Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

Abstract

In this paper, we consider the generalized porous medium equation. For small initial data $u_0$ belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution.

1. Introduction

We consider the three dimensional generalized porous medium (GPM) equation:

$$\left\{\begin{array}{c}{u}_t+\mu {\Lambda }^{\beta }u=\nabla ·(u\nabla Pu)\mathrm{in}{\mathbb{R}}^3\times {\mathbb{R}}^{+},\hfill \\ u(x,0)=u_0\mathrm{in}{\mathbb{R}}^3,\hfill \end{array}\right.$$

(1)

where $u=u(x,t)$ denotes the concentration or density. The dissipative coefficient $\mu =0$ is the inviscid case, while $\mu >0$ represents the viscous case. P is an abstract operator and the fractional Laplacian term ${\Lambda }^{\beta }$ is the Fourier Transform defined by $\widehat{{\Lambda }^{\beta }u}={\left|\xi \right|}^{\beta }\widehat{u}$. For simplicity we chose $\mu =1$.

The Equation (1) was first introduced by Zhou et al. [1]. Actually, by applying the fractional dissipative term $\mu {\Lambda }^{\beta }$ to the continuity equation $u_t+\nabla ·(uV)=0$, Caffarelli et al. [2] obtained the Equation (1), where the velocity V derives from the potential, $V=-\nabla p$, and the velocity potential or pressure is associated with u by the abstract operator $p=Pu$ [3]. There are a number of physical applications of porous medium of equations, mainly to describe processes involving fluid flow, heat transfer or diffusion. May be the best known of them is the description of the flow of an isentropic gas through a porous medium. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory and other fields. For detailed study related to the physical significance of (1), we refer the reader to [4] and the references therein.

The abstract form pressure term $Pu$ gives a good suitability in many cases. The simplest case comes from a model in ground water in filtration [5,6]. For more general case $\mu =0$ and $Pu={(-\Delta )}^{-s}u={\Lambda }^{-2s}$, $0<s<1$, Zhou et al. [7] obtained the strong solution for Equation (1) in Besov spaces $B_{p,\infty }^{\beta }$ and for any initial data in $B_{1,\infty }^{\beta }$ the local solution is obtained. Lin and zhang [8] considered (1) for $s=1$ (critical case), which gives the mean field equation. For more details in this direction we refer the reader to [1,9] and the references therein.

Another similar model appears in the aggregation equation, which explains a aggregation phenomena and collective motion in mechanics of continuous media and biology [10,11]. This equation has many applications in various applied sciences such as chemistry, biology, physics and population dynamics. In the aggregation equation, the operator P can also be written as convolution operator with kernel $\mathcal{K}$ as $Pu=\mathcal{K}*u$. The typical kernels are exponential potential $-e^{-\left|x\right|}$ [12] and the Newton Potential ${\left|x\right|}^{\gamma }$ [13]. For more results related to the well-posedness and blowup criterion, we refer the reader to [14,15] and the references therein.

Furthermore, using the same initial data, we can write the Equation (1) as:

$$\begin{array}{c}{u}_t+\mu {\Lambda }^{\beta }u+v·\nabla u=-u(\nabla ·v);\\ v=-\nabla Pu.\end{array}$$

(2)

Then the Equation (2) can be compared to the geostrophic model and the convective velocity for the generalized porous medium equation is not absolutely divergence-free. In addition, if the divergence-free vector function v satisfies the equation $(\nabla ·v=0)$, the Equation (2) will hold the quasi-geostrophic (Q-G) equation [16,17].

One of the main problems to the Equation (1) is the singularity of the abstract form pressure term $Pu$ that establish the well-posedness or gives the blow up solution. Zhou et al. [1] established the local well-posedness for large initial data in Besov spaces and the global solution for small inital data. They also given a blowup criterion for the solution. Li and Rodrigo [12] obtained well-posedness and the blownup criterion for the solution of Equation (1) associated with the pressure $Pu=\mathcal{K}*u$, where $\mathcal{K}(x)=e^{-\left|x\right|}$ belongs to Sobolev space. Moreover, their work is further extended by Wu and Zhang [18] to the case $\mathcal{K}(x)=e^{-\left|x\right|}$ and $\nabla \mathcal{K}\in W^{1,1}$. Xiao and Zhou [3] have shown the local well-posedness for large initial data in Fourier-Besov spaces and obtained the global well-posedness for small initial data for $\nabla \mathcal{K}\in L^1$. The controllability of convolution $\mathcal{K}*u$ and its gradient $\nabla \mathcal{K}*u$ in Besov spaces leads to these result.

Inspired by the above works, we obtain the global well-posedness and analyticity for Equation (1) and show the Gevery class regularity of the solution in homogenous Fourier-Besov-Morrey spaces with variable exponent by considering $\nabla \mathcal{K}\in L^1.$ The Fourier-Besov spaces goes back to the work of Konieczny and Yoneda [19] in the context of Navier-Stokes equation (NSE) with Coriolis force. Fourier-Besov spaces and Fourier-Besov-Morrey spaces have been extensively considered by many authors in order to deal with the well-posedness, self-similar solution, regularity etc. For more study in this direction, we refer the reader to [20,21,22] and the references therein.

The variable exponent Lebesgue spaces $L^{p(·)}$, comes from Orlicz [23] and further developed by Musielak [24] and Nakano [25]. However the recent improvement begins with Kovacik and Rakosnik [26] and further advancement by Cruzuribe [27] and Diening [28]. The main reason of variable exponent function spaces is their applications in fluid dynamics [29], image processing [30] and partial differential equations [31]. For detailed study related to Besov spaces with variable exponent and Besov-Morrey spaces with variable exponent, we refer the reader to [32,33,34,35,36,37] and the references therein.

2. Preliminaries

In this section, we recall some definitions of various functions spaces with variable exponent, basic facts about dyadic decomposition, and some useful propositions. Throughout this paper, $f\lesssim g$ denotes that $f\le Cg$ for some constant $C>0$. We obtained the global well-posedness result and prove Gevrey class regularity in Section 3 and Section 4, respectively. The conclusion is given in Section 5.

Definition 1.

Let ${\mathcal{P}}_0$ denotes the set of all measurable functions $p(·):{\mathbb{R}}^n\to (0,\infty )$ such that

$$\begin{array}{c}0<p_{-}=\underset{\mathrm{x}\in {\mathbb{R}}^n}{\mathrm{essinf}}p(x),\underset{\mathrm{x}\in {\mathbb{R}}^n}{\mathrm{esssup}}p(x)=p_{+}<\infty .\end{array}$$

The Lebesgue space with variable exponent is defined by

$$\begin{array}{c}{L}^{p(·)}({\mathbb{R}}^n)=\left\{f:{\mathbb{R}}^n\to \mathbb{R}\mathit{is}\mathit{measurable},{\int }_{{\mathbb{R}}^n}{\left|f(x)\right|}^{p(x)}dx<\infty \right\},\end{array}$$

with Luxemburg-Nakano norm

$$\begin{array}{c}{\parallel f\parallel }_{L^{p(·)}}=inf\left\{\lambda >0:{\int }_{{\mathbb{R}}^n}{\left(\frac{|f(x)|}{\lambda }\right)}^{p(x)}dx\le 1\right\}.\end{array}$$

In order to differentiate between constant and variable exponent, we denote constant exponent by p, and variable exponent by $p(·)$. Also $(L^{p(·)}({\mathbb{R}}^n),\parallel f{\parallel }_{L^{p(·)}})$ is a Banach space.

Since the $L^{p(·)}$ does not have the same desired properties like $L^p$. So, to ensure that the Hardy-Littlewood maximal operator M is bounded on $L^{p(·)}({\mathbb{R}}^n)$, we postulate the following standard conditions:

(1) (locally log-Hölder continuous) There exists a constant $C_{log}(p)$ such that

$$\begin{array}{c}|p(x)-p(y)|\le \frac{C_{log}(p)}{log(e+|x-y{|}^{-1})},\mathrm{for}\mathrm{any}x,y\in {\mathbb{R}}^n\mathrm{and}x\ne y.\end{array}$$

(2) (locally log-Hölder continuous) There exists a constant $C_{log}(p)$ and some constant independent of x such that

$$\begin{array}{c}|p(x)-p_{\infty }|\le \frac{C_{log}(p)}{log(e+|x|)},\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^n.\end{array}$$

$C^{log}({\mathbb{R}}^n)$ denote the set of all functions $p(·):{\mathbb{R}}^n\to \mathbb{R}$ satisfying (1) and (2).

Definition 2.

Let $p(·),h(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)$ with $0<p^{-}\le p(x)\le h(x)\le \infty$, the Morrey space with variable exponent $M_{p(·)}^{h(·)}$:= $M_{p(·)}^{h(·)}{\mathbb{R}}^n$ be the set of measurable functions on ${\mathbb{R}}^n$ with finite quasinorm

$${\parallel f\parallel }_{M_{p(·)}^{h(·)}}:=\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\parallel r^{\frac{n}{h(x)}-\frac{n}{p(x)}}f{\chi }_{B(x_0,r)}\parallel }_{L^{p(·)}}.$$

By the definition of $L^{p(·)}$ quasinorm ${\parallel f\parallel }_{M_{p(·)}^{h(·)}}$ also has the following form

$${\parallel f\parallel }_{M_{p(·)}^{h(·)}}:=\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}inf\{\lambda >0:{\varrho }_{p(·)}(r^{\frac{n}{h(x)}-\frac{n}{p(x)}}\frac{f}{\lambda }{\chi }_{B(x_0,r)})\le 1\}.$$

Now we present some essential lemmas from [36].

Lemma 1.

Let $p(·),h(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)$ with $p(x)\le h(x)$ and f be any measurable function. Then

$${\parallel f\parallel }_{M_{p(·)}^{h(·)}}:=inf\{\lambda >0:\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\varrho }_{p(·)}(r^{\frac{n}{h(x)}-\frac{n}{p(x)}}\frac{f}{\lambda }{\chi }_{B(x_0,r)})\le 1\}.$$

Lemma 2.

$p(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)$. For any measurable function f

$$\underset{x\in {\mathbb{R}}^n,r>0}{sup}{\varrho }_{p(·)}(f{\chi }_{B(x_0,r)})={\varrho }_{p(·)}(f).$$

Lemma 3.

If $p(·)\in {\mathcal{P}}_0^n$, then ${\parallel f\parallel }_{M_{p(·)}^{p(·)}}={\parallel f\parallel }_{L^{p(·)}}$.

Now we recall dyadic decomposition of ${\mathbb{R}}^n$. Let $\chi \in {\mathcal{C}}_0^{\infty }({\mathbb{R}}^n)$ and $\phi \in {\mathcal{C}}_0^{\infty }({\mathbb{R}}^n\backslash \left\{0\right\})$ be the two non negative radial functions such that

$$\begin{array}{c}supp\chi \subset \{\xi \in {\mathbb{R}}^n:|\xi |\le \frac{4}{3}\},\\ supp\phi \subset \{\xi \in {\mathbb{R}}^n:\frac{3}{4}\le |\xi |\le \frac{8}{3}\},\\ \chi (\xi )+\sum _{i\ge 0}\phi (2^{-i}\xi )=1,\xi \in {\mathbb{R}}^n,\\ \sum _{i\in \mathbb{Z}}\phi (2^{-i}\xi )=1,\xi \in {\mathbb{R}}^n\backslash \left\{0\right\}.\end{array}$$

We denote ${\phi }_i(\xi )=\phi (2^{-i}\xi )$ and $h_i={\mathcal{F}}^{-1}{\phi }_i$ to define the frequency localization as follows:

$$\begin{array}{c}{\Delta }_{i}f:={\mathcal{F}}^{-1}{\phi }_i\mathcal{F}f={\int }_{{\mathbb{R}}^n}{h}_i(y)f(x-y)dy,\\ S_{i}f=\sum _{j\le i-1}{\Delta }_{j}f,\end{array}$$

where ${\Delta }_i=S_i-S_{i-1}$ is a frequency projection to the annulus $\left\{\right|\xi |\sim 2^i\}$ and $S_i$ is a frequency to the ball $\left\{\right|\xi |\lesssim 2^i\}$. we can easily obtain that

$$\begin{array}{c}{\Delta }_i{\Delta }_j\equiv 0,\mathrm{if}|i-j|\ge 2\mathrm{and}{\Delta }_i({\Delta }_{j-1}{\Delta }_j)\equiv 0,\mathrm{if}|i-j|\ge 5.\end{array}$$

Definition 3.

Let $p(·)$, $q(·)$, $h(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)$, the mixed Lebesgue-sequence space is defined as the set of all sequences ${\left\{f_i\right\}}_{i\in \mathbb{Z}}$ of measurable functions in ${\mathbb{R}}^n$ such that

$$\begin{array}{c}{\parallel \left\{f\right\}}_{i\in \mathbb{Z}}{\parallel }_{l^{q(·)}\left(M_{p(·)}^{h(·)}\right)}:=inf\left\{\mu >0,{\varrho }_{l^{q(·)}\left(M_{p(·)}^{h(·)}\right)}\left({\left\{\frac{f_i}{\mu }\right\}}_{i\in \mathbb{Z}}\right)\le 1\right\}<\infty ,\end{array}$$

where

$$\begin{array}{c}{\varrho }_{l^{q(·)}\left(M_{p(·)}^{h(·)}\right)}\left({\left\{f\right\}}_{i\in \mathbb{Z}}\right):=\sum _{i\in Z}inf\left\{{\lambda }_i>0,{\int }_{{\mathbb{R}}^n}{(\frac{|r^{\frac{n}{h(x)}-\frac{n}{p(x)}}{f}_i{\chi }_{B(x_0,r)}|}{{\lambda }^{\frac{1}{q(x)}}})}^{p(x)}dx\le 1\right\}.\end{array}$$

Notice that if $q_{+}<\infty$ and $p(x)\le q(x)$, then

$${\varrho }_{{\ell }^{q(·)}(M_{p(·)}^{h(·)})}({\left\{f_j\right\}}_{j\in {\mathbb{N}}_0})=\sum _{j\in {\mathbb{N}}_0}\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}\parallel (|r^{\frac{n}{h(x)}-\frac{n}{p(x)}}{f}_j|{\chi }_{B(x_0,r)}{{)}^{q(·)}\parallel }_{L^{\frac{p(·)}{q(·)}}}.$$

Definition 4.

Let $s(·)\in C^{log}({\mathbb{R}}^n)$ and $p(·),q(·),h(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)\cap C^{log}({\mathbb{R}}^n)$ with $0<p^{-}\le p(x)\le h(x)\le \infty$. We define the homogeneous Besov-Morrey space with variable exponent ${\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}$ by

$${\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}=\{f\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n){:\parallel f\parallel }_{{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}}<\infty \},$$

with norm

$${\parallel f\parallel }_{{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}}:={\parallel {\left\{2^{js(·)}{\Delta }_{j}f\right\}}_{j\in \mathbb{Z}}\parallel }_{{\ell }^{q(·)}(M_{p(·)}^{h(·)})}.$$

The space ${\mathcal{D}}^{\prime }({\mathbb{R}}^n)$ is the dual space of

$$\mathcal{D}({\mathbb{R}}^n)=\left\{f\in \mathcal{S}({\mathbb{R}}^n):\left(D^{\alpha }f\right)(0)=0,\forall \alpha \right\},$$

Definition 5.

Let $s(·)\in C^{log}({\mathbb{R}}^n)$ and $p(·),q(·),h(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)\cap C^{log}({\mathbb{R}}^n)$ with $0<p^{-}\le p(x)\le h(x)\le \infty$. We define the homogeneous Fourier-Besov-Morrey space with variable exponent $\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}$ by

$$\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}=\left\{f\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n);{\parallel f\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}}<\infty \right\},$$

with norm

$${\parallel f\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}}:={\parallel {\left\{2^{js(·)}{\phi }_j\widehat{f}\right\}}_{-\infty }^{\infty }\parallel }_{{\ell }^{q(·)}(M_{p(·)}^{h(·)})}.$$

Definition 6.

Let $T\in (0,T]$ and $1\le r,\gamma \le \infty$. We define the Chemin-Lerner type homogeneous Fourier-Besov-Morrey space with variable exponent ${\mathcal{L}}^{\gamma }\left(0,T;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}\right)$ by

$${\mathcal{L}}^{\gamma }\left(0,T;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}\right)=\left\{f\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n);{\parallel f\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)})}<\infty \right\},$$

with norm

$${\parallel f\parallel }_{{\mathcal{L}}^{\gamma }\left(0,T;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q(·)}^{s(·)}\right)}={\left(\sum _{i\in \mathbb{Z}}\parallel 2^{is(·)}{\dot{\Delta }}_{i}f{\parallel }_{L_T^{\gamma }{M}_{p(·)}^{h(·)}}^r\right)}^{\frac{1}{r}}.$$

Proposition 1.

The following inclusions are established for the Morrey spaces with variable exponent.

(1) ([37]) let $p(·),p_1(·),p_2(·),h(·),h_1(·),h_2(·)\in {\mathcal{P}}_0({\mathbb{R}}^n)$, such that $\frac{1}{p(x)}=\frac{1}{p_1(x)}+\frac{1}{p_2(x)},\frac{1}{h(x)}=\frac{1}{h_1(x)}+\frac{1}{h_2(x)}$. Then there exists a constant C depending only on $p_{-}$ and $p_{+}$ such that

$${\parallel fg\parallel }_{M_{p(·)}^{h(·)}}\le C_{p,p_1}{\parallel f\parallel }_{M_{p_1(·)}^{h_1(·)}}{\parallel g\parallel }_{M_{p_2(·)}^{h_2(·)}}$$

holds for every $f\in M_{p_1(·)}^{h_1(·)}$ and $g\in M_{p_2(·)}^{h_2(·)}$.

(2) [37]) Let $p,p_0,p_1,h,h_0,h_1\in {\mathcal{P}}_0$, $0<q<\infty$ and $s_0,s_1\in L^{\infty }\cap C^{log}({\mathbb{R}}^n)$ with $s_0>s_1$. If $\frac{1}{q}$ and

$$s_0(x)-\frac{n}{p_0(x)}=s_1(x)-\frac{n}{p_1(x)}$$

are locally log- Hölder continuous, then

$${\mathcal{N}}_{p_0(·),h_0(·),q}^{s_0(·)}\hookrightarrow {\mathcal{N}}_{p_1(·),h_1(·),q}^{s_1(·)}.$$

(3) ([36]) For $p(·)\in C^{log}({\mathbb{R}}^n)$ and $\psi \in L^1({\mathbb{R}}^n),$ assume $\Psi (x)=\underset{y\notin B(0,|x|)}{sup}\left|\psi (y)\right|$ is integrable. Then

$$\parallel f*{\psi }_{\epsilon }{\parallel }_{M_{p(·)}^{h(·)}({\mathbb{R}}^n)}\le {C\parallel f\parallel }_{M_{p(·)}^{h(·)}({\mathbb{R}}^n)}{\parallel \Psi \parallel }_{L^1({\mathbb{R}}^n)},$$

for all $f\in M_{p(·)}^{h(·)}({\mathbb{R}}^n)$, where ${\psi }_{\epsilon }=\frac{1}{{\epsilon }^n}\psi (\frac{1}{\epsilon })$ and C depends only on n.

Next, we recall the paradifferential calculus which enables us to define a generalized product between distributions. The paraproduct between u and v is defined by

$$T_{u}v:=\sum _{j\in \mathbb{Z}}{S}_{j-1}u{\Delta }_{j}v.$$

Then we have the formal decomposition:

$$uv=T_{u}v+T_{v}u+R(u,v),$$

with

$$R(u,v)=\sum _{j\in \mathbb{Z}}{\Delta }_{j}u{\tilde{\Delta }}_{j}v\mathrm{and}{\tilde{\Delta }}_j={\Delta }_{j-1}+{\Delta }_j+{\Delta }_{j+1}.$$

This decomposition is called Bony’s paraproduct decomposition.

Proposition 2.

Let $s>0$, $1\le \rho \le \infty$, $p(·),h(·),q(·)\in C^{log}\cap {\mathcal{P}}_0({\mathbb{R}}^n)$, $\frac{1}{h(.)}=\frac{1}{h_1(·)}+\frac{1}{h_2(·)}$ and $\frac{1}{\rho }=\frac{1}{p(·)}+\frac{1}{q(·)}$. Then we have

$$\parallel u{\partial }_k{Pv\parallel }_{{\dot{\mathcal{N}}}_{\rho ,h(·),1}^s}\lesssim {\parallel u\parallel }_{{\dot{\mathcal{N}}}_{p(·),h_1(·),1}^0}{\parallel v\parallel }_{{\dot{\mathcal{N}}}_{q(·),h_2(·),1}^s}+{\parallel v\parallel }_{{\dot{\mathcal{N}}}_{p(·),h_1(·),1}^0}{\parallel v\parallel }_{{\dot{\mathcal{N}}}_{q(·),h_2(·),1}^s}.$$

Proof. 

Applying Bony’s paraproduct decomposition for some fixed $i\in \mathbb{Z}$, we can write

$$\begin{array}{cc}\hfill {\Delta }_i(u{\partial }_{k}Pv)& =\sum _{|j-i|\le 4}{\Delta }_i(S_{j-1}u{\Delta }_j({\partial }_{k}Pv))+\sum _{|j-i|\le 4}{\Delta }_i(S_{j-1}({\partial }_{k}Pv){\Delta }_{j}u)\hfill \\ \hfill & +\sum _{j\ge i-2}{\Delta }_i({\Delta }_{j}u{\tilde{\Delta }}_j({\partial }_{k}Pv))\hfill \\ \hfill & =I_1+I_2+I_3.\hfill \end{array}$$

To prove this proposition, we estimate the above three terms separately. According to the Proposition 1, we have

$$\parallel 2^{is}{\Delta }_i(S_{j-1}u{\Delta }_j({\partial }_{k}Pv)){\parallel }_{M_{\rho }^{h(·)}}\lesssim \parallel S_{j-1}{u\parallel }_{M_{p(·)}^{h_1(·)}}{\parallel 2^{is}{\Delta }_j({\partial }_{k}Pv)\parallel }_{M_{q(·)}^{h_2(·)}}.$$

Then we have

$$\parallel 2^{is}{\Delta }_i(S_{j-1}u{\Delta }_j({\partial }_{k}v)){\parallel }_{M_{\rho }^{h(·)}}\lesssim \parallel S_{j-1}{u\parallel }_{M_{p(·)}^{h_1(·)}}{\parallel 2^{is}{\Delta }_{j}v\parallel }_{M_{q(·)}^{h_2(·)}}.$$

Considering $I_1$, we have

$$\parallel 2^{is}{I}_1{\parallel }_{M_{\rho }^{h(·)}}\lesssim \sum _{|j-i|\le 4}\parallel S_{j-1}{u\parallel }_{M_{p(·)}^{h_1(·)}}{\parallel 2^{is}{\Delta }_{j}v\parallel }_{M_{q(·)}^{h_2(·)}}.$$

Similarly, for $I_2$ we have

$$\parallel 2^{is}{I}_2{\parallel }_{M_{\rho }^{h(·)}}\lesssim \sum _{|j-i|\le 4}\parallel S_{j-1}{u\parallel }_{M_{p(·)}^{h_1(·)}}{\parallel 2^{is}{\Delta }_{j}v\parallel }_{M_{q(·)}^{h_2(·)}}.$$

To estimate $I_3$, using Proposition 1 and obtain

$$\parallel {\Delta }_i({\Delta }_{j}u{\tilde{\Delta }}_j)({\partial }_{k}Pv){\parallel }_{M_{\rho }^{h(·)}}\lesssim \parallel {\Delta }_j{u\parallel }_{M_{q(·)}^{h_2(·)}}{\parallel {\tilde{\Delta }}_j({\partial }_{k}Pv)\parallel }_{M_{p(·)}^{h_1(·)}}.$$

Then we get

$$\parallel {\Delta }_i({\Delta }_{j}u{\tilde{\Delta }}_j)({\partial }_{k}Pv){\parallel }_{M_{\rho }^{h(·)}}\lesssim \parallel {\Delta }_j{u\parallel }_{M_{q(·)}^{h_2(·)}}{\parallel {\tilde{\Delta }}_{j}v\parallel }_{M_{p(·)}^{h_1(·)}}.$$

Considering $I_3$, we have

$$\begin{array}{cc}\hfill \parallel 2^{is}{I}_3{\parallel }_{M_{\rho }^{h(·)}}& \lesssim \sum _{j\ge i-2}\parallel {\Delta }_j{u\parallel }_{M_{q(·)}^{h_2(·)}}{\parallel {\tilde{\Delta }}_{j}v\parallel }_{M_{p(·)}^{h_1(·)}}\hfill \\ \hfill & =\sum _{j\ge i-2}{2}^{(i-j)s}\parallel {\Delta }_j{u\parallel }_{M_{q(·)}^{h_2(·)}}{\parallel {\tilde{\Delta }}_{j}v\parallel }_{M_{p(·)}^{h_1(·)}}.\hfill \end{array}$$

By combining the above estimates yields the result. □

Proposition 3.

Let $s>0$, $1\le \gamma ,{\gamma }_1,{\gamma }_2,h,h_1,h_2,p,q,r,\rho \le \infty$, $\frac{1}{\rho }=\frac{1}{r}+\frac{1}{q}$, $\frac{1}{h}=\frac{1}{h_1}+\frac{1}{h_2}$ and $\frac{1}{\gamma }=\frac{1}{{\gamma }_1}+\frac{1}{{\gamma }_2}$. Then we have

$${\parallel uv\parallel }_{{\mathcal{L}}^{\gamma }{\dot{\mathcal{N}}}_{\rho ,h,q}^s}\lesssim {\parallel u\parallel }_{{\mathcal{L}}^{{\gamma }_1}{\dot{\mathcal{N}}}_{p,h_1,q}^s}{\parallel v\parallel }_{{\mathcal{L}}^{{\gamma }_2}{M}_r^{h_2}}+{\parallel v\parallel }_{{\mathcal{L}}^{{\gamma }_1}{\dot{\mathcal{N}}}_{p,h_2,q}^s}{\parallel u\parallel }_{{\mathcal{L}}^{{\gamma }_2}{M}_r^{h_1}}.$$

Proof. 

Replacing $M_{\rho }^{h(·)}$ in the proof of Proposition 2 by ${\mathcal{L}}^{\gamma }{M}_{\rho }^{h(·)}$, we can obtain the desired inequality. □

3. The Well-Posedness

In this section, we obtain the global well-posedness of generalized porous equations by using contraction mapping in critical Banach spaces. To insure the global well-posedness for small initial data $u_0$, the following lemma is very important.

Lemma 4.

(Lemma 5.5, [38]) Let X be a Banach space with norm $\parallel ·\parallel$ and $B:X\times X\to X$ be a bounded linear operator satisfying $\parallel B(u,v)\parallel \le \eta \parallel u\parallel \parallel v\parallel$ for any $u,v\in X$ and a constant $\eta >0$. Then for any $y\in X$ such that $4\eta \parallel y\parallel <1$, the equation $x=y+B(u,v)$ has a solution $x\in X$. In particular, the solution is such that $\parallel x\parallel \le 2\parallel y\parallel$ and it is the only one such that $\parallel x\parallel <\frac{1}{2\eta }$.

The main tool to to prove the result is to obtain the priori estimates of the Equation (1). First, we prove the linear estimates of (1). To obtain this, we consider the linear homogeneous dissipative equation

$$\left\{\begin{array}{c}{u}_t+\mu {\Lambda }^{\beta }=f(x,t)\mathrm{in}{\mathbb{R}}^3\times {\mathbb{R}}^{+},\hfill \\ u(x,0)=u_0\mathrm{in}{\mathbb{R}}^3,\hfill \end{array}\right.$$

(3)

for which we show the following lemma.

Lemma 5.

Let $I=[0,T)$, $T\in (0,\infty ]$, $1\le \gamma ,q\le \infty$, $p(·),p_1(·),h(·)\in \mathcal{P}({\mathbb{R}}^n)$ for $p_1(·)\le p(·)$, $p(·)\le h(·)<\infty$ and $s(·)\in C^{log}({\mathbb{R}}^n)$. Assume that $u_0\in \mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}}$ and $f\in {\mathcal{L}}^{\gamma }(I;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }-\beta })$. Then the Cauchy problem (3) has a unique solution $u\in {\mathcal{L}}^{\infty }(I;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p_1^{\prime }(·)}})\cap {\mathcal{L}}^{\gamma }(I;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})$ such that for all ${\gamma }_1\in [\gamma ,\infty ]$

$$\parallel u{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;\mathcal{F}{\dot{\mathcal{N}}}_{p_1(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}+\frac{\beta }{{\gamma }_1}})}\lesssim \parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}}}+\parallel f{\parallel }_{{\mathcal{L}}^{\gamma }(I;\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }-\beta })}.$$

(4)

Moreover, if $q<\infty$, then $u\in \mathcal{C}(I;\mathcal{F}{\dot{\mathcal{N}}}_{p_1(·),h(·),q}^{s(·)+\frac{3}{p^{\prime }(·)}})$.

Proof. 

Clearly, solution to the Cauchy problem (3) can be written by the following integral equation:

$$u(x,t)=e^{-t{\Lambda }^{\beta }}{u}_0+{\int }_0^t{e}^{-(t-\tau ){\Lambda }^{\beta }}f(x,t)d\tau .$$

(5)

The Fourier Transform of Equation (5) gives

$$\widehat{u}(x,t)=e^{{-t|·|}^{\beta }}{\widehat{u}}_0+{\int }_0^t{e}^{{-(t-\tau )|·|}^{\beta }}\widehat{f}(·,t)d\tau .$$

(6)

Multiplying $2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i$ both sides of Equation (6), we have

$$\begin{array}{cc}{2}^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i\widehat{u}(x,t)& =2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i{e}^{{-t|·|}^{\beta }}{\widehat{u}}_0\hfill \\ & +{\int }_0^t{2}^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i{e}^{{-(t-\tau )|·|}^{\beta }}\widehat{f}(·,t)d\tau .\hfill \end{array}$$

(7)

Applying ${\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})$ on both sides of Equation (7), we have

$$\begin{array}{cc}\hfill & \parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i\widehat{u}{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})}\hfill \\ \hfill & \le \parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i{e}^{{-t|·|}^{\beta }}{\widehat{u}}_0{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})}\hfill \\ \hfill & +\parallel {\int }_0^t{2}^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{e}^{{-(t-\tau )|·|}^{\beta }}{\phi }_i\widehat{f}(·,t)d\tau {\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})}\hfill \\ & =I_1+I_2.\hfill \end{array}$$

(8)

We estimate the above two terms separately. Let $p^{*}=\frac{p(x)p_1(x)}{p(x)-p_1(x)}$, using Proposition 1 and considering $p_1(·)\le p(·)$, we get

$$\begin{array}{cc}\hfill & \parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_i{e}^{{-t|·|}^{\beta }}{\widehat{u}}_0{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})}\hfill \\ \hfill & \lesssim \sum _{j=0,\pm 1}\parallel 2^{i(s(·)+\frac{3}{p^{\prime }(·)})}{\phi }_i{\widehat{u}}_0{\parallel }_{M_{p(·)}^{h(·)}}\parallel r^{-\frac{3}{p^{*}}}{2}^{\frac{3}{p_1^{\prime }(·)}-\frac{3}{p^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{\phi }_{i+j}{e}^{{-t|·|}^{\beta }}{\widehat{u}}_0{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;L^{p^{*}})}\hfill \\ \hfill & \lesssim \sum _{j=0,\pm 1}\parallel 2^{i(s(·)+\frac{3}{p^{\prime }(·)})}{\phi }_i{\widehat{u}}_0{\parallel }_{M_{p(·)}^{h(·)}}\parallel r^{-\frac{3}{p^{*}}}{2}^{i\frac{\beta }{{\gamma }_1}}{e}^{-t{2}^{\beta (i+j)}}{\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I)}\parallel 2^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\phi }_{i+j}{\parallel }_{L^{p^{*}}}\hfill \\ \hfill & \lesssim \sum _{j=0,\pm 1}\parallel 2^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\phi }_{i+j}{\parallel }_{L^{p^{*}}}\parallel 2^{i(s(·)+\frac{3}{p^{\prime }(·)})}{\phi }_i{\widehat{u}}_0{\parallel }_{M_{p(·)}^{h(·)}}\hfill \\ \hfill & \lesssim \parallel 2^{i(s(·)+\frac{3}{p^{\prime }(·)})}{\phi }_i{\widehat{u}}_0{\parallel }_{M_{p(·)}^{h(·)}},\hfill \end{array}$$

(9)

where we have used the following fact in the above estimate

$$\begin{array}{cc}\hfill & \sum _{j=0,\pm 1}\parallel 2^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\phi }_{i+j}{\parallel }_{L^{p^{*}}}\hfill \\ \hfill & =\sum _{j=0,\pm 1}\parallel 2^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\phi }_{i+j}{\parallel }_{L^{\frac{p(·)p_1(·)}{p(·)-p_1(·)}}}\hfill \\ \hfill & =\sum _{j=0,\pm 1}inf\left\{\lambda >0:\int |\frac{2^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\phi }_{i+j}}{\lambda }{|}^{\frac{p(·)p_1(·)}{p(·)-p_1(·)}}dx\le 1\right\}\hfill \\ \hfill & =\sum _{j=0,\pm 1}inf\left\{\lambda >0:\int |\frac{{\phi }_{i+j}}{\lambda }{|}^{\frac{p(·)p_1(·)}{p(·)-p_1(·)}}{2}^{-3i}dx\le 1\right\}\hfill \\ \hfill & =\sum _{j=0,\pm 1}inf\left\{\lambda >0:\int |\frac{{\phi }_{i+j}}{\lambda }{|}^{\frac{p(2^i·)p_1(2^i·)}{p(2^i·)-p_1(2^i·)}}dx\le 1\right\}\lesssim C.\hfill \end{array}$$

To estimate $I_2$, using Proposition 1, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_0^t{2}^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{{\gamma }_1})}{e}^{{-(t-\tau )|·|}^{\beta }}{\phi }_i\widehat{f}(·,t)d\tau {\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I;M_{p_1(·)}^{h(·)})}\hfill \\ & \lesssim \sum _{j=0,\pm 1}\parallel {\int }_0^t{\parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{\gamma }-\beta )}{\phi }_i\widehat{f}(·,t)\parallel }_{M_{p(·)}^{h(·)}}\hfill \\ & ·\parallel r^{\frac{-3}{p^{*}}}{\phi }_{i+j}{e}^{-(t-\tau )2^{\beta (i+j)}}{2}^{\beta i(1+\frac{1}{{\gamma }_1}-\frac{1}{\gamma })+3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\parallel }_{L^{p^{*}}}d\tau {\parallel }_{{\mathcal{L}}^{{\gamma }_1}(I)}\hfill \\ & \lesssim \sum _{j=0,\pm 1}\parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{\gamma }-\beta )}{\phi }_i\widehat{f}{(·,t)\parallel }_{{\mathcal{L}}^{\gamma }(I;M_{p_1(·)}^{h(·)})}{\parallel {\phi }_{i+j}{2}^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}\parallel }_{L^{p^{*}}}\hfill \\ & ·\parallel r^{\frac{-3}{p^{*}}}{e}^{-t{2}^{\beta (i+j)}}{2}^{\beta i(1+\frac{1}{{\gamma }_1}-\frac{1}{\gamma })}{\parallel }_{{\mathcal{L}}^{{(1+\frac{1}{{\gamma }_1}-\frac{1}{\gamma })}^{-1}}({\mathbb{R}}_{+})}\hfill \\ & \lesssim \sum _{j=0,\pm 1}\parallel {\phi }_{i+j}{2}^{3i(\frac{1}{p_1^{\prime }(·)}-\frac{1}{p^{\prime }(·)})}{\parallel }_{L^{p^{*}}}{\parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{\gamma }-\beta )}{\phi }_i\widehat{f}(·,t)\parallel }_{{\mathcal{L}}^{\gamma }(I;M_{p_1(·)}^{h(·)})}\hfill \\ & \lesssim \parallel 2^{i(s(·)+\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{\gamma }-\beta )}{\phi }_i\widehat{f}{(·,t)\parallel }_{{\mathcal{L}}^{\gamma }(I;M_{p_1(·)}^{h(·)})}.\hfill \end{array}$$

(10)

Using the inequalities (9), (10) in (8) and taking the $l^q$-norm, we can obtain the desired estimate (4). □

Theorem 1.

Let $p(·),h(·)\in C^{log}({\mathbb{R}}^n)\cap {\mathcal{P}}_0({\mathbb{R}}^n)$, $h(·)\le p(·)$, $2\le p(·)\le \frac{6}{5-2\beta }$, $1\le q<\frac{3}{\beta -1}$, $1<\beta \le \frac{5}{2}$ and $1\le \gamma \le \infty$ then there exists a constant $\delta >0$ such that for any $u_0\in \mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}({\mathbb{R}}^n)$ satisfies $\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}({\mathbb{R}}^n)}<\delta$, the Equation (1) has a unique global solution

$$u\in {\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta }),$$

such that

$${\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}\lesssim {\parallel u_0\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}.$$

(11)

Proof. 

For $H_{\beta }(t)=e^{-t{\Lambda }^{\beta }}$, we can write the solution $u(x,t)$ of the Equation (1) in the following integral form

$$u=H_{\beta }(t)u_0+{\int }_0^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau .$$

(12)

Next we define

$$Y:={\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })$$

and consider the mapping below

$$\psi :u\to H_{\beta }(t)u_0+{\int }_0^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau .$$

(13)

We have to show that the above mapping is a contraction mapping. First, we can write

$$\begin{array}{cc}\hfill \parallel \psi (u){\parallel }_Y& \le \parallel H_{\beta }(t)u_0{\parallel }_Y+\parallel {\int }_o^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau {\parallel }_Y\hfill \\ & =I_1+I_2.\hfill \end{array}$$

(14)

To estimate $I_1$, using Lemma 5 with $f\equiv 0$ and considering the hypothesis $p(·)\ge 2$, we obtain

$$\parallel H_{\beta }(t)u_0{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+},\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})}\lesssim \parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}},$$

$$\parallel H_{\beta }(t)u_0{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+},\mathcal{F}{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}\lesssim \parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}},$$

$$\parallel H_{\beta }(t)u_0{\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+},\mathcal{F}{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}\lesssim \parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}},$$

Hence, we get

$$\parallel H_{\beta }(t)u_0{\parallel }_Y\lesssim \parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}<\delta .$$

(15)

To estimate $I_2$, let ${\overline{p}}_{\alpha }(·)=\frac{6p(·)}{6-(5-2\beta )p(·)}$, using Propositions 1 and 3, then

$$\begin{array}{cc}\hfill & \parallel {\int }_o^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})}\hfill \\ & \lesssim \parallel \parallel {\int }_o^t{2}^{i(1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i{e}^{{-(t-\tau )|·|}^{\beta }}\widehat{\nabla ·(u\nabla Pu)}d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}{\parallel }_{l^q}\hfill \\ & \lesssim \parallel \parallel {\int }_o^t\parallel r^{\frac{-3}{{\overline{p}}_{\alpha }(·)}}{2}^{i(2-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i{e}^{{-(t-\tau )|·|}^{\beta }}{\parallel }_{L^{{\overline{p}}_{\alpha }(·)}}{\parallel {\Delta }_i(u\nabla Pu)\parallel }_{M_{\frac{6}{2\beta +1}}^{h(·)}}d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+})}{\parallel }_{l^q}\hfill \\ & \lesssim \parallel \parallel {\int }_o^t{2}^{i(\frac{\beta }{\gamma }+\frac{5}{2})}{e}^{-(t-\tau )2^{\beta i}}\parallel r^{\frac{-3}{{\overline{p}}_{\alpha }(·)}}{2}^{-3i\frac{1}{{\overline{p}}_{\alpha }(·)}}{\phi }_i{\parallel }_{L^{{\overline{p}}_{\alpha }(·)}}{\parallel {\Delta }_i(u\nabla Pu)\parallel }_{M_{\frac{6}{2\beta +1}}^{h(·)}}d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+})}{\parallel }_{l^q}\hfill \\ & \lesssim \parallel \parallel 2^{i(\frac{\beta }{\gamma }+\frac{5}{2}-\beta )}{\parallel {\Delta }_i(u\nabla Pu)\parallel }_{M_{\frac{6}{2\beta +1}}^{h(·)}}{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+})}\parallel e^{-t{2}^{\beta i}}{2}^{\beta i}{\parallel }_{{\mathcal{L}}^1({\mathbb{R}}_{+})}{\parallel }_{l^q}\hfill \\ & \lesssim {\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}{\parallel u\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+};L^{\frac{3}{\beta -1}})}\hfill \\ & \lesssim {\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}{\parallel u\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{B}}_{2,q}^{\frac{5}{2}-\beta })}\hfill \\ & \lesssim {\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}{\parallel u\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}.\hfill \end{array}$$

(16)

Similarly, we have

$$\begin{array}{cc}\hfill & \parallel {\int }_o^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}\hfill \\ & \lesssim {\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}{\parallel u\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}\hfill \end{array}$$

(17)

and

$$\begin{array}{cc}\hfill & \parallel {\int }_o^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{\frac{5}{2}-\beta })}\hfill \\ & \lesssim {\parallel u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,h(·),q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})}{\parallel u\parallel }_{{\mathcal{L}}^{\infty }({\mathbb{R}}_{+};{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}.\hfill \end{array}$$

(18)

Hence, we obtain

$$\begin{array}{cc}\hfill & \parallel {\int }_o^t{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau {\parallel }_Y\lesssim {\parallel u\parallel }_Y{\parallel u\parallel }_Y.\hfill \end{array}$$

(19)

Using the estimates (15) and (19) in (14), we have

$$\begin{array}{cc}\hfill \parallel \psi (u){\parallel }_Y& \le C_1\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}+C_2{\parallel u\parallel }_Y{\parallel u\parallel }_Y\hfill \\ & \le C_1\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}+C_2{\delta }_0^2.\hfill \end{array}$$

Choosing ${\delta }_0<\frac{1}{2max\{C_1,C_2\}}$ and for any $u_0\in \mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}$ with

$$\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}<\frac{{\delta }_0}{2max\{C_1,C_2\}},$$

we get

$$\parallel \psi (u){\parallel }_Y\le \frac{{\delta }_0}{2}+\frac{{\delta }_0}{2}={\delta }_0.$$

Using Lemma 4, we can directly obtain that Equation (1) has a unique global solution with $\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·)q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}<\delta$ for sufficiently small $\delta$. □

4. Gevrey Class Regularity

In this section, we show the gevrey class regularity for Equation (1). Many researchers have studied analyticity with respect to Navier-Stokes equations. For more study related to Gevrey class regularity we refer the reader to to [39,40]. In order to prove the spatial analyticity, the following lemma is very helpful.

Lemma 6.

(Lemma 4.1 [41]) Let $0<s\le t<\infty$ and $0\le \beta \le 2$. Then the following inequality holds

$${t\left|x\right|}^{\frac{\beta }{2}}-\frac{1}{2}(t^2-s^2){\left|x\right|}^{\beta }{-s|x-y|}^{\frac{\beta }{2}}-s{\left|y\right|}^{\frac{\beta }{2}}\le \frac{1}{2}$$

for all $x,y\in {\mathbb{R}}^n$.

Theorem 2.

Let $p(·),h(·)\in C^{log}({\mathbb{R}}^n)\cap {\mathcal{P}}_0({\mathbb{R}}^n)$,$h(·)\le p(·)$, $2\le p(·)\le \frac{6}{5-2\beta }$, $1\le q<\frac{3}{\beta -1}$, then there exist a constant $\epsilon >0$ such that for all initial data $u_0\in \mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}$ satisfying $\parallel u_0{\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}<\epsilon$. The Equation (1) has a unique analytic solution u in the sense that

$$\parallel e^{\sqrt{t}{\left|D\right|}^{\frac{\beta }{2}}}{u\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{2,h(·)q}^{\frac{5}{2}-\beta +\frac{\beta }{\gamma }})\cap {\mathcal{L}}^{\infty }({\mathbb{R}}_{+};\mathcal{F}{\dot{\mathcal{N}}}_{2,2,q}^{\frac{5}{2}-\beta })}\lesssim {\parallel u_0\parallel }_{\mathcal{F}{\dot{\mathcal{N}}}_{p(·),h(·),q}^{1-\beta +\frac{3}{p^{\prime }(·)}}}$$

Proof. 

Assume $\omega (x,t)=e^{\sqrt{t}{\left|D\right|}^{\frac{\beta }{2}}}u(x,t)$ and using Equation (11), we obtain

$$\omega (x,t)=e^{\sqrt{t}{\left|D\right|}^{\frac{\beta }{2}}}{H}_{\beta }(t)u_0+{\int }_0^t{e}^{\sqrt{t}{\left|D\right|}^{\frac{\beta }{2}}}{H}_{\beta }(t-\tau )\nabla ·(u\nabla Pu)d\tau .$$

It is easy to obtain

$$\begin{array}{cc}\hfill & \parallel 2^{i(1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i\widehat{\omega }{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}\hfill \\ & \lesssim \parallel e^{\sqrt{t}{\left|\xi \right|}^{\frac{\beta }{2}}-t{\left|\xi \right|}^{\beta }}{2}^{i(1-\beta +\frac{3}{p_1^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i\widehat{u_0}{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}\hfill \\ & +\parallel 2^{i(2-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i{\int }_0^t{e}^{-\frac{1}{2}(t-\tau ){\left|\xi \right|}^{\beta }}{\int }_{{\mathbb{R}}^n}{e}^{\sqrt{t}{\left|\xi \right|}^{\frac{\beta }{2}}-\frac{1}{2}{(t-\tau )\left|\xi \right|}^{\beta }-\sqrt{\tau }{(|\xi -\eta |}^{\frac{\beta }{2}}+{\left|\eta \right|}^{\frac{\beta }{2}})}\hfill \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\widehat{\left(\nabla ·(u(\xi -\eta ,\tau )\nabla Pu(\eta ,\tau )\right)}d\eta d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}.\hfill \end{array}$$

Using $e^{\sqrt{t}{\left|\xi \right|}^{\frac{\beta }{2}}-\frac{1}{2}t{\left|\xi \right|}^{\beta }}=e^{\frac{1}{2}(\sqrt{t}{\left|\xi \right|}^{\frac{\beta }{2}}{-1)}^2+\frac{1}{2}}\le e^{\frac{1}{2}}$ and Lemma 6, we obtain

$$\begin{array}{cc}\hfill & \lesssim \parallel e^{-\frac{1}{2}t{\left|\xi \right|}^{\beta }}{2}^{i(1-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i\widehat{u_0}{\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}\hfill \\ & +\parallel 2^{i(2-\beta +\frac{3}{p^{\prime }(·)}+\frac{\beta }{\gamma })}{\phi }_i{\int }_0^t{e}^{-\frac{1}{2}(t-\tau ){\left|\xi \right|}^{\beta }}\widehat{\left(\nabla ·(u(\xi -\eta ,\tau )\nabla Pu(\eta ,\tau )\right)}d\tau {\parallel }_{{\mathcal{L}}^{\gamma }({\mathbb{R}}_{+};M_{p(·)}^{h(·)})}.\hfill \end{array}$$

The remaining part of the proof is similar to the proof of Theorem 3.3, therefore the details can be omitted. □

5. Conclusions

In this paper, we considered the generalized porous medium equations. Young’s inequality is one of the important tools to obtain the global well-posedness result of such equations. In the previous work of Xiao and Zhou [3], they obtained the local well-posedness for large initial data and the global well-posedness for small initial data in Fourier-Besov spaces. We can’t use Young’s inequality in variable exponent function spaces in order to obtain the global well-posedness result. We overcame with this problem and obtained the global well-posedness results of this equation for small initial data $u_0$ belonging to the homogeneous Fourier-Besov-Morrey spaces with variable exponent. In addition, we also shown the Gevrey class regularity of the solution.