一类抛物方程在Fourier-Besov-Morrey空间内解的适定性

The Well-Posedness of Solutions to a Class of Parabolic Equations in Fourier-Besov-Morrey Spaces

摘要: 本文应用傅里叶局部化方法和Littlewood-Paley定理，在临界Fourier-Besov-Morrey空间$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^3\right)$ 对一类抛物方程小初值解的全局适定性问题进行研究，其中$s=2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}$ 。

Abstract: This paper applies the Fourier localization method and the Littlewood-Paley theorem to study the global well-posedness of small initial value solutions for a class of parabolic equations in the critical Fourier-Besov-Morrey spaces $ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^3\right)$ where $s=2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}$ .

1. 引言

本文研究分数阶一类抛物方程在Fourier-Besov-Morrey空间的适定性

$\{\begin{array}{ll}{u}_t+{\left(-\Delta \right)}^{\alpha }u=\nabla \cdot \left(u\nabla u\right)\hfill & \left(t,x\right)\in {ℝ}^{+}\times {ℝ}^3,\hfill \\ u\left(0,x\right)=u_0\left(x\right)\hfill & x\in {ℝ}^3,\hfill \end{array}$ (1)

其中$u\left(t,x\right)=\left(u_1\left(t,x\right),u_2\left(t,x\right),u_3\left(t,x\right)\right)$ 表示速度场，算子${\left(-\Delta \right)}^{\alpha }$ 是傅里叶乘子其符号为${\left|\xi \right|}^{2\alpha }$ 。Leray [1]和Kato [2]以及多位研究者对局部存在性进行研究、Leray [1]和Hopf [3]研究了弱解的全局存在性、Fujita和Kato [4]在临界Sobolev空间${\dot{H}}^{\frac{1}{2}}$ ，Cannone [5]在当$s>\frac{1}{2}$ 的空间${\dot{H}}^s$ ，Kato [6]在Lebesgue空间$L^3$ ，Koch和Tataru [7]。在空间$BM{O}^{-1}$ 以及Cannone [5]在空间${\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}\left({ℝ}^3\right)$ 中都对小初始数据(初值)强解的适定性问题进行研究。Lei-Lin [7]研究了一个新空间

${\chi }^{-1}=\left\{u\in {\mathcal{S}}^{\prime }\left({ℝ}^3\right);{\displaystyle {\int }_{{ℝ}^3}{\left|\xi \right|}^{-1}\left|\widehat{u}\right|\text{d}\xi }<\infty \right\},$

此空间包含于$BM{O}^{-1}$ 中且等价于Fourier-Herz空间${\dot{B}}_1^{-1}$ [3]。他们证明了当初值$u_0\in {\chi }^{-1}$ 且${\Vert u_0\Vert }_{{\chi }^{-1}}<1$ ，分数阶Navier-Stokes系统存在全局温和解，当${\Vert u\Vert }_{L^{\infty }\left({ℝ}^{+}; {\chi }^{-1}\right)}<1$ 时，解唯一。Cannone和Wu [3]给出在临界Fourier-Herz空间${\dot{B}}_q^{-1}\left(q\in \left[1,2\right]\right)$ 中小初值时解的全局适定性，且${\Vert u\Vert }_{L^{\infty }\left({ℝ}^{+};{\dot{B}}_q^{-1}\right)\cap L^1\left({ℝ}^{+};{\dot{B}}_q^{-1}\right)}<\frac{9}{5}{\Vert u_0\Vert }_{{\dot{B}}_q^{-1}}$ ，存在唯一温

和解。由${\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)$ 空间的定义可知，其包含了一些经典的空间。例如${\dot{\mathcal{N}}}_{p,0,q}^s={\dot{B}}_{p,q}^s$ 。由Fourier-Besov-Morrey空间$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s$ 定义及性质，以及Navier-Stokes方程在Fourier-Besov空间$ℱ{\dot{B}}_{p,q}^s$ 内的研究。例如Cannone和Wu在Fourier-Herz研究了在空间${\dot{B}}_q^s=ℱ{\dot{\mathcal{N}}}_{p,1,q}^s$ 中的情况、肖伟良和陈杰成研究了分数阶Navier-Stokes方程在$ℱ{\dot{B}}_{p,q}^s$ 空间中的情况，由定义可知${\dot{B}}_q^s= ℱ{\dot{B}}_{1,q}^s=ℱ{\dot{\mathcal{N}}}_{1,0,q}^s$ 。拓展了新思路，本文研究当$\alpha >1$ 超临界情况下广义的Navier-Stokes方程，其中$u_0\in ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\left({ℝ}^3\right)$ 。本文的结果拓展了[8]和[9]的结果，用${\dot{B}}_{p,q}^s$ 表示经典的齐次Besov空间，C表示常数，在不同的地方可以取不同值，$U\lesssim V$ 表示存在常数$C>0$ ，使得$U\le CV$ ，$p^{\prime }$ 是$p$ 的共轭，满足$\frac{1}{p}+\frac{1}{p^{\prime }}=1$ ，当$1\le p\le \infty$ 。当$s=2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}$ 时在临界空间$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^3\right)$ 内可保证解

的伸缩不变形。

2. 预备知识和主要结论

本论文证明主要依靠傅里叶变换的二进制分割，也称为齐次Littlewood-Paley分解，下面我们将简要说明。首先对${ℝ}^n$ 空间进行分割。

假设$\mathcal{X}\in C_0^{\infty }\left({ℝ}^n\right)$ ，$\phi \in C_0^{\infty }\left({ℝ}^n\setminus \left\{0\right\}\right)$ 满足

$\text{supp}\mathcal{X}\subset \left\{\xi \in {ℝ}^n:\left|\xi \right|\le \frac{4}{3}\right\},$

$\text{supp}\phi \subset \left\{\xi \in {ℝ}^n:\frac{4}{3}\le \left|\xi \right|\le \frac{8}{3}\right\},$

$\mathcal{X}\left(\xi \right)+{\displaystyle \sum }_{j>0}\phi \left(2^{-j}\xi \right)=1,\xi \in {ℝ}^n,$

${\displaystyle \sum }_{j\in \mathbb{Z}}\phi \left(2^{-j}\xi \right)=1,\xi \in {ℝ}^n\setminus \left\{0\right\},$

其中${\phi }_j\left(\xi \right)=\phi \left(2^{-j}\xi \right)$ 且$$ 是所有多项式的集合，$S^{\prime }$ 表示缓增空间。对$\forall j\in \mathbb{Z}$ ，齐次二元块${\dot{\Delta }}_j$ 与齐次低频截止算子${\dot{S}}_j$ 定义为

${\dot{\Delta }}_{j}u=\phi \left(2^{-j}D\right)u=2^{jn}{\displaystyle \int h}\left(2^{j}y\right)u\left(x-y\right)\text{d}y,$

${\dot{S}}_{j}u={\displaystyle \sum }_{k\le j-1}{\dot{\Delta }}_{k}u=\mathcal{X}\left(2^{-j}D\right)u=2^{jn}{\displaystyle \int \tilde{h}}\left(2^{j}y\right)u\left(x-y\right)\text{d}y,$

其中$h={ℱ}^{-1}\phi$ ，$\tilde{h}={ℱ}^{-1}\mathcal{X}$ 。

定义2.1 若$1\le p\le \infty$ ，$0\le \lambda <n$ ，Morrey空间$M_p^{\lambda }=M_p^{\lambda }\left({ℝ}^n\right)$ 定义为函数$f\in L_{loc}^p\left({ℝ}^n\right)$ 的集合

${\Vert f\Vert }_{M_p^{\lambda }}=\underset{x_0\in {ℝ}^n}{\sup }\underset{r>0}{\sup }{r}^{-\frac{\lambda }{p}}{\Vert f\Vert }_{L^p\left(B\left(x_0,r\right)\right)}<\infty ,$ (2)

其中$B\left(x_0,r\right)$ 表示${ℝ}^n$ 中以$x_0$ 为中心，为半径的球。空间$M_p^{\lambda }$ 是一个巴拿赫空间。

当$\frac{n-\mu }{p_2}\ge \frac{n-\lambda }{p_1}$ 且$p_2\le p_1$ 时，有$M_{p_1}^{\lambda }$ ↪$M_2^{\mu }$ 且$M_p^0=L^p$ 。若$1\le p_1,p_2,p_3<\infty$ ，且$0\le {\lambda }_1,{\lambda }_2,{\lambda }_3<n$ ，有$\frac{1}{p_3}=\frac{1}{p_2}+\frac{1}{p_1}$ 且$\frac{{\lambda }_3}{p_3}=\frac{{\lambda }_2}{p_2}+\frac{{\lambda }_1}{p_1}$ ，由Hölder不等式得

${\Vert fg\Vert }_{M_{p_3}^{{\lambda }_3}}\le {\Vert f\Vert }_{M_{p_1}^{{\lambda }_1}}{\Vert g\Vert }_{M_{p_2}^{{\lambda }_2}}.$

类似的，当$1\le p<\infty$ 且$0\le \lambda <n$ ，对$\forall \phi \in L^1$ 且$\forall g\in M_p^{\lambda }$ ，可得

${\Vert \phi \ast g\Vert }_{M_p^{\lambda }}\le {\Vert \phi \Vert }_{L^1}{\Vert g\Vert }_{M_p^{\lambda }}.$ (3)

定义2.2 若$s\in ℝ$ ，$1\le p<+\infty$ ，$1\le q\le +\infty$ 且$0\le \lambda <n$ ，则${\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)$ 空间定义为

${\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)=\left\{u\in {\mathcal{Z}}^{\prime }\left({ℝ}^n\right);{\Vert u\Vert }_{{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)}<\infty \right\}.$

其中${\Vert u\Vert }_{{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)}$ 定义为

${\Vert u\Vert }_{{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)}=\{\begin{array}{ll}{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{jqs}{\Vert {\dot{\Delta }}_{j}u\Vert }_{M_p^{\lambda }}^q\right\}}^{1/q}\hfill & �q<\infty ,\hfill \\ \underset{j\in \mathbb{Z}}{\sup }{2}^{jqs}{\Vert {\dot{\Delta }}_{j}u\Vert }_{M_p^{\lambda }}\hfill & �q=\infty .\hfill \end{array}$

空间${\mathcal{Z}}^{\prime }\left({ℝ}^n\right)$ 表示空间$\mathcal{Z}\left({ℝ}^n\right)=\left\{f\in S\left({ℝ}^n\right);{\partial }^{\alpha }\widehat{f}\left(0\right)=0,\forall \alpha \right\}$ 的拓扑对偶空间。

定义2.3 若$s\in ℝ,1\le p\le +\infty ,1\le q\le +\infty ,0\le \lambda <n$ ，齐次Fourier-Besov-Morrey空间$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)$ 定义为

${\Vert u\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)}={\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{M_p^{\lambda }}^q\right\}}^{1/q}<+\infty ,$ (4)

当$q=\infty$ 时，适当修改。当$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)$ 空间赋予范数(4)时，则为巴拿赫空间。

定义2.4 若$s\in ℝ,1\le p<\infty ,1\le q,\rho \le \infty ,0\le \lambda <n,I=\left[0,T\right),T\in \left(0,\infty \right]$ 。对$u\left(t,x\right)$ 定义时空范数为

${\Vert u\left(t,x\right)\Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)}={\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{1/q},$

其中${ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)$ 表示分布在中且范数${\Vert \cdot \Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)}$有限的集合。

定义2.5 若$s\in ℝ, 1\le p,q\le \infty$ ，齐次Fourier-Besov空间$ℱ{\dot{B}}_{p,q}^s$ 定义为

其中

${\Vert u\Vert }_{ℱ{\dot{B}}_{p,q}^s}=\{\begin{array}{ll}{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{L^p}^q\right\}}^{1/q}\hfill & �q<\infty ,\hfill \\ \underset{j\in \mathbb{Z}}{\text{sup}}{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{L^p}^q\hfill & �q=\infty .\hfill \end{array}$

当$p=1$ 时Cannone和Wu引入Fourier-Herz空间${\dot{B}}_q^s$ [3]，其范数为

${\Vert u\Vert }_{{\dot{B}}_{p,q}^s}=\{\begin{array}{ll}{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{L^1}^q\right\}}^{1/q}\hfill & �q<\infty ,\hfill \\ \underset{j\in \mathbb{Z}}{\sup }{2}^{jqs}{\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert }_{L^1}^q\hfill & �q=\infty .\hfill \end{array}$

有${\dot{B}}_q^s=ℱ{\dot{\mathcal{N}}}_{1,q}^s$ 。

由Lei and Lin [7]引入了空间${\chi }^{-1}$

${\chi }^{-1}=\left\{u\in {\mathcal{S}}^{\prime }\left({ℝ}^3\right);{\displaystyle {\int }_{{ℝ}^3}{\left|\xi \right|}^{-1}\left|\widehat{u}\right|\text{d}\xi }<\infty \right\}.$

并且得到${\chi }^{-1}=ℱ{\dot{B}}_{1,1}^{-1}={\dot{B}}_1^{-1}$ 。

引理2.1 [10] 若$s_1,s_2\in ℝ,1\le p_1,p_2<+\infty ,1\le q_1,q_2\le +\infty ,0\le {\lambda }_1,{\lambda }_2<n,0\le \theta \le 1$ 。当$\frac{1}{p}=\frac{1-\theta }{p_1}+\frac{\theta }{p_2}$ ，$\frac{1}{q}=\frac{1-\theta }{q_1}+\frac{\theta }{q_2}$ ，$s=\left(1-\theta \right)s_1+\theta s_2$ ，$\lambda =\left(1-\theta \right){\lambda }_1+\theta {\lambda }_2$ ，则

${\Vert u\Vert }_{ℱ{\dot{B}}_{p,q}^s}<{\Vert u\Vert }_{ℱB_{p_1,{\lambda }_1,q_1}^{s_1}}^{1-\theta }{\Vert u\Vert }_{ℱB_{p_2,{\lambda }_2,q_2}^{s_2}}^{\theta }.$

引理2.2 [10] 导数$D_{\xi }^{\alpha }\mu$ 是一个由$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{s+\left|\alpha \right|}\left({ℝ}^n\right)$ 到$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\left({ℝ}^n\right)$ 的有界算子。

引理2.3 [10] 若$1\le q\le p<\infty ,0\le {\lambda }_1,{\lambda }_2<n,\frac{n-{\lambda }_1}{p}\le \frac{n-{\lambda }_2}{q}$ ，$\gamma$ 是一多重指标。若$\text{supp}\left(\widehat{f}\right)\subset \left\{\left|\xi \right|\le A{2}^j\right\}$，则存在与f和j无关的常数C > 0，使得

${\Vert {\left(i\xi \right)}^{\gamma }\widehat{f}\Vert }_{M_q^{{\lambda }_2}}\le C{2}^{j\left|\gamma \right|+j\left(\frac{n-{\lambda }_2}{q}-\frac{n-{\lambda }_1}{p}\right)}{\Vert \widehat{f}\Vert }_{M_q^{{\lambda }_1}}.$ (5)

定理2.1 若$s\in ℝ,1\le p<\infty ,1\le q\le \infty ,0\le \lambda <n,1\le \alpha <\frac{3+\frac{3}{p^{\prime }}+\frac{\lambda }{p}}{4-\frac{2\alpha }{\rho }}$ ，对$\forall u_0\in ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\left({ℝ}^3\right)$ 满足$\nabla \cdot u_0=0$ 且${\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}<C_0$ ，则存在一个常数$C_0\left(\alpha ,p,q\right)$ ，使得(1)有唯一的全局温和解u，

$u\in C\left(\left[0,\infty \right);ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)\cap {ℒ}^1\left(\left[0,\infty \right);ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2+\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right),$

并且满足

${\Vert u\Vert }_{{ℒ}^{\infty }\left(\left[0,\infty \right);ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}+{\Vert u\Vert }_{{ℒ}^1\left(\left[0,\infty \right),ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2+\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\le \left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right){\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}.$

注记2.1 其中Fourier-Besov-Morrey空间与${\phi }_j$ 的选择无关，并且与经典的Besov-Morrey空间相比选择此空间为研究空间优势在于，该空间可以不使用Bernstein不等式，可直接通过Hölder's不等式估计双线性仿积。

注记2.2 Fourier-Besov-Morrey空间$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\left({ℝ}^3\right)$ 对方程(1)是关键的，若$u_{0,\gamma }\left(\xi \right)={\gamma }^{2\alpha -2}{u}_0\left(\gamma \xi \right)$ ，

则$\widehat{u_{0,\gamma }}\left(\xi \right)={\gamma }^{2\alpha -5}\widehat{u_0}\left({\gamma }^{-1}\xi \right)$ 。设$f_j\left(\xi \right)=\phi \left(2^{-j+\left[{\log }_2\gamma \right]-{\log }_2\gamma }\xi \right)\widehat{u_{0,\gamma }}\left(\xi \right)$ ，有

$\begin{array}{c}{2}^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\Vert f_j\Vert }_{M_p^{\lambda }}=2^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}\underset{x_0\in {ℝ}^3}{\sup }\underset{r>0}{\sup }{r}^{-\frac{\lambda }{p}}{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]-{\log }_2\gamma }\xi \right)\widehat{u_{0,\gamma }}\left(\xi \right)\Vert }_{L^p\left(B\left(x_0,r\right)\right)}\\ =2^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}\underset{x_0\in {ℝ}^3}{\sup }\underset{r>0}{\sup }{r}^{-\frac{\lambda }{p}}{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]-{\log }_2\gamma }\xi \right){\gamma }^{2\alpha -5}\widehat{u_0}\left({\gamma }^{-1}\xi \right)\Vert }_{L^p\left(B\left(x_0,r\right)\right)}\\ =2^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\gamma }^{2\alpha -5}\underset{x_0\in {ℝ}^3}{\sup }\underset{r>0}{\sup }{r}^{-\frac{\lambda }{p}}{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]}{\gamma }^{-1}\xi \right)\widehat{u_0}\left({\gamma }^{-1}\xi \right)\Vert }_{L^p\left(B\left(x_0,r\right)\right)}\\ =2^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\gamma }^{2\alpha -5}{\gamma }^{\frac{3}{p}}{r}^{-\frac{\lambda }{p}}\underset{x_0\in {ℝ}^3}{\sup }\underset{r>0}{\sup }{\left({\gamma }^{-1}r\right)}^{-\frac{\lambda }{p}}{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]}\eta \right)\widehat{u_0}\left(\eta \right)\Vert }_{L^p\left(B\left({\gamma }^{-1}{x}_0,{\gamma }^{-1}r\right)\right)}\\ =2^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{2}^{\left(2\alpha -2-\frac{3}{p^{\prime }}-\frac{\lambda }{p}\right){\log }_2\gamma }{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]}\eta \right)\widehat{u_0}\left(\eta \right)\Vert }_{M_p^{\lambda }}\\ =2^{\left(\left[{\log }_2\gamma \right]-{\log }_2\gamma \right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{2}^{\left(j-\left[{\log }_2\gamma \right]\right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]}\eta \right)\widehat{u_0}\left(\eta \right)\Vert }_{M_p^{\lambda }}\\ \approx 2^{\left(j-\left[{\log }_2\gamma \right]\right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}\times {\Vert \phi \left(2^{-j+\left[{\log }_2\gamma \right]}\eta \right)\widehat{u_0}\left(\eta \right)\Vert }_{M_p^{\lambda }}.\end{array}$

则说明

${\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)q}{\Vert f_j\Vert }_{M_p^{\lambda }}^q\right\}}^{1/q}\approx {\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{},$

由于

${\phi }_j\left(\xi \right)\widehat{{\mu }_{0,\gamma }}\left(\xi \right)={\displaystyle \sum }_{\left|k-j\right|\le 2}{\phi }_j\left(\xi \right)f_k\left(\xi \right),$

则

${\Vert u_{0,\gamma }\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}\approx {\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}.$

3. 定理证明

首先，考虑非齐次线性分数阶方程

$\{\begin{array}{ll}{u}_t+{\left(-\Delta \right)}^{\alpha }u=f\left(t,x\right)\hfill & \left(t,x\right)\in {ℝ}^{+}\times {ℝ}^3,\hfill \\ u\left(0,x\right)=u_0\left(x\right)\hfill & x\in {ℝ}^3,\hfill \end{array}$ (6)

有以下结论。

引理3.1 [10] 若$I=\left[0,T\right),1\le T<\infty ,s\in ℝ,1\le p<\infty ,1\le q\le \infty ,0\le \lambda <n$ ，当$f\in {ℒ}^1\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)$ 且$u_0\in ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s$ ，则柯西问题(6)的解$u\left(t,x\right)$ 满足

${\Vert u\Vert }_{{ℒ}^{\infty }\left(I;ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)}+{\Vert u\Vert }_{{ℒ}^1\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{s+2\alpha }\right)}\le \left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right)\left({\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s}+{\Vert f\Vert }_{{ℒ}^1\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^s\right)}\right).$

命题3.1 若$1\le p<\infty ,1\le \rho \le \infty ,1\le q\le 2,1\le \alpha <\frac{3+\frac{3}{p^{\prime }}+\frac{\lambda }{p}}{4-\frac{2\alpha }{\rho }},0\le \lambda <n$ ，设

$X={ℒ}^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)\cap {ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right),$

其范数为

${\Vert u\Vert }_X={\Vert u\Vert }_{{ℒ}^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}+{\Vert u\Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}.$

存在一个常数$C=C\left(\alpha ,p,q\right)>0$ 与$\alpha ,p,q$ 有关，使得

${\Vert \nabla \cdot \left(v\nabla u\right)\Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-4\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}\le C{\mu }^{-1}{\Vert u\Vert }_X{\Vert v\Vert }_X$ (7)

证明. 首先引入一些标准逻辑算子，设

$u_j={\widehat{\Delta }}_{j}u,{\dot{S}}_{j}u={\displaystyle \sum }_{k\le j-1}{\Delta }_{k}u,{\tilde{\dot{\Delta }}}_{j}u={\displaystyle \sum }_{\left|k-j\right|\le 1}{\dot{\Delta }}_{k}u$

利用Bony’s仿积分解与quasi-orthogonality性质对Littlewood-Paley分解，固定j，有

$\begin{array}{c}{\dot{\Delta }}_j\left(v\nabla u\right)={\displaystyle \sum }_{\left|k-j\right|\le 4}{\dot{\Delta }}_j\left({\dot{S}}_{k-1}v{\dot{\Delta }}_k\nabla u\right)+{\displaystyle \sum }_{\left|k-j\right|\le 4}{\dot{\Delta }}_j\left({\dot{S}}_{k-1}\nabla u{u}_{k}v\right)+{\displaystyle \sum }_{k\ge j-3}{\dot{\Delta }}_j\left({\dot{\Delta }}_{k}v{\tilde{\dot{\Delta }}}_k\nabla u\right)\\ =I_j+I{I}_j+II{I}_j .\end{array}$

则

$\begin{array}{l}{\Vert \partial \left(v\nabla u\right)\Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-4\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}\\ \lesssim {\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{{\dot{\Delta }}_j\left(v\nabla u\right)}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\\ \lesssim {\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{I_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\\ +{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{I{I}_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\\ +{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{II{I}_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\end{array}$ (8)

由Morrey空间的Young’s不等式(3)和引理2.3当$\left|\gamma \right|=0$ 时，可以得到

$\begin{array}{c}{\Vert \widehat{I_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\le {\displaystyle \sum }_{\left|k-j\right|\le 4}{\Vert \widehat{{\dot{S}}_{k-1}v{\dot{\Delta }}_k\nabla u}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}\\ \le {\displaystyle \sum }_{\left|k-j\right|\le 4}{\Vert \widehat{v_k}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\displaystyle \sum }_{l\le k-2}{\Vert \widehat{\nabla u_l}\Vert }_{L^{\infty }\left(I,L^1\right)}\\ \le {\displaystyle \sum }_{\left|k-j\right|\le 4}{\Vert \widehat{v_k}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\displaystyle \sum }_{l\le k-2}{2}^{l\left(\frac{3}{p^{\prime }}+\frac{\lambda }{p}+1\right)}{\Vert \widehat{u_l}\Vert }_{L^{\infty }\left(I,M_p^{\lambda }\right)}\\ \le {\displaystyle \sum }_{\left|k-j\right|\le 4}{\Vert \widehat{v_k}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\displaystyle \sum }_{l\le k-2}{2}^{l\left(\frac{3}{p^{\prime }}+\frac{\lambda }{p}+1\right)}{2}^{-l\left(2\alpha -1\right)}{2}^{l\left(2\alpha -1\right)}{\Vert \widehat{u_l}\Vert }_{L^{\infty }\left(I,M_p^{\lambda }\right)}\\ \le {\displaystyle \sum }_{\left|k-j\right|\le 4}{\Vert \widehat{v_k}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\left({\displaystyle \sum }_{l\le k-2}{2}^{l\left(2\alpha -1\right)q^{\prime }}\right)}^{\frac{1}{q}}{\Vert u\Vert }_{{\ell }^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\\ \lesssim {\displaystyle \sum }_{\left|k-j\right|\le 4}{2}^{k\left(2\alpha -1\right)}{\Vert \widehat{v_k}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\Vert u\Vert }_{{\ell }^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\end{array}$

等式两端同时乘以$2^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}\right)}$ 并且取$l^q$ 模，可得到

$\begin{array}{l}{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{I_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\\ \lesssim {\Vert v\Vert }_{{\ell }^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}{\Vert u\Vert }_{{\ell }^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}.\end{array}$ (9)

同理可得，第二部分可得

$\begin{array}{l}{\left\{{\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{I_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right\}}^{\frac{1}{q}}\\ \lesssim {\Vert u\Vert }_{{\ell }^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}{\Vert v\Vert }_{{\ell }^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}.\end{array}$ (10)

估计第三部分$II{I}_j$ ，设

$II{I}_{jk}:={\dot{\Delta }}_j\left({\displaystyle \sum }_{\left|i-k\right|\le 1}{\dot{\Delta }}_{i}v{\dot{\Delta }}_i\nabla u\right)={\displaystyle \sum }_{i=-1}^1{\dot{\Delta }}_j\left({\dot{\Delta }}_k\nabla u{\dot{\Delta }}_{i+k}v\right).$

由Morrey空间的Young’s不等式(3)和引理2.3当$\left|\gamma \right|=0$ 时，可得到

$\begin{array}{l}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{\Vert \widehat{II{I}_{jk}}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}\\ \le {\displaystyle \sum }_{i=-1}^1{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{\Vert \widehat{{\dot{\Delta }}_{k}u{\dot{\Delta }}_{i+k}\nabla v}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}\\ \le {\displaystyle \sum }_{i=-1}^1{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{\Vert {\widehat{u}}_k\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\Vert {\widehat{\nabla v}}_{i+k}\Vert }_{L^{\infty }\left(I,L^1\right)}\\ \le {\displaystyle \sum }_{i=-1}^1{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{2}^{\left(i+k+1\right)\left(\frac{3}{p\text{'}}+\frac{\lambda }{p}\right)}{\Vert {\widehat{u}}_k\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{\Vert {\widehat{v}}_{i+k}\Vert }_{L^{\infty }\left(I,M_p^{\lambda }\right)}\\ ={\displaystyle \sum }_{i=-1}^1{2}^{\left(j-k\right)\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{2}^{i\left(2\alpha -1\right)}\left(2^{k\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{\Vert {\widehat{u}}_k\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}\right)\times \left(2^{\left(i+k\right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\Vert {\widehat{v}}_{i+k}\Vert }_{L^{\infty }\left(I,M_p^{\lambda }\right)}\right).\end{array}$

对等式两端同时取${\ell }^q$ 的范数，当$\alpha <\frac{3+\frac{3}{p^{\prime }}+\frac{\lambda }{p}}{4-\frac{2}{\rho }}$ 时，$3-4\alpha +\frac{2\alpha }{\rho }+\frac{3}{p^{\prime }}+\frac{\lambda }{p}>0$ ，由Hölder’s不等式，可得

$\begin{array}{l}{\left({\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{II{I}_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right)}^{\frac{1}{q}}\\ \le (({\displaystyle \sum }_{l\le 3}{\displaystyle \sum }_{i=-1}^1{2}^{l\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{2}^{\left(j-l\right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{2}^{i\left(2\alpha -1\right)}\\ {{\times {\Vert {\widehat{u}}_{j-l}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}{2}^{\left(i+j-l\right)\left(2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}\right)}{\Vert {\widehat{v}}_{j+i-l}\Vert }_{L^{\infty }\left(I,M_p^{\lambda }\right)})}^q)}^{\frac{1}{q}}\\ \le {\displaystyle \sum }_{i=-1}^1{\displaystyle \sum }_{l\le 3}{2}^{l\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)}{2}^{i\left(2\alpha -1\right)}{\Vert u\Vert }_{{\ell }^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}{\Vert v\Vert }_{{\ell }^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\end{array}$

由$1\le q\le 2$ ，有$q\le q^{\prime }$ 且$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}$ ↪$ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q^{\prime }}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}$ ，则

$\begin{array}{l}{\left({\displaystyle \sum }_{j\in \mathbb{Z}}{2}^{j\left(3-4\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}+\frac{2\alpha }{\rho }\right)q}{\Vert \widehat{II{I}_j}\Vert }_{L^{\rho }\left(I,M_p^{\lambda }\right)}^q\right)}^{\frac{1}{q}}\\ \lesssim {\Vert u\Vert }_{{ℒ}^{\rho }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{2\alpha }{\rho }+\frac{\lambda }{p}}\right)}^{}{\Vert v\Vert }_{{ℒ}^{\infty }\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\end{array}$ (11)

由(8)，(9)，(10)，(11)联立可得等式(7)成立。

引理3.2 设$\left(X,\Vert \cdot \Vert \right)$ 为Banach空间，$B:X\times X\mapsto X$ 是一个有界线性算子，对$\forall u,v\in X$ ，存在一个常数$\eta >0$ 使得${\Vert B\left(u,v\right)\Vert }_X\le \eta {\Vert u\Vert }_X$ 。若$0<\epsilon <\frac{1}{4\eta }$ 且$y\in X$ 使得${\Vert y\Vert }_X\le \epsilon$ ，方程$x:=y+B\left(x,x\right)$ 存在一个解$\overline{x}\in X$ 且${\Vert \overline{x}\Vert }_X\le 2\epsilon$ 。$\overline{x}$ 是球$\overline{B}\left(0,2\epsilon \right)$ 内唯一解。此外，该解$\overline{x}$ 对$y$ 是连续依赖的。

即当${\Vert y^{\prime }\Vert }_X<\epsilon$ ，$x^{\prime }:=y^{\prime }+B\left(x^{\prime },x^{\prime }\right)$ ，${\Vert x^{\prime }\Vert }_X\le 2\epsilon$ ，则${\Vert \overline{x}-x^{\prime }\Vert }_X\le \frac{1}{1-4\epsilon \eta }{\Vert \overline{y}-y^{\prime }\Vert }_X$ 。

定理2.1证明. 由引理3.2可知当确初值很小时，该函数是一个向量场，其范数是三个分量范数之和。因此对方程(1)由常数变易法可得

$\begin{array}{c}u\left(t,x\right)={\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}{u}_0-{\displaystyle {\int }_0^t{\text{e}}^{-\left(t-\tau \right){\left(-\Delta \right)}^{\alpha }}\nabla \cdot \left(u\nabla u\right)\left(\tau ,x\right)\text{d}\tau }\\ ={\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}{u}_0+B\left(u,u\right).\end{array}$ (12)

当$u_0=0$ 且$f=\nabla \cdot \left(u\nabla u\right)$ 时，$B\left(u,v\right)$ 是热方程(6)的解。当$s=2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}$ 由引理3.1和当$\rho =1$ ，

由命题3.1可得

$\begin{array}{c}{\Vert B\left(u,v\right)\Vert }_X\le \left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right){\Vert \nabla \cdot \left(u\nabla u\right)\Vert }_{{ℒ}^1\left(I,ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}\right)}^{}\\ \le \left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right)C{\Vert u\Vert }_X{\Vert v\Vert }_X.\end{array}$

由引理3.2知，当${\Vert {\text{e}}^{-\mu t{\left(-\Delta \right)}^{\alpha }}{u}_0\Vert }_X<R$ ，其中$R=\frac{1}{4\left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right)C}$ 时，方程(12)在

$B\left(0,2R\right)�=\left\{x\in X:{\Vert x\Vert }_X\le 2R\right\}$ 内存在唯一解。当$u_0=u_0$ 且$f=0$ 时，${\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}{u}_0$ 是耗散方程(6)的解。所以，由定理3.1有

${\Vert {\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}{u}_0\Vert }_X\le \left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right){\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}.$

因此，若${\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}\le C_0$ ，其中$C_0=\frac{1}{4{\left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right)}^{2}C}$ ，则方程(13)存在唯一解$u\in X$ ，并且满足

${\Vert u\Vert }_X\le 2\left(1+{\left(\frac{16}{9}\right)}^{\alpha }\right){\Vert u_0\Vert }_{ℱ{\dot{\mathcal{N}}}_{p,\lambda ,q}^{2-2\alpha +\frac{3}{p^{\prime }}+\frac{\lambda }{p}}}^{}.$