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

Boundary Value Problems20192019:10

https://doi.org/10.1186/s13661-019-1125-0

  • Received: 6 September 2018
  • Accepted: 10 January 2019
  • Published:

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

MSC

  • 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.
The hyper-Bessel operator reads
$$ L=t^{a_{1}}\frac{d}{dt}t^{a_{2}} \frac{d}{dt}\cdots\frac {d}{dt}t^{a_{n+1}},\quad t>0, $$
(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, 2528]. 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, 3438] 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:
$$ \bigl\vert E_{\delta,\beta}(y) \bigr\vert \leq\frac{M}{1+|y|}, \quad \mu \leq|\arg y|\leq\pi. $$
where M denotes 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 operator A in \(\mathbb {R}^{n}\). Suppose that \(a(\xi)\in C^{\infty }(\mathbb {R}^{n}\setminus\{0\})\), and there is some positive constant M for 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, A is a bounded linear operator from \(L^{p}(\mathbb {R}^{n})\) into itself for \(1< p<+\infty\), and its operator norm depends only on M, n and p.
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 some ν, the asymptotic representation of M-Wright function for large y is
$$ M_{\nu}\biggl(\frac{y}{\nu}\biggr)\sim a(\nu)y^{\frac{\nu-\frac{1}{2}}{1-\nu}}e^{-b(\nu)y^{\frac{1}{1-\nu}}}. $$

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 constant T.

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.

Declarations

Acknowledgements

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

Availability of data and materials

Not applicable.

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).

Authors’ contributions

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

Ethics approval and consent to participate

Not applicable.

Competing interests

The author declares to have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, China

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