Hardy-Sobolev spaces of higher order associated to Hermite operator

1.   Introduction

In this paper, we study an initial boundary value problem of the nonlinear diffusion equation with space-time fractional derivative

with a modified initial datum condition, where $\nabla$ is the gradient operator, $\Delta = \sum_{i = 1}^{n}\partial_{x_i}^2$ is the Laplace operator, ${_*D_\beta^{\alpha, \gamma}}$ is the Caputo-type modification of the Erdélyi-Kober fractional differential operator with $\gamma$-th order, parameters $\alpha\in(-1, +\infty)$, $\beta\in\mathbb (0, \infty)$, $\gamma\in (0, 1]$, $\mu\in(0, 1]$, and $\Omega\subseteq \mathbb R^n$ is a bounded domain.

Fractional calculus is an important subject in mathematics, physics, biology, economics, and many other different fields since it is usually used to describe the property of memory and heredity of many materials ^[1,2]. Riemann-Liouville derivatives, Caputo derivatives, and Erdélyi-Kober derivatives are the well-known ones. The Riemann-Liouville fractional derivative is always employed in mathematical texts and not frequently used in applications. The Caputo definition of a fractional derivative is more useful in modeling reality. The Erdélyi-Kober fractional derivative is often used in both mathematical texts and applications. The so-called Caputo type modification of the Erdélyi-Kober fractional derivative is a generalization of these types of fractional derivatives, and its operations attracts much attentions. Gorenflo, Luchko, and Mainardi first introduced and applied it to investigate of the scale-invariant solutions of the diffusion-wave equation in ^[3]. Kiryakova and Luchko investigated its general properties in the sense of multiple Erdélyi-Kober fractional derivatives and studied some examples of Cauchy problems of fractional differential equations involving these type operators in ^[4,5]. For more meaningful results and useful applications of the Erdélyi-Kober fractional derivative, one can find in ^[6,7,8,9].

In this paper, we investigate the local well-posedness of the solution of the nonlinear problem (1.1) and prepare to establish the theoretical basis for finding efficient numerical approaches later. Some researchers with similar interest are finding solutions using numerical methods on diffusion models involving fractional derivatives. Recently, Hoang Luc N. and his collaborators studied a diffusion equation involving a regularized hyper-Bessel operator in ^[10], and Van Au V. and his collaborators established the existence and blowup results of a similar model with gradient nonlinearity in ^[11]. For more results, one can refer to ^{[12,13,14,15,16,17]} and the references therein. In order to show our results of the nonlinear problems, we should first point out the singularity of the initial datum is generated by the the Erdélyi-Kober fractional derivative ^[18], then construct and estimate the solution of the linear problem based on the Mittag–Leffler function. Through a direct observation with the estimates, the smooth effects of the Caputo type modification of the Erdélyi-Kober fractional derivative on the solution were confirmed in the inner of the domain. Based on these, by applying embedding theorem between Hilbert scales spaces and Lebesgue spaces, fixed point theory, and the Picard-iteration method, we establish the existence, uniqueness, and stability of the solution of fractional diffusion equation for the source term with two types different nonlinearities. Under this framework, the establishment of global well-posedness results of the problem (1.1) doesn't given and deserve to be considered with new ideas and approaches in another paper.

The rest of this paper is organized as follows: In $\S$2, the basic knowledge of some interpolation function spaces and some established useful results are given. In $\S$3, we construct a solution of the linear inhomogeneous diffusion equation and give related estimates in terms of Mittag-Leffler functions in Hilbert scales spaces. Meanwhile, the smooth effects of the Erdélyi-Kober operator are shown. In $\S$4, the well-posedness is established for the nonlinear source function that satisfies the Lipschitz condition or with gradient nonlinearity.

2.   Preliminary

First, we recall the results of the eigenvalue problem ^[19]

Then there exists a system of standard and complete orthogonal basis $\{\Theta_i(x)\}_{i\in \mathbb N^+}$ in $\mathbb L^2(\Omega)$, the corresponding eigenvalues $\{\lambda_i\}_{i\in \mathbb N^+}$ which satisfies $\lim_{i\to+\infty}\lambda_i = \infty$.

For any $\nu\in \mathbb R$, set

is equipped with the norm

It is obviously that $\mathbb D^0(\Omega) = \mathbb L^2(\Omega)$. Define

where $u_i = \int_\Omega u(x)\Theta_i(x)dx$. Since $\int_\Omega \Theta_i^2(x)dx = 1$, then it is easy to verify that

In fact, $\mathbb D^\nu(\Omega)$ is a Hilbert scale space, for more introductions of these spaces, one can refers to ^[20].

In the following, we recall some embedding results between Hilbert scales spaces and Lebesgue spaces.

Lemma 2.1. ^[11] Assume $\Omega\subset \mathbb R^n$ is a smooth bounded domain, then

Based on Lemma 2.1, btain the following nonlinear estimate, although the result had been established in ^[10]. Here we rewrite the proof by constructing an analytic function.

Lemma 2.2. Prescribed $u, v\in\mathbb{D}^\nu(\Omega)$, $\Omega\subset \mathbb R^n$ is a smooth bounded domain, and $\nu, \rho, p$ satisfy

then there exists

Remark 2.3. Here and in the following of this paper, $|a|\lesssim |b|$ means there exists a constant $C > 0$ such that $|a|\leq C|b|$.

Proof. Consider the function

then by computing the derivative, one has

Furthermore, for $0 < p < 1$, there is

with the zero point $x_0 = 1+\frac{1}{1-p}$, and $f'(x)$ is monotonically increasing for $x\in(0, 1+\frac{1}{p-1})$, monotonically decreasing for $x\in (1+\frac{1}{p-1}, \infty)$. Hence

This yields that $f(x)$ is monotonically decreasing for $x\in (0, +\infty)$, and then $f(x)\le 0$ since $f(1) = 0$. For $p = 1$, we obtain $f(x) = 1-x\le 0$ for $x\in (0, +\infty)$. For $p > 1$, we also have $f'(x) < 0$, and $f(x)$ is monotonically decreasing for $x\in (0, \infty)$ by a similar analysis, then $f(x)\le 0$ is also holding. Therefore, there exists $f(x)\le 0$ for any $x\in (0, \infty)$, $p > 0$.

Set $x = \left\lvert \nabla u \right\rvert(\left\lvert \nabla v \right\rvert)^{-1}$, and substitute it into $f(x)\le 0$ in terms of (2.5), then we derive

which is equivalent to

Take any $q \in [\max\{1, \frac{2n}{n-4(\nu-\rho)}\}p, \frac{2n}{n-4(\nu-\frac{1}{2})}]$ and integrate (2.6) on $\Omega$, we have

in terms of of h$\ddot{o}$lder inequality, where

Besides, it follows Minkovsiki inequality and (2.8) that

and applying Cauchy inequality, we have

Then, substituting (2.9) and (2.10) into (2.7), we arrive at

Based on (2.11) and Lemma 2.1 (2.3), one has

Then, (2.5) is derived in terms of (2.12) and Lemma 2.1 (2.2) for $\frac{2n}{n-4(\nu-\rho)}\le\frac{q}{p}$.

Next, we recall some definitions of Erdélyi-Kober fractional integral and differential operators.

Definition 2.4. ^[1,2] For a function $f(t)\in \mathbb C_\mu$, $\mu\in \mathbb R$, then the integral

is called the Erdélyi-Kober fractional integral of $f(t)$ with arbitrary parameters $\alpha\in \mathbb R$, $\beta\in \mathbb R^+$ and $\gamma\in \mathbb R^+$, where the weighted space of continuous functions $f(t)$ is defined by

The Erdélyi-Kober fractional integral is defined as the identity operator for $\gamma = 0$ and is reduced to the well-known Riemann-Liouville fractional integral with a power weight for $\alpha = 0$, $\beta = 1$.

Definition 2.5. ^[4,5] The Riemann-Liouille type modification of the Erdélyi-Kober fractional derivative of a function $f(t)\in \mathbb C_\mu^{(1)}$ with the order $\gamma$ is defined by

Compared with the definition of the Riemann-Liouille type modification of the Erdélyi-Kober fractional derivative of a function $f(t)\in \mathbb C_\mu^{(1)}$ with the order $\gamma$, the Caputo type modification also can be defined.

Definition 2.6. ^[4,5] The Caputo type modification of the Erdélyi-Kober fractional derivative of a function $f(t)\in \mathbb C_\mu^{(1)}$ with the order $\gamma$ is defined by

The singular initial value problem of fractional differential equation with the Caputo-type modification of the Erdélyi-Kober derivative

had been studied and the following result can be directly derived from Theorem 3.4 in ^[18] with parameters $\alpha\in(-1, +\infty)$, $\beta\in\mathbb (0, \infty)$, $\gamma\in (0, 1]$, $\mu\in(0, 1]$.

Lemma 2.7. ^[18] Given a function $f\in C_{\beta\delta}$, $\delta\ge \max\{0, -\alpha-\gamma\}-1$, then there exists an explicit solution $u\in C_{\beta\delta}$ of the problem (2.13), which is given in the form

At last, the Mittag-Leffler function is an entire function, which is represented by the convergent series

where $\alpha\in \mathbb C$, $\beta\in\mathbb C$, $\Re(\cdot)$ denoting the real part of a complex number. The asymptotic expansion of the Mittag-Leffler function is given in the following.

Lemma 2.8. ^[21] Given $\alpha\in(0, 1)$, $\beta\in\mathbb{R}$, and $\gamma\in(\frac{\pi\alpha}{2}, \pi)$, then there exists

for any $z\in\mathbb{C}$ such that $\gamma\le\left\lvert argz \right\rvert < \pi$.

3.   Estimates of the solution of the linear fractional diffusion equation

In this section, we consider the singular initial value problem of the linear fractional diffusion equation

with the initial datum $u_0(x)\in\mathbb D^\nu(\Omega)$. In order to obtain the formal solution of (3.1), applying the method of separation of variables, we obtain the spectral problem

and the corresponding Cauchy problem

where $f_i(t) = \int_\Omega f(t, x)\Theta_i(x)dx$ and $u_{0i} = \int_\Omega u_0(x)\Theta_i(x)dx$.

Take

then, in terms of (2.1) and Lemma 2.7 (2.14), we obtain a formal solution of the problem (3.1), which is represented by

Lemma 3.1. Given $g\in\mathbb{D}^\nu (\Omega)$ and $0\leq\theta\leq 1$, then there exist

Proof. In terms of (3.2), it is easy to derive that

Based on (2.16), we have

then, substituting (3.7) into (3.8), we obtain

This yields (3.5).

Similarly, we derive

Hence, (3.6) is holding.

Theorem 3.2 Given $\theta\in[0, 1]$, $u_0(x)\in\mathbb{D}^\nu (\Omega)$, and $f(t, x)\in C((0, +\infty); \mathbb{D}^{\nu}(\Omega))$ with a bounded norm in the sense of $\underset{\tau\in(0, t)}{sup}\left\lVert t^{\beta(\alpha+1+\theta\gamma)}f(t, x) \right\rVert_{\mathbb{D}^{\nu}(\Omega)} < M$ for some positive constant $M$, then there exists a solution $u\in C((0, +\infty); \mathbb{D}^{\nu+\theta\mu}(\Omega))$ of the problem (3.1), which is expressed by (3.4) and satisfies

In particular, we have

Moreover, there exists ${_{*}\mathbb{D}_{\beta}^{\alpha, \gamma}}u(t, x)\in C(0, +\infty; \mathbb{D}^{\nu+(\theta-1)\mu}(\Omega))$ and

Proof. Based on Lemma 3.1 (3.6), we have

Then, applying (3.4) and Lemma 3.1 (3.5), we obtain

Hence, we derive that $u\in C(0, +\infty; \mathbb{D}^{\nu+\theta\mu}(\Omega))$ which satisfies (3.9) and (3.10). Furthermore, by a direct computation, there exists

Besides, in terms of the first Eq (3.1), we have

Hence, we obtain (3.11) and complete the proof.

Remark 3.3. It is easy to verify that $f(t, x)\in C_{-\beta(\alpha+1+\theta\gamma)-\epsilon}((0, +\infty); \mathbb{D}^{\nu}(\Omega))$ for any $\epsilon > 0$ in Theorem 3.2, then $f(t, x)\in C_{\beta\delta}((0, +\infty); \mathbb{D}^{\nu}(\Omega))$ which is given in Theorem 2.8.

Remark 3.4. The condition $\underset{\tau\in(0, t)}{sup}\left\lVert t^{\beta(\alpha+1+\theta\gamma)}f(t, x) \right\rVert_{\mathbb{D}^{\nu}(\Omega)} < M$ is natural for $\alpha\in (-1, +\infty)$, $\beta\in (0, +\infty)$, $\gamma\in (0, 1]$ and $\theta\in[0, 1]$, because $\beta(\alpha+1+\theta\gamma) > 0$ for $\theta = 0$.

Remark 3.5. The Caputo type modification of the Erdélyi-Kober fractional differential operator has smooth effects; the regularity of the solution is higher than the initial datum with $\theta\mu$ order.

4.   Local well-posedness for space-time fractional diffusion equation with nonlinear source term

First, we consider

with a gradient nonlinearity, where $\alpha\in(-1, +\infty)$, $\beta\in\mathbb (0, \infty)$, $\gamma\in (0, 1]$, $\mu\in(0, 1)$, $\Omega\subseteq \mathbb R^n$, $n\in\mathbb N^+$. Given $\nu, \theta, p$ satisfy

where $m = \max\{1, \frac{2n}{n-4(\nu-\theta\mu)}\}$.

Based on Lemma 2.2, we show a similar result for the problem (4.1) in the same space.

Theorem 4.1. (Uniqueness and stability) Under the conditions (4.2), prescribed $u_0\in\mathbb{D}^\nu(\Omega)$, if $u$ is a solution of the problem (4.1), which satisfies

for some given positive constants $M$ and $T$, then the solution $u\in C((0, T];\mathbb{D}^{\nu+\theta\mu}(\Omega))$ is uniqueness and stability.

Proof. Assume $u_i(t, x)$ is a solution of the following problem

for $u_{01}(x) = u_{02}(x)+\epsilon$.

Uniqueness If $\epsilon = 0$, and $u_1(t, x)\neq u_2(t, x)$, then set $U(t, x) = u_1(t, x)-u_2(t, x)$, we obtain

Under the condition $0 < p < \frac{\alpha+\gamma+1}{\alpha+1+\theta\gamma}$, there exist

According to Lemma 2.2, the nonlinear term satisfies

Then, in terms of (4.6) and (4.7) and Lemma 3.1 (3.6), we have

Besides, for some small $T_0\in[0, 1)$, there exists $T\in[0, T_0]$ such that

Then, for all $t\in[0, T_0]$, it follows

in terms of (4.8). This yields

which means $U(t, x)\equiv 0$ for all $t\in(0, T]$. Then the uniqueness of the solution is established.

Stability If $|\epsilon|\ll 1$, set $U(t, x) = u_1(t, x)-u_2(t, x)$, we consider

Then, by a direct computation, we have

Similarly, we can find a small $T\in(0, 1)$ such that

This yields that the solution of the problem (4.1) continuously depends on the initial datum.

Theorem 4.2. (Existence) Under the conditions (4.2), assume $u_0\in\mathbb{D}^\nu(\Omega)$, then there exists a solution $u\in C((0, T];\mathbb{D}^{\nu+\theta\mu}(\Omega))$ of the problem (4.1) which satisfies (4.3) with $M = M(\left\lVert u_0(x) \right\rVert_{\mathbb{D}^{\nu}(\Omega)})$. Besides, if $p < 1$, then there exists ${_{*}\mathbb{D}_{\beta}^{\alpha, \gamma}}u(t, x)\in C((0, T]; \mathbb{D}^{\nu+(\theta-1)\mu}(\Omega))$.

Proof. Denotes a set $\mathbb S = \{u(t, x)| \underset{t\in(0, T]}{sup}\left\lVert t^{\beta(\alpha+1+\theta\gamma)}u(t, x) \right\rVert_{\mathbb{D}^{\nu+\theta\mu}(\Omega)} < M\}$ for some $M > 0$ and $T > 0$. Consider the sequence $\{u_j(t, x)\}_{j\in\mathbb N}\subset \mathbb S$ expressed by

Set $v_i(t, x) = t^{\beta{\alpha+1+\theta\gamma}}u_i(t, x)$ in the following.Then applying Lemma 3.1 (3.5), we have

This yields $u_1\in \mathbb S$.

In the following, by use of induction methods, we prove $u_j\in S$ for $j\ge 2$. By use of a direct computation, there exists

Taking $u = u_j, v = 0$ in Lemma 2.2, we have

Substituting (4.14) into (4.13), it follows that

Since $u_j\in \mathbb{S}$, and

Then, in terms of (4.15) and (4.16), there exists some small $T$ such that

This yields $u_{j+1}\in \mathbb {S}$. In terms of induction methods, we confirm $\{u_{j}\}_{j\in\mathbb{N}}\in \mathbb{S}$.

In the following, we show $\{u_{j}\}_{j\in\mathbb{N}}\in \mathbb {S}$ is a Cauchy convergent sequence. Consider

Then, for some small $T$, (4.17) becomes into

This implies $\{v_j\}_{j\in\mathbb N^+}$ is a Cauchy convergent sequence, which implies that there exists a $u\in \mathbb S$ such that

Then, we derive

Hence, we established the existence of the solution to the problem (4.1).

Moreover, by using a similar analysis in (4.8) with the first equation of (4.1), we obtain

for some small $T > 0$.

Finally, we complete the proof of Theorem 4.2.

At last, we consider the problem (1.1) with the nonlinear source term $f(u, \nabla u)$, which satisfies $f(0, 0) = 0$, and the Lipschitz condition, that is

Based on Theorem 3.2, applying the fixed point theorem or a similar method used in Theorems 4.1 and 4.2, we give the following results.

Theorem 4.3. (Uniqueness and stability) Given $u_0\in\mathbb{D}^\nu(\Omega)$ and $\theta\in[0, 1)$, prescribed $u\in C((0, T];\mathbb{D}^{\nu+\theta\mu}(\Omega))$ is a solution of the problem (1.1) under the condition (4.19), which satisfies

for some positive constants $M$ and $T$, then the solution is unique and stable.

Proof. Assume $u_i(t, x)$, $i = 1, 2$ is a solution of the following problem

for $u_{01}(x) = u_{02}(x)+\epsilon$. Set $U(t, x) = u_1(t, x)-u_2(t, x)$, then there exists

Uniqueness If $\epsilon = 0$, and $u_1(t, x)\neq u_2(t, x)$, then there exists a nonzero solution $U$ solves the problem

According to the Lipschitz condition (4.19), the nonlinear term satisfies

Then, in terms of (3.9) in Lemma 3.2, we have

Besides, for some small $T_0\in[0, 1)$, there exists $T\in[0, T_0]$ such that

Then, for all $t\in[0, T]$, it follows

in terms of (4.24). This yields

which means $U(t, x)\equiv 0$ for all $t\in(0, T]$. This is a contradiction with $u_1(t, x)\neq u_2(t, x)$.Then the uniqueness of the solution is established.

Stability Consider the problem (4.22) for $|\epsilon|\ll 1$, by a similar computation as deriving (3.9), we have

Then, based on the condition (4.20) and $\theta\in(0, 1)$, we can find a small $T\in(0, 1)$ such that

Finally, we complete the proof of the stability of the solution.

Theorem 4.4. (Existence) Given $u_0\in\mathbb{D}^\nu(\Omega)$ and $\theta\in [0, 1)$, then there exists a solution $u\in C((0, T];\mathbb{D}^{\nu+\theta\mu}(\Omega))$ of the problem (1.1), which satisfies (4.20) for $M = M(\left\lVert u_0(x) \right\rVert_{\mathbb{D}^{\nu}(\Omega)})$. Moreover, there exists ${_{*}\mathbb{D}_{\beta}^{\alpha, \gamma}}u(t, x)\in C((0, T]; \mathbb{D}^{\nu+(\theta-1)\mu}(\Omega))$.

Proof. Using a set $\mathbb S$ as defined in Theorem 4.2 for some $M > 0$ and $T > 0$. Define a mapping $F$ by

where

Then applying Lemma 3.2 (3.9), we have

for some small $T > 0$. This means mapping $F$ maps $\mathbb S$ into itself.

In the following, consider $Fu_1-Fu_2$ for any $u_1, u_2\in \mathbb S$. Based on Lemma 3.2 (3.9) and the Lipschitz condition (4.19), we have

for some small $T$, which implies mapping $F$ is a contraction.

In terms of (4.28) and (4.29), we confirm that the mapping $F$ has one fixed point in $\mathbb S$, and the point is the solution of the problem; then we established the existence of the solution.

Moreover, by use of (3.11) and (4.19), we obtain

for some $T > 0$. Finally, we complete the proof of Theorem 4.4.

5.   Conclusions

This research on the initial boundary value problem of nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative is an continuation of the work ^[17]. Through meticulous calculations, the smooth effects of the Caputo-type modification of the Erdélyi-Kober fractional derivative are established for the first time. Then based on this and the embedding theorem between Hilbert scales spaces and Lebesgue spaces, the well-posedness results are obtained with the nonlinear source term satisfying the Lipschitz condition or the gradient nonlinearity. Compared with the diffusion problems involving a regularized hyper-Bessel operator considered in ^[10,11], we improved the interior regularity of the solution $u\in C((0, T];\mathbb{D}^{\nu+\theta\mu}(\Omega))$ with order $\theta\mu$ if the initial datum $u_0\in\mathbb{D}^\nu(\Omega)$ in our research. These results seem to be meaningful in potential applications and numerical calculations since the Caputo-type fractional models are easy to be interpreted in physical reality, whose initial datum is described with functions and their integer order derivatives, not any other fractional order derivatives.

Author contributions

All authors contributed equally to this work.

Use of AI tools declaration

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

Acknowledgements

Wei Fan was supported by the Practical Innovation Training Program Projects for the University Students of Jiangsu Province of China (No. 202311276102Y). Kangqun Zhang was supported by NSF (No. 23KJB110012), Qinglan Project of Education Department of Jiangsu Province of China and NNSF of China (No. 11326152).

Conflict of interest

The authors declare no competing interests.