Boundary Value Problems

, 2019:10

# Existence results for a generalization of the time-fractional diffusion equation with variable coefficients

• Kangqun Zhang
Open Access
Research

## Abstract

In this paper we consider the Cauchy problem of a generalization of time-fractional diffusion equation with variable coefficients in $$\mathbb {R}_{+}^{n+1}$$, where the time derivative is replaced by a regularized hyper-Bessel operator. The explicit solution of the inhomogeneous linear equation for any $$n\in\mathbb {Z}^{+}$$ and its uniqueness in a weighted Sobolev space are established. The key tools are Mittag-Leffler functions, M-Wright functions and Mikhlin multiplier theorem. At last, we obtain the existence of solution of the semilinear equation for $$n=1$$ by using a fixed point theorem.

## Keywords

Fractional diffusion equation Mittag-Leffler function M-Wright function Mikhlin multiplier theorem Weighted Sobolev space Existence

34A08 35R11

## 1 Introduction

In this paper we study the existence of solutions for the following generalization of the time-fractional diffusion equation with variable coefficients:
$$\textstyle\begin{cases} {}^{\mathcal{C}} (t^{\theta}\frac{\partial}{\partial t} )^{\alpha}u-\triangle u=f\quad \mbox{in } \mathbb {R}_{+}^{n+1}, \\ u(0,x)=\varphi(x), \end{cases}$$
(1)
where $$\mathbb {R}_{+}^{n+1}=(0,+\infty)\times\mathbb {R}^{n}$$, $$\triangle =\sum_{i=1}^{n}\partial_{x_{i}}^{2}$$ is the Laplace differential operator, $${}^{\mathcal{C}} (t^{\theta}\frac{\partial}{\partial t} )^{\alpha}$$ stands for a Caputo-like counterpart to hyper-Bessel operator of order $$\alpha\in(0,1)$$ and the parameter $$\theta<1$$.
Fractional models are proved to be more adequate than those of integer order for some problems in science and engineering. Fractional differential equations play a very important role in the mathematical modeling of various physical systems [8, 10, 14, 20, 30]. The investigation of (1) is inspired by the fractional extension of the diffusion equation governing the law of the fractional Brownian motion [3, 22]:
$$\biggl(t^{1-2H}\frac{\partial}{\partial t} \biggr)^{\alpha}u(t,x)=H^{\alpha}\frac{\partial^{2}}{\partial x^{2}}u(t,x), \quad \alpha\in (0,1), H\in(0,1), x\in\mathbb {R},$$
(2)
where $$(t^{1-2H}\frac{\partial}{\partial t} )^{\alpha}$$ is a hyper-Bessel type operator. Set $$y=H^{\frac{\alpha}{2}}x$$ and $$1-2H=\theta$$, then (2) is reduced into
$$\biggl(t^{\theta}\frac{\partial}{\partial t} \biggr)^{\alpha}u(t,y)-\frac{\partial^{2}}{\partial y^{2}}u(t,y)=0,\quad \alpha\in (0,1), \theta\in(-1,1), x\in \mathbb {R},$$
(3)
which is a special case of (1). For the general case, [11, 12] provided the definition of the operator $$(t^{\theta}\frac{\partial }{\partial t} )^{\alpha}$$ for $$\alpha\in(0,1]$$ and $$\theta\in\mathbb {R}$$ when studying the fractional diffusions and fractional relaxation.
$$L=t^{a_{1}}\frac{d}{dt}t^{a_{2}} \frac{d}{dt}\cdots\frac {d}{dt}t^{a_{n+1}},\quad t>0,$$
(4)
where $$a_{i}$$, $$i=1,2,\ldots,n+1$$, are real numbers and $$n\in\mathbb {Z}^{+}$$. To the best of our knowledge, the fractional power $$L^{\alpha}$$ of the hyper-Bessel operator was first introduced by Dimovski [9] and developed by McBride and Lamb [19, 23, 24]. The theory of $$L^{\alpha}$$ has been applied to solve various problems, such as diffusive transport [11, 12, 29], Brownian motion [3, 22, 25, 26, 27, 28]. Recently, Al-Musalhi, Al-Salti, and Karimov generalized $$(t^{\theta}\frac{d}{d t} )^{\alpha}$$ to the Caputo-like counterpart of hyper-Bessel operator $${}^{\mathcal{C}} (t^{\theta}\frac{\partial }{\partial t} )^{\alpha}$$ in [1] defined by
$${}^{\mathcal{C}} \biggl(t^{\theta}\frac{d}{d t} \biggr)^{\alpha}f(t)= \biggl(t^{\theta}\frac{d}{dt} \biggr)^{\alpha}f(t)-\frac{f(0)t^{-\alpha (1-\theta)}}{(1-\theta)^{-\alpha}\varGamma(1-\theta)}, \quad 0< \alpha < 1, \theta< 1.$$
They used Erdélyi–Kober fractional integral to express the hyper-Bessel operator and established the series solution by considering both direct and inverse source problem in a rectangular domain. In [2], Al-Saqabi and his collaborators considered Volterra integral equation of the second kind and a fractional differential equation, involving Erdélyi–Kober fractional integral or differential operator. The explicit solutions of these equations were derived by use of transmutation method. For a special case of $$\theta=0$$ and $$\alpha>1$$, the existence of unique solution was established by use of a perturbation argument and Green’s function in [4, 5]. In [13], applying a direct variational approach and the theory of the fractional derivative spaces, the existence of infinitely many distinct positive solutions were given. For more results related to hyper-Bessel operator and Erdélyi–Kober fractional integral or differential operator, see [6, 29, 31] and references therein. However, these methods and techniques cannot be directly employed to the multidimensional or the nonlinear case in Sobolev space. In this paper, we will go a step further to form the explicit solution in multidimensional space, then use Mittag-Leffler functions and Mikhlin’s multiplier theorem to obtain the weighted $$\dot{H}^{s,p}$$, $$1< p<+\infty$$ and $$L^{\infty}$$ estimate of the solution. At last, we form a contractible mapping to show the existence of solution of the semilinear problem in a suitable fractional derivative Sobolev space. The main idea is motivated in the proof of [32, 33]. The existence of solutions in Banach spaces were also investigated in [7, 13, 34, 35, 36, 37, 38] and the necessary and sufficient conditions on the initial data for the solvability of a space-fractional semilinear parabolic equation were obtained in [17].

This paper is organized as follows: In Sect. 2, the related results of Mittag-Leffler functions and M-Wright functions are recalled. The explicit solution of a related time-fractional ordinary differential equation is established. In Sect. 3, in terms of the explicit solution given in Sect. 2, we derive the existence and uniqueness of solution $$u\in C([0,+\infty),L^{p}(\mathbb {R}^{n}))\cap C((0,+\infty),\dot{H}^{k,p}(\mathbb {R}^{n}))\cap C^{\alpha}((0,+\infty),L^{p}(\mathbb {R}^{n}))$$, $$k=1,2$$ of the corresponding linear problem. In the last section, by use a fixed point theorem we show the existence of solution $$u\in C([0,T),L^{p}(\mathbb {R}))\cap C((0,T),\dot{H}^{k,p}(\mathbb {R}))\cap C^{\alpha}((0,T),L^{p}(\mathbb {R}))$$, $$k=1,2$$ of the semilinear problem for a fixed positive number T.

## 2 Preliminaries

In this section we present some necessary definitions and auxiliary results for the convenience of the reader, then establish the explicit solution of the Cauchy problem of a time-fractional ordinary differential equation.

First, we recall Mittag-Leffler function $$E_{\delta,\beta}(z)$$ with two parameters, which can be found in [15, 16] or [30],
$$E_{\delta,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\varGamma (\delta k+\beta)},\quad \Re(\delta)>0, \Re(\beta)>0.$$
(5)

### Lemma 2.1

\begin{aligned}& \frac{d}{dy}E_{\delta,\beta}(y)=\frac{E_{\delta,\beta -1}(y)-(\beta-1)E_{\delta,\beta}(y)}{\delta y}, \end{aligned}
(6)
\begin{aligned}& \frac{d^{m}}{dy^{m}} \bigl(y^{\beta-1}E_{\delta,\beta }\bigl(y^{\delta}\bigr) \bigr)=y^{\beta-m-1}E_{\delta,\beta-m}\bigl(y^{\alpha}\bigr),\quad \Re (\beta-m)>0, m\in\mathbb {N}. \end{aligned}
(7)

### Lemma 2.2

Let$$\delta<2$$, $$\beta\in\mathbb {R}$$and$$\frac {\pi \delta}{2}<\mu<\min\{\pi,\pi\delta\}$$. Then we have the following estimate:
$$\bigl\vert E_{\delta,\beta}(y) \bigr\vert \leq\frac{M}{1+|y|}, \quad \mu \leq|\arg y|\leq\pi.$$
whereMdenotes a positive constant.

### Lemma 2.3

For each$$k\in\mathbb {Z}^{+}$$and any$$\Re(\alpha)>0$$, $$\beta\in\mathbb {R}$$, $$0\leq\delta\leq1$$, there exists a positive constant$$C_{k}$$such that
$$|y|^{k} \biggl\vert \frac{d^{k}}{dy^{k}} \bigl(y^{\delta}E_{\alpha,\beta }(y) \bigr) \biggr\vert \leq C_{k}.$$
(8)

### Proof

For $$k=1$$, (8) directly follows from (6) in Lemma 2.1 and Lemma 2.2.

For $$k=2$$, $$y^{2}\frac{d^{2}}{dy^{2}}=(y\frac{d}{dy})^{2}-y\frac{d}{dy}$$. Then it is enough to show $$(y\frac{d}{dy})^{2} (y^{\delta}E_{\alpha,\beta }(y) )$$ is bounded. By a direct computation in terms of (6), we get that
\begin{aligned} &\biggl(y\frac{d}{dy}\biggr)^{2} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr) \\ &\quad =\frac{1}{\alpha}y\frac{d}{dy} \bigl(y^{\delta}\bigl(E_{\alpha,\beta -1}(y)-(\beta-1)E_{\alpha,\beta}(y)\bigr) \bigr)+\delta y \frac{d}{dy} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr). \end{aligned}
This reduces to $$k=1$$. Hence, (8) holds for $$k=2$$. Furthermore, following the same idea, we conclude that $$(y\frac{d}{dy})^{k} (y^{\delta}E_{\alpha,\beta}(y) )$$ is bounded for any $$k\in\mathbb {Z}^{+}$$.
By induction, assume for $$k-1$$ that
\begin{aligned}& |y|^{k-1} \biggl\vert \frac{d^{k-1}}{dy^{k-1}} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr) \biggr\vert \leq C_{k-1}, \end{aligned}
(9)
\begin{aligned}& y^{k-1}\frac{d^{k-1}}{dy^{k-1}}=\sum_{i=1}^{k-1}b_{i} \biggl(y\frac{d}{dy}\biggr)^{i}, \end{aligned}
(10)
where $$b_{i}$$ are constants. Then by use of (6) or (7), we have
\begin{aligned} &y^{k}\biggl(\frac{d}{dy}\biggr)^{k} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr) \\ &\quad =y\frac{d}{dy} \Biggl(\sum_{i=1}^{k-1}b_{i} \biggl(y\frac{d}{dy}\biggr)^{i} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr) \Biggr) \\ &\quad =\sum_{i=1}^{k}d_{i} \biggl(y\frac{d}{dy}\biggr)^{i} \bigl(y^{\delta}E_{\alpha,\beta }(y) \bigr). \end{aligned}
(11)
It follows from (9) and (11) that (8) holds. □

From (8) we can prove the following.

### Corollary 2.4

For each$$\gamma\in Z^{+}$$and any$$\alpha>0$$, $$\beta\in\mathbb {R}$$, $$0\leq\delta\leq1$$, there exists a positive constant$$C_{\gamma}$$such that
$$\biggl\vert |\xi|^{\gamma}\frac{\partial^{\gamma}}{\partial\xi ^{\gamma}} \bigl(y^{\delta}E_{\alpha,\beta}(y) \bigr) \biggr\vert \leq C_{\gamma},$$
(12)
where$$y=-\rho^{-\alpha}|\xi|^{2}t^{\rho\alpha}$$.

Next, we choose the version of Mikhlin’s multiplier theorem given in [18] as our lemma.

### Lemma 2.5

Let$$a(\xi)$$be the symbol of a singular integral operatorAin$$\mathbb {R}^{n}$$. Suppose that$$a(\xi)\in C^{\infty }(\mathbb {R}^{n}\setminus\{0\})$$, and there is some positive constantMfor all$$\xi\neq0$$such that
$$|\xi|^{|\gamma|} \biggl\vert \frac{\partial^{\gamma}a(\xi )}{\partial\xi^{\gamma}} \biggr\vert \leq M,\quad 0\leq|\gamma|\leq1+\frac{[n]}{2}.$$
Then, Ais a bounded linear operator from$$L^{p}(\mathbb {R}^{n})$$into itself for$$1< p<+\infty$$, and its operator norm depends only onM, nandp.
Based on expression (5), the explicit solution of the following problem of the inhomogeneous time-fractional differential equation
$$\textstyle\begin{cases} {}^{C} (t^{\theta}\frac{d}{dt} )^{\alpha}u(t)=-\lambda u(t)+f(t),\quad t>0, \\ u(0)=u_{0}, \end{cases}$$
(13)
is obtained, where $$u_{0}$$ is a constant number, $$\theta<1$$, $$0<\alpha<1$$.

### Theorem 2.6

Consider problem (13). Then there is an explicit solution, which is given in the integral form
$$u(t)=u_{0}E_{\alpha,1}\bigl(\lambda^{*} t^{\rho\alpha}\bigr)+\frac {1}{\rho^{\alpha}} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1}E_{\alpha,\alpha }\bigl(\lambda^{*}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr) f(s)\, d\bigl(s^{\rho}\bigr),$$
(14)
where$$\rho=1-\theta$$and$$\lambda^{*}=-\frac{\lambda}{\rho^{\alpha}}$$.

### Proof

In terms of Lemma 2.7 given in [1], the expression of $$u(t)$$ is written as
\begin{aligned} u(t)&=u_{0}E_{\alpha,1}\bigl(\lambda^{*} t^{\rho\alpha}\bigr)+ \frac {1}{\rho^{\alpha}\varGamma(\alpha)} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha -1}f(s)\, d\bigl(s^{\rho}\bigr) \\ &\quad {}+\frac{\lambda^{*}}{\rho^{\alpha}} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{2\alpha -1}E_{\alpha,2\alpha}\bigl(\lambda^{*}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr)f(s)\, d\bigl(s^{\rho}\bigr) \\ &=u_{0}E_{\alpha,1}\bigl(\lambda^{*} t^{\rho\alpha}\bigr)+ \frac{1}{\rho^{\alpha}\varGamma (\alpha)} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1} \\ &\quad {}\times \bigl(1+\varGamma(\alpha)\lambda^{*}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}E_{\alpha ,2\alpha}\bigl(\lambda^{*}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr) \bigr)f(s)\, d\bigl(s^{\rho}\bigr), \end{aligned}
(15)
Besides, the integrand in the last integral of (16) satisfies
\begin{aligned} &1+\varGamma(\alpha)y^{\alpha}E_{\alpha,2\alpha}\bigl(y^{\alpha}\bigr) \\ &\quad =1+\varGamma(\alpha)\sum_{k=0}^{\infty}\frac{y^{(k+1)\alpha}}{\varGamma(k\alpha +2\alpha)} \\ &\quad =1+\varGamma(\alpha)\sum_{k=1}^{\infty}\frac{y^{k\alpha}}{\varGamma(k\alpha +\alpha)} \\ &\quad =\varGamma(\alpha)\sum_{k=0}^{\infty}\frac{y^{k\alpha}}{\varGamma(k\alpha +\alpha)} \\ &\quad =\varGamma(\alpha)E_{\alpha,\alpha}\bigl(y^{\alpha}\bigr). \end{aligned}
(16)
Then substituting (16) into (15) with $$y^{\alpha}=\lambda^{*}(t^{\rho}-s^{\rho})^{\alpha}$$, the explicit solution (14) is established.

Hence, we complete the proof of Theorem 2.6. □

Last, we recite the asymptotic behavior of M-Wright function derived in [21], which is defined as
$$M_{\nu}(y)=\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!\varGamma(-n\nu +1-\nu)},\quad \nu\in(0,1).$$

### Lemma 2.7

Given$$a(\nu)=\frac{1}{\sqrt{2\pi(1-\nu)}}>0$$, $$b(\nu)=\frac{1-\nu}{\nu}>0$$for someν, the asymptotic representation ofM-Wright function for largeyis
$$M_{\nu}\biggl(\frac{y}{\nu}\biggr)\sim a(\nu)y^{\frac{\nu-\frac{1}{2}}{1-\nu}}e^{-b(\nu)y^{\frac{1}{1-\nu}}}.$$

## 3 Existence and uniqueness of solution of the linear problem

In this section, based on Theorem 2.6, Mattag-Leffler function, M-Wright functions and Mikhlin multiplier theorem, we show the existence of $$L^{p}$$ solution of the corresponding linear problem (1) for any $$n\in\mathbb {Z}^{+}$$.

We first consider the linear problem
$$\textstyle\begin{cases} {}^{\mathcal{C}} (t^{\theta}\frac{\partial}{\partial t} )^{\alpha}u-\triangle u=f(t,x)\quad \mbox{in } \mathbb {R}_{+}^{n+1}, \\ u(0,x)=\varphi(x). \end{cases}$$
(17)
Taking partial Fourier transformation with respect to x in Eq. (17) yields the following problem:
$$\textstyle\begin{cases} {}^{\mathcal{C}} (t^{\theta}\frac{\partial}{\partial t} )^{\alpha}\hat{u}(t,\xi)=-|\xi|^{2}\hat{u}(t,\xi)+\hat{f}(t,\xi) \quad \mbox{in } \mathbb {R}_{+}^{n+1}, \\ \hat{u}(0,\xi)=\hat{\varphi}(\xi), \end{cases}$$
where $$\hat{u}(t,\xi)=\mathfrak{F}(u(t,x))=\int_{\mathbb {R}^{n}}e^{-ix\cdot \xi }u(t,x)\,dx$$.
Set $$\lambda=|\xi|^{2}$$ in (11). According to Theorem 2.6, the solution of (17) is given by
$$u(t,x)= u_{0}(t,x)+\frac{1}{\rho^{\alpha}} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1}\mathfrak{F^{-1}} \bigl(E_{\alpha,\alpha}\bigl(-\rho ^{-\alpha}|\xi|^{2}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr) f(s,\xi) \bigr)\, d\bigl(s^{\rho}\bigr),$$
(18)
where
$$u_{0}(t,x)=\mathfrak{F^{-1}} \bigl(\hat{\varphi}(\xi )E_{\alpha,1}\bigl(-\rho^{-\alpha}|\xi|^{2} t^{\rho\alpha}\bigr) \bigr).$$
(19)

### Theorem 3.1

Set$$1< p<+\infty$$, $$\alpha\in(0,1)$$, $$\theta <1$$. Suppose$$\varphi\in C^{\infty}_{0}(\mathbb {R}^{n})$$, $$f\in C^{\infty}_{0}(\mathbb {R}_{+}^{n+1})$$, then there exists a unique solution$$u\in C([0,+\infty ),L^{p}(\mathbb {R}^{n}))\cap C((0,+\infty),\dot{H}^{k,p}(\mathbb {R}^{n}))\cap C^{\alpha}((0,+\infty), L^{p}(\mathbb {R}^{n}))$$of problem (17), which is represented by (18) under Fourier transformation and satisfies
\begin{aligned} &\sum_{k=0}^{2} \bigl\Vert t^{\delta_{k}}u(t,\cdot) \bigr\Vert _{\dot {H}^{k,p}(\mathbb {R}^{n})}+ \biggl\Vert t^{\delta_{2}} {}^{\mathcal{C}} \biggl(t^{\theta}\frac {\partial}{\partial t} \biggr)^{\alpha}u(t,\cdot) \biggr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad \lesssim \Vert \varphi \Vert _{L^{p}(\mathbb {R}^{n})}+t^{\delta_{2}} \int_{0}^{1}\sum_{k=0}^{2} \bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac{k\alpha}{2}} \bigl\Vert f(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\, d\bigl(s^{\rho}\bigr), \end{aligned}
(20)
where$$\dot{H}^{k,p}(\mathbb {R}^{n})$$denotes the homogeneous Sobolev space, $$\delta_{k}=\frac{\rho\alpha k}{2}$$, $$\rho=1-\theta$$.

### Proof

It follows from (18)–(19) that
\begin{aligned} & \bigl\Vert u(t,\cdot) \bigr\Vert _{\dot{H}^{\delta,p}(\mathbb {R}^{n})} \\ &\quad = \bigl\Vert \mathfrak {F}^{-1} \bigl(|\xi|^{\delta}\hat{u}(t,\xi) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad \leq \bigl\Vert \mathfrak{F}^{-1} \bigl(|\xi|^{\delta}\hat{u}_{0}(t,\xi) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\qquad {} + \biggl\Vert \mathfrak{F}^{-1} \biggl( \frac{|\xi|^{\delta}}{\rho^{\alpha}} \int _{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1}E_{\alpha,\alpha}\bigl(-\rho^{-\alpha}|\xi |^{2}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr) \hat{f}(s,\xi)\,d\bigl(s^{\rho}\bigr) \biggr) \biggr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad \lesssim \bigl\Vert \mathfrak{F}^{-1} \bigl(\hat{\varphi}(\xi) t^{-\frac{\rho\alpha \delta}{2}}\bigl(-\rho^{-\alpha}|\xi|^{2} t^{\rho\alpha} \bigr)^{\frac{\delta }{2}}E_{\alpha,1}\bigl(-\rho^{-\alpha}|\xi|^{2} t^{\rho\alpha}\bigr) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}+ \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1-\frac{\delta \alpha }{2}} \\ &\qquad {} \times \bigl\Vert \mathfrak{F}^{-1} \bigl( \bigl(- \rho^{-\alpha}|\xi|^{2}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr)^{\frac{\delta}{2}}E_{\alpha,\alpha}\bigl(- \rho^{-\alpha}|\xi |^{2}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\bigr)\hat{f}(s,\xi) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\,d \bigl(s^{\rho}\bigr). \end{aligned}
(21)
Let $$y=-\rho^{-\alpha}|\xi|^{2}(t^{\rho}-s^{\rho})^{\alpha}$$, then (12) yields
$$|\xi|^{\gamma}\biggl\vert \frac{\partial^{\gamma}}{\partial\xi ^{\gamma}} \bigl(y^{\frac{\delta}{2}}E_{\alpha,\beta}(y) \bigr) \biggr\vert \leq C_{\gamma}.$$
According to Lemma 2.5, we have
\begin{aligned}& \bigl\Vert \mathfrak{F}^{-1} \bigl(\hat{\varphi}(\xi) t^{-\frac{\rho \alpha\delta}{2}}y^{\frac{\delta}{2}}E_{\alpha,1}(y) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\lesssim t^{-\frac{\rho\alpha\delta}{2}} \Vert \varphi \Vert _{L^{p}(\mathbb {R}^{n})}, \end{aligned}
(22)
\begin{aligned}& \bigl\Vert \mathfrak{F}^{-1} \bigl(y^{\frac{\delta}{2}}E_{\alpha ,\alpha}(y) \hat{f}(s,\xi) \bigr) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\lesssim \bigl\Vert f(s,\cdot ) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}. \end{aligned}
(23)
Substituting (22)–(23) into (21), we get
$$\bigl\Vert u(t,\cdot) \bigr\Vert _{\dot{H}^{\delta,p}(\mathbb {R}^{n})}\lesssim t^{-\frac{\rho\alpha\delta}{2}} \biggl( \Vert \varphi \Vert _{L^{p}(\mathbb {R}^{n})}+t^{\rho\alpha} \int_{0}^{1}\bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac{\delta\alpha }{2}} \bigl\Vert f(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\, d \bigl(s^{\rho}\bigr) \biggr).$$
Summing up with $$\delta=0,1,2$$, we arrive at the following estimate:
\begin{aligned} \sum_{k=0}^{2} \bigl\Vert t^{\delta_{k}}u(t,\cdot) \bigr\Vert _{\dot {H}^{k,p}(\mathbb {R}^{n})}&\lesssim \Vert \varphi \Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad {}+t^{\rho\alpha} \int_{0}^{1}\sum_{k=0}^{2} \bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac{k\alpha }{2}} \bigl\Vert f(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\, d\bigl(s^{\rho}\bigr) \end{aligned}
(24)
with $$\delta_{k}=\frac{\rho\alpha k}{2}$$.
For the term $${}^{\mathcal{C}} (t^{\theta}\frac{\partial}{\partial t} )^{\alpha}u(t,\cdot)$$, we will use Eq. (17) to estimate as follows:
\begin{aligned} & \biggl\Vert {}^{\mathcal{C}} \biggl(t^{\theta}\frac{\partial}{\partial t} \biggr)^{\alpha}u(t,\cdot) \biggr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad = \bigl\Vert \triangle u+f(t,x) \bigr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad \lesssim \bigl\Vert u(t,\cdot) \bigr\Vert _{\dot{H}^{2,p}(\mathbb {R}^{n})}+ \bigl\Vert f(t,\cdot) \bigr\Vert _{L^{p}(\mathbb {R}^{n})} \\ &\quad \lesssim t^{-\rho\alpha} \Biggl( \Vert \varphi \Vert _{L^{p}(\mathbb {R}^{n})}+t^{\rho \alpha} \int_{0}^{1}\sum_{k=0}^{2} \bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac{k\alpha}{2}} \bigl\Vert f(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R}^{n})}\, d\bigl(s^{\rho}\bigr) \Biggr). \end{aligned}
(25)
Combing (24) and (25), we arrive at (20), which implies the existence and uniqueness of solution $$u\in C([0,+\infty),L^{p}(\mathbb {R}^{n}))\cap C((0,+\infty),\dot{H}^{k,p}(\mathbb {R}^{n}))\cap C^{\alpha}((0,+\infty ),L^{p}(\mathbb {R}^{n}))$$, $$k=1,2$$.

Thus, we complete the proof of Theorem 3.1. □

## 4 Existence of solution of the semilinear problem

In this section, we consider the semilinear problem (1) in the half-space $$\mathbb {R}^{2}_{+}$$ and show the existence of a solution by use of a fixed point theorem.

We assume a condition on the nonlinear term with a positive constant C so that
$$\bigl\vert f(u) \bigr\vert \lesssim \vert u \vert ^{\mu},\qquad \bigl\vert f^{(k)}(u) \bigr\vert \lesssim C, \quad \mu>1, k=1,2.$$
(26)

The $$L^{\infty}$$-norm estimate of $$u_{0}(t,x)$$ is necessary, with $$u_{0}(t,x)$$ defined in (19).

### Theorem 4.1

$$\bigl\Vert u_{0}(t,\cdot) \bigr\Vert _{L^{\infty}(\mathbb {R}_{+}^{2})}\lesssim \Vert \varphi \Vert _{L^{\infty}(\mathbb {R})}.$$
(27)

### Proof

It follows from (19) that
$$u_{0}(t,x)=\mathfrak{F^{-1}} \bigl(E_{\alpha,1}\bigl(- \rho ^{-\alpha}|\xi|^{2} t^{\rho\alpha}\bigr) \bigr)*\varphi(x),$$
and then we arrive
$$\bigl\Vert u_{0}(t,\cdot) \bigr\Vert _{L^{\infty}(\mathbb {R}_{+}^{2})}\lesssim \bigl\Vert \mathfrak{F^{-1}} \bigl(E_{\alpha,1}\bigl(-\rho^{-\alpha}|\xi|^{2} t^{\rho\alpha }\bigr) \bigr) \bigr\Vert _{L^{\infty}((0,+\infty),L^{1}(\mathbb {R}))} \Vert \varphi \Vert _{L^{\infty}(\mathbb {R})}.$$
(28)
The Fourier transformation of M-Wright function given by (4.15) in [12] is
$$\mathfrak{F} \bigl(M_{\nu}\bigl( \vert x \vert \bigr) \bigr)=2E_{2\nu,1}\bigl(-|\xi|^{2}\bigr),$$
which implies
$$\mathfrak{F^{-1}} \bigl(E_{\alpha,1}\bigl(-\rho^{-\alpha} | \xi |^{2}t^{\rho\alpha}\bigr) \bigr)=\frac{\rho^{\frac{\alpha}{2}}}{2t^{\frac{\rho\alpha }{2}}}M_{\frac{\alpha}{2}} \bigl(\rho^{\frac{\alpha}{2}} |x|t^{-\frac{\rho \alpha}{2}}\bigr).$$
Then by a direct computation in terms of the analytic expression of M-Wright function and the asymptotics for large variables given in Lemma 2.7, we have
\begin{aligned} & \bigl\Vert \mathfrak{F^{-1}} \bigl(E_{\alpha,1}\bigl(- \rho^{-\alpha}|\xi |^{2} t^{\rho\alpha}\bigr) \bigr) \bigr\Vert _{L^{\infty}((0,+\infty),L^{1}(\mathbb {R}))} \\ &\quad \leq \biggl\Vert \frac{\rho^{\frac{\alpha}{2}}}{2t^{\frac{\rho\alpha}{2}}}M_{\frac {\alpha}{2}}\bigl( \rho^{\frac{\alpha}{2}} |x|t^{-\frac{\rho\alpha}{2}}\bigr) \biggr\Vert _{L^{\infty}((0,+\infty),L^{1}(\mathbb {R}))} \\ &\quad \leq C. \end{aligned}
(29)
Substituting (29) into (28), we obtain (27).

This concludes the proof of Theorem 4.1. □

### Theorem 4.2

Set$$1< p<+\infty$$, $$\alpha\in(0,1)$$, $$\theta <1$$. Suppose$$\varphi\in C^{\infty}_{0}(\mathbb {R})$$and let$$f(t,x,\cdot)$$satisfy (26), then there exists a solution$$u\in C([0,T),L^{p}(\mathbb {R}))\cap C((0,T),\dot{H}^{k,p}(\mathbb {R}))\cap C^{\alpha}((0,T),L^{p}(\mathbb {R}))$$, $$k=1,2$$to problem (1) for some positive constantT.

### Proof

Set $$S_{M}$$ denote a closed set given by
\begin{aligned} S_{M}&\equiv \Bigl\{ u\in C\bigl([0,T),L^{p}(\mathbb {R})\bigr)\cap C\bigl((0,T),\dot{H}^{k,p}(\mathbb {R})\bigr) \\ &\quad {}\cap C^{\alpha}\bigl((0,T),L^{p}(\mathbb {R})\bigr):\sup _{t\in(0,T)} \bigl\Vert u(t,\cdot) \bigr\Vert _{S_{M}} \leq M \Bigr\} , \end{aligned}
where
$$\bigl\Vert u(t,\cdot) \bigr\Vert _{S_{M}}=\sum _{k=0}^{2} \bigl\Vert t^{\delta _{k}}u(t,\cdot) \bigr\Vert _{\dot{H}^{k,p}(\mathbb {R})}+ \biggl\Vert t^{\delta_{2}} {}^{\mathcal{C}} \biggl(t^{\theta}\frac{\partial}{\partial t} \biggr)^{\alpha}u(t,\cdot) \biggr\Vert _{L^{p}(\mathbb {R})}$$
and $$\delta_{k}=\frac{\rho\alpha k}{2}$$, $$\rho=1-\theta$$, the positive constants T and M will be given in the following.
Consider the nonlinear mapping F in $$S_{M}$$ such that
\begin{aligned} Fu&=\mathfrak{F^{-1}} (\hat{\varphi}(\xi)E_{\alpha ,1}\bigl(- \rho^{-\alpha}|\xi|^{2} t^{\rho\alpha}\bigr) \\ &\quad {}+\frac{1}{\rho^{\alpha}} \int_{0}^{t}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha-1}E_{\alpha ,\alpha}\bigl(-\rho^{-\alpha}| \xi|^{2}\bigl(t^{\rho}-s^{\rho}\bigr)^{\alpha}\hat{f}\bigl(s,\xi ,u(s,\xi)\bigr)\, d\bigl(s^{\rho}\bigr) \bigr). \end{aligned}
On the one hand, in terms of a modified result of Theorem 3.1 and Theorem 4.1, we arrive at
\begin{aligned} \bigl\Vert Fu(t,\cdot) \bigr\Vert _{S_{M}}&\lesssim \Vert \varphi \Vert _{L^{p}(\mathbb {R})}+t^{\delta_{2}} \int_{0}^{1} \Biggl(\sum _{k=0}^{2}\bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac {k\alpha}{2}} \bigl\Vert f(u) (st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})} \Biggr)\,d\bigl(s^{\rho}\bigr) \\ &\lesssim \Vert \varphi \Vert _{L^{p}(\mathbb {R})}+t^{\delta_{2}} \int_{0}^{1} \Biggl(\sum _{k=0}^{1}\bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac{k\alpha}{2}}{ \bigl\Vert u(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})} \bigl\Vert u(st,\cdot) \bigr\Vert ^{\mu-1}_{L^{\infty}(\mathbb {R}_{+}^{2})}} \\ &\quad {} +\bigl(1-s^{\rho}\bigr)^{\frac{\alpha}{2}-1}(st)^{-\delta_{1}} \bigl\Vert {(st)^{\delta _{1}}\partial_{i}u(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})}} \Biggr)\,d\bigl(s^{\rho}\bigr) \\ &\lesssim \Vert \varphi \Vert _{L^{p}(\mathbb {R})}+t^{\delta_{2}} \Vert \varphi \Vert ^{\mu -1}_{L^{\infty}(\mathbb {R})}\sum_{k=0}^{1} \int_{0}^{1}\bigl(1-s^{\rho}\bigr)^{\alpha -1-\frac {k\alpha}{2}} \bigl\Vert u(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})} \\ &\quad {} +t^{\delta_{1}} \int_{0}^{1}\bigl(1-s^{\rho}\bigr)^{\frac{\alpha}{2}-1}s^{-\delta _{1}} \bigl\Vert {(st)^{\delta_{1}} \partial_{i}u(st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})}}\,d \bigl(s^{\rho}\bigr) \\ &\leq C_{0} \Vert \varphi \Vert _{L^{p}(\mathbb {R})}+C_{1} \bigl(t^{\delta_{2}} \Vert \varphi \Vert ^{\mu -1}_{L^{\infty}(\mathbb {R})}+t^{\delta_{1}} \bigr)\sup_{t\in(0,T)} \bigl\Vert u(t,\cdot) \bigr\Vert _{S_{M}}. \end{aligned}
(30)
Take T such that
$$\frac{1}{2}-C_{1}\bigl(T^{\delta_{2}}\|\varphi \|^{\mu -1}_{L^{\infty}(\mathbb {R})}+T^{\delta_{1}}\bigr)>0,$$
(31)
then for $$M=2C_{0}\|\varphi\|_{L^{p}(\mathbb {R}^{n})}$$, (30)–(31) yield
$$\sup_{t\in(0,T)} \bigl\Vert Fu(t,\cdot) \bigr\Vert _{S_{M}}\leq M.$$
(32)
This demonstrates that the mapping F maps $$S_{M}$$ into itself.
On the other hand, for any $$u\in S_{M}$$, $$v\in S_{M}$$, by a direct computation, we have
\begin{aligned} & \bigl\Vert (Fu-Fv) (t,\cdot) \bigr\Vert _{S_{M}} \\ &\quad \lesssim t^{\delta_{2}} \int_{0}^{1}\sum_{k=0}^{1} \bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac {k\alpha}{2}} \bigl\Vert f(u) (st,\cdot)-f(v) (st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})}\,d\bigl(s^{\rho}\bigr) \\ &\qquad {} +t^{\delta_{2}} \int_{0}^{1}\bigl(1-s^{\rho}\bigr)^{\frac{\alpha}{2}-1} \bigl\Vert \partial _{i}\bigl(f(u)-f(v)\bigr) (st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})}\,d\bigl(s^{\rho}\bigr) \\ &\quad \lesssim t^{\delta_{2}} \int_{0}^{1}\sum_{k=0}^{1} \bigl(1-s^{\rho}\bigr)^{\alpha-1-\frac {k\alpha}{2}} \bigl\Vert (u-v) (st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})} \bigl( \Vert u \Vert ^{\mu -1}_{L^{\infty}(\mathbb {R}_{+}^{2})}+ \Vert v \Vert ^{\mu-1}_{L^{\infty}(\mathbb {R}_{+}^{2})} \bigr)\,d\bigl(s^{\rho}\bigr) \\ &\qquad {} +t^{\delta_{1}} \int_{0}^{1}\bigl(1-s^{\rho}\bigr)^{\frac{\alpha}{2}-1}s^{\delta _{1}} \bigl( \bigl\Vert \partial_{i}(u-v) (st,\cdot) \bigr\Vert _{L^{p}(\mathbb {R})}+ \bigl\Vert (u-v) (st,\cdot ) \bigr\Vert _{L^{p}(\mathbb {R})} \bigr)\,d\bigl(s^{\rho}\bigr) \\ &\quad \leq C_{1} \bigl(T^{\delta_{2}} \Vert \varphi \Vert ^{\mu-1}_{L^{\infty}(\mathbb {R})}+T^{\delta_{1}} \bigr)\sup_{t\in(0,T)} \bigl\Vert (u-v) (t,\cdot) \bigr\Vert _{S_{M}}. \end{aligned}
(33)
According to (31) and (33), one has
$$\sup_{t\in(0,T)} \bigl\Vert (Fu-Fv) (t,\cdot) \bigr\Vert _{S_{M}}< \sup_{t\in (0,T)} \bigl\Vert (u-v) (t, \cdot) \bigr\Vert _{S_{M}},$$
(34)
which implies that mapping F is a contraction.

In terms of (32) and (34), we confirm that mapping F has one fixed point in $$S_{M}$$. This concludes the proof of Theorem 4.2. □

## 5 Conclusions

In this paper, the Cauchy problem (1) has been considered. By means of Mikhlin’s multiplier theorem, in terms of Mittag-Leffler functions and M-Wright functions, we obtained an explicit solution $$u\in C([0,+\infty),L^{p}(\mathbb {R}^{n}))\cap C((0,+\infty),\dot{H}^{k,p}(\mathbb {R}^{n}))\cap C^{\alpha}((0,+\infty),L^{p}(\mathbb {R}^{n}))$$, $$k=1,2$$ for the linear equation with a source term. Meanwhile, the local existence of a solution of the semilinear equation in $$\mathbb {R}_{+}^{2}$$ was obtained by a fixed point theorem.

## Notes

### Acknowledgements

The authors wish to thank the editor and anonymous referees for their valuable suggestions.

Not applicable.

### Authors’ contributions

The author discussed, read and approved the final version of the manuscript.

### Funding

This work was supported by NNSF of China (No. 11326152), NSF of Jiangsu Province of China (No. BK20130736) and NSF of Nanjing Institute of Technology (CKJB201709).

Not applicable.

### Competing interests

The author declares to have no competing interests.

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