, 50]$, where the parameter $\alpha$ takes $\alpha = 1.2, 1.5, 1.9$, respectively.

4.   Conclusions

This work investigates asymptotic behaviors of solutions of Cauchy problems for superdiffusion equation (1.1) and subdiffusion equation (1.2) with integral initial conditions in the sense of $L^{p}(\mathbb{R}^{d})$ and $L^{p, \infty}(\mathbb{R}^{d})$. For these two kinds of equations, we construct their fundamental solutions and solutions, analyze asymptotic behaviors of solutions, and study gradient estimates and large time behaviors. In particular, the optimal $L^{2}$ decay estimate of solution is derived for Eq (1.2). Compared with the cases of Caputo derivative in the time direction ^[7,18] in Eqs (1.1) and (1.2), it is not difficult to see that the asymptotic rates are faster in the cases of Riemann-Liouville derivative.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The work was supported in part by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2021L573), the Key Research and Development Project of Lvliang city (No. 2022RC11), the Fundamental Research Program of Shanxi Province (No. 202103021224317), the Innovation and Entrepreneurship Training Program of College Students for Higher Education Institutions in Shanxi Province (No. 20221257).

Conflicts of interest

The authors declare no conflicts of interest.

Appendix

In the appendix we recall several concepts of integral transforms ^[5] such as Laplace transform, Mellin transform and Fourier transform, which play an important role on the process of deriving the solutions of Eqs (1.1) and (1.2). Moreover, we introduce the definition of the Fox $H$-function too.

The standard Laplace transform of a function $f(t)$ is defined by

Correspondingly, the inverse Laplace transform is given by

In particular, the Laplace transforms of the Riemann-Liouville fractional integral (1.3) and derivative (1.4) with the starting point $a = 0$ can be written as ^[28]

and

for $n-1 < \alpha < n\in \mathbb{N}$.

The standard Mellin transform of a function $f(t)$ is defined by

The inverse Mellin transform is correspondingly represented as

The relation connecting the Laplace transform and the Mellin transform has the following form ^[6]:

The Fourier transform of a function $f(x)$ is defined by

while the corresponding inverse Fourier transform can be written as

Then the Fourier transform of the integral fractional Laplacian (1.5) is given by ^[8]

Next, let us briefly recall the definition of the Fox $H$-function. The Fox $H$-function are special functions, where the fundamental solutions of Eqs (1.1) and (1.2) are written by means of these functions, and they paly a basic role in asymptotic analysis of the solutions.

According to the contour integral of Mellin-Barnes type, the Fox $H$-function can be represented as

where

and $m, n, \mu, \nu$ are nonnegative integers with $0\leq m \leq \nu$ and $0\leq n \leq \mu$, and $\alpha_{l}, \beta_{j}$ are positive real numbers and $a_{l}, b_{j}$ are complex numbers for $l = 1, \ldots, \mu; \ j = 1, \ldots, \nu$. All the poles

of the gamma functions $\Gamma(b_{j}+\beta_{j}\tau)$ and

of the gamma functions $\Gamma(1-a_{l}-\alpha_{l}\tau)$ are not equal, i.e.,

The contour $\mathfrak{L}$ is an infinite contour in the complex plane which separates all the poles $b_{j\sigma}$ from all the poles $a_{lk}$, and it may take $\mathfrak{L} = \mathfrak{L}_{-\infty}$ or $\mathfrak{L} = \mathfrak{L}_{+\infty}$ or $\mathfrak{L} = \mathfrak{L}_{i\gamma\infty}$ with $\gamma\in \mathbb{R}$ and $i^{2} = -1$. Besides, we denote

A comprehensive and detailed description for the Fox $H$-function can be available from ^[2,14,15,30].