Optimal Time-Decay Estimates in the Critical Framework for a Chemotaxis Model

Abstract

In this paper, we investigate the Cauchy problem of a multidimensional hyperbolic–parabolic system which derived from the Keller–Segel model describing chemotaxis in the critical Besov spaces with initial data close to a constant equilibrium state. The time-decay estimates of global strong solutions are established in more general $L^{p}$ framework comparing with Xu et al. (J Math Phys 60:091509, 2019) and Xu and Li (Commun Contemp Math, 2021. https://doi.org/10.1142/S0219199721500784), and a new regularity assumption is proposed that the low frequencies have more freedom. The proof mainly depends on tricky and non-classical Besov product estimates about various Sobolev embeddings.

1 Introduction

It is well known that chemotaxis is a common biological phenomenon and one of the most basic physiological responses of cells. It describes the tendency of cells, bacteria or other organisms to move according to the distribution of related chemicals in the environment. As far as we know, the mathematical chemotactic modeling has developed into many diverse disciplines since Keller and Segel (see [3,4,5]) built an effective mathematical modeling to describe chemical tropism. Just as the authors of [6] wrote, Keller–Segel modeling is a cornerstone for many works investigating the biological and mathematical properties of chemotaxis, due to its intuitive simplicity, analytical tractability and capability to model the basic dynamics of chemotactic populations. Precisely, the following model with logarithmic sensitivity is proposed:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u=\mathrm {div}\left( D\nabla u-\chi u\nabla \mathrm {ln}v\right) ,\\ \tau \partial _{t}v=D_{1}\Delta v-\mu uv-\sigma v \end{array} \right. \end{aligned}$$

(1)

for $(t,x)\in \mathbb {R}_{+}\times \mathbb {R}^{d}(d\ge 1)$. Here, u(t, x) and v(t, x) are the density of a cellular population and the concentration of a chemical signal, respectively. The constant $D>0$ stands for the diffusion coefficient of cellular population and the constant $D_{1}\ge 0$ is the diffusion coefficient of chemical signal. The constant $\chi \ne 0$ is the coefficient of chemotactic sensitivity, where $|\chi |$ measures the strength of chemical signals. The constant $\tau \ge 0$ denotes a relaxation time scale such that $1/\tau$ is the rate toward the equilibrium. The constant $\mu \ne 0$ is the coefficient of density-dependent production/degradation rate of chemical signal, and $\sigma \ge 0$ is the natural degradation rate of chemical signal.

In this paper, we only consider the limiting case that the diffusion of the chemical substance is so small ($D_{1}\rightarrow 0$) despite there are two major limiting cases in (1)(see [7] for the case $\tau \rightarrow 0$). It is not difficult to find that, when $\tau \rightarrow 0$, system (1) is reduced to a parabolic–elliptic equations, while the system (1) becomes a partial differential equation coupled with an ordinary differential equation when $D_{1}\rightarrow 0$, that is

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u=\mathrm {div}\left( D\nabla u-\chi u\nabla \,\mathrm {ln} \,v \right) ,\\ \partial _{t}v=-\mu uv-\sigma v, \end{array} \right. \end{aligned}$$

(2)

with $\tau =1$. In fact, what we are most interested in is the hyperbolic–parabolic case $\chi \mu >0$ (see [8, 9] for more explanations). This includes two scenarios: $\chi >0$ and $\mu >0$, or $\chi <0$ and $\mu <0$. The former case indicates that organisms are attracted to and consume the chemicals while the latter case describes the movement of organisms that deposit a chemical signal to amendment the local environment for succeeding information (see [10]). In order to eliminate the singularity caused by $\mathrm {in}\,v$, a couple of new variables in terms of the Hopf–Cole transformation was introduced (see [11]):

$$\begin{aligned} u=\varrho , \ \ w=-\nabla \mathrm {ln}\,v. \end{aligned}$$

Under the rescaled and dimensionless variables:

$$\begin{aligned} {\widetilde{t}}=\frac{\chi \mu }{D} t, \ \ {\widetilde{x}}=\frac{\sqrt{\chi \mu }}{D}x \ \ \hbox {and} \ \ {\widetilde{w}}=\mathrm {sign}(\chi )\sqrt{\frac{\chi }{\mu }} w, \end{aligned}$$

one can write (2) as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}\varrho -\Delta \varrho =\mathrm {div}\left( \varrho w\right) ,\\ \partial _{t}w-\nabla \varrho =0, \end{array} \right. \end{aligned}$$

(3)

after dropping the tilde accent.

Due to the powerful biological background and important potential applications, the rigorous analysis of (3) and its related system has gradually become one of the focal points and hot spots in many diverse disciplines such as computational mathematics and applied mathematics in recent years. In particular, the existence and decay of solutions are always concerned in applied mathematics. In one dimension, Zhang and Zhu [12] considered the initial-boundary value problem for (3) where the initial data are chosen to be small in $H^{2}([0,1])$. Subsequently, Li, Pan and Zhao [6] improve the result obtained in [12] and established the global existence of classical solutions with large initial data in $H^{2}([0,1])$. When the large initial perturbations around constant equilibrium states in $H^{2}(\mathbb {R})$, the global existence and quantitative decay of solutions to the Cauchy problem for the system was shown by Li, Pan and Zhao [13]. What should be mentioned is that Fan and Zhao [14] and give out the blow-up criterion and presented the global well-posedness of classical solutions with large initial data in $H^{s}(\mathbb {R})$ and $s>\frac{1}{2}$. For the multidimensional cases of (3), the initial-boundary value problems have been investigated in [6] and references therein. In [15], the local and global existences of classical solutions to the Cauchy problem are investigated for the initial data in $H^{s}(\mathbb {R}^{d})$ ($d\ge 2$) with $s>\frac{d}{2}+1$. Xie et al. [16] proved the global existence of strong solutions when $H^{2}(\mathbb {R}^{3})$-norm of the initial perturbation around a constant state is small enough. Meanwhile, the large-time asymptotic behavior of global solutions to (3) has also attracted many researcher’s interest. For example, Li, Li and Zhao [15] also established the large-time dynamics and temporal decay rates of classical solutions if the initial data has high Sobolev regular index and the dimension $d\ge 4$. Xie et al. [16] exhibited a detailed analysis concerning the optimal time-decay rates of strong solutions to the 3D chemotaxis model where the initial data belong to $H^{2}(\mathbb {R}^{3})\cap L^{1}(\mathbb {R}^{3})$. For other related results, the readers can refer to, for instance, [17, 18] for bounded domains, and [19,20,21] for nonlinear stabilities of traveling waves and references therein.

Let us emphasize that the above results mentioned are established in Sobolev spaces with higher regularity. However, to the best of our knowledge, so far there are few results (except for [1, 2, 22]) for (3) from the point of view of scaling invariance. Indeed, observe that (3) is obviously invariant for all $l >0$ by the following transformation

$$\begin{aligned} \varrho ( t,x) \rightsquigarrow l ^{2} \varrho ( l ^{2}t,l x), \ \ w(t,x) \rightsquigarrow l w(l ^{2}t,l x). \end{aligned}$$

Inspired by the scaling property, Hao [22] obtained the global existence and uniqueness of strong solutions to (3) with initial data close to a constant equilibrium state in the $L^{2}$ critical homogeneous Besov spaces by using the compactness arguments. Recently, under the additional smallness assumption of low frequencies of initial data, Xu, Li and Wang [1] established the large-time asymptotic description of the constructed solution in [22] if the initial data are close to a stable equilibrium state in the $L^{2}$ critical regularity framework. The natural next problem is to extend the results of [1, 22] to more general $L^{p}$ critical Besov spaces. To do this, we here introduce an appropriate transformation, which is initiated by Deng and Li [23]. Namely, let $q=\Lambda ^{-1}\mathrm {div}\,w$ with $\Lambda ^{-1}=(-\Delta )^{-\frac{1}{2}}$, we get from (3) that

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}\varrho -\Delta \varrho =-\mathrm {div}\left( \varrho \nabla \Lambda ^{-1} q\right) ,\\ \partial _{t}q+\Lambda \varrho =0. \end{array} \right. \end{aligned}$$

(4)

Indeed, we would like to mention that (4) is equivalent to (3), since $\mathrm {curl} \, w=0$ and $w=-\nabla \Lambda ^{-1}\,q$. System (4) is supplemented with the initial data

$$\begin{aligned} (\varrho ,q)|_{t=0}=(\varrho _{0}(x),q_{0}(x)), \ \ x\in \mathbb {R}^{d}, \end{aligned}$$

(5)

and we focus on solutions that are close to some constant state $(\bar{\varrho },0)$ with $\bar{\varrho }>0$, at infinity. More recently, Xu and Li [2] assumed that $\bar{\varrho }=1$ for the sake of simplicity, and constructed the unique global strong solution to (4)–(5) in the general $L^{p}$ critical Besov spaces. Precisely,

Theorem 1

Let $d\ge 2$ and p fulfill

$$\begin{aligned} 2\le p\le \min (4,2d/(d-2)) \ \hbox {and}, \ \hbox {additionally}, \ p\ne 4 \ \hbox {if} \ d=2. \end{aligned}$$

(6)

There exists a small positive constant $c=c(p,d)$ and a universal integer $k_{0}\in \mathbb {Z}$ such that if $a^{h}_{0}\triangleq (\varrho _{0}-1)^{h} \in \dot{B}^{\frac{d}{p}-2}_{p,1}$ and $q^{h}_{0} \in \dot{B}^{\frac{d}{p}-1}_{p,1}$ with $(a_{0},q_{0})^{\ell }$ in $\dot{B}^{\frac{d}{2}-2}_{2,1}$ satisfy

$$\begin{aligned} \mathcal {E}_{p,0}\triangleq \Vert (a_{0},q_{0})\Vert ^{\ell }_{\dot{B}^{\frac{d}{2}-2}_{2,1}} +\Vert a_{0}\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}+\Vert q_{0}\Vert ^{h}_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\le c, \end{aligned}$$

then Cauchy problem (4, 5) admits a unique global-in-time solutions $(\varrho ,q)$ with $\varrho =1+a$ and (a, q) in the space $X_{p}$ defined by:

$$\begin{aligned}&(a,q)^{\ell } \in \widetilde{\mathcal {C}_{b}}(\mathbb {R_{+}};\dot{B}_{2,1}^{\frac{d}{2}-2})\cap L^{1}(\mathbb {R_{+}};\dot{B}_{2,1}^{\frac{d}{2}}), \ \ \ a^{h}\in \widetilde{\mathcal {C}_{b}}(\mathbb {R_{+}};\dot{B}_{p,1}^{\frac{d}{p}-2})\cap L^{1}(\mathbb {R_{+}};\dot{B}_{p,1}^{\frac{d}{p}}),\\&q^{h}\in \widetilde{\mathcal {C}_{b}}(\mathbb {R_{+}};\dot{B}_{p,1}^{\frac{d}{p}-1})\cap L^{1}(\mathbb {R_{+}};\dot{B}_{p,1}^{\frac{d}{p}-1}). \end{aligned}$$

Furthermore, we have for some constant $C=C\left( p,d\right)$ and any $t>0$,

$$\begin{aligned} \mathcal {E}_{p}(t)\le C\mathcal {E}_{p,0} \end{aligned}$$

with

$$\begin{aligned} \mathcal {E}_{p}(t)\triangleq & {} \Vert (a,q)\Vert _{{\widetilde{L}}^{\infty }_{t} (\dot{B} _{2,1}^{\frac{d}{2}-2})}^{\ell }+\Vert (a,q)\Vert _{L^{1}_{t}(\dot{B}_{2,1}^{\frac{d}{2}})}^{\ell }+\Vert a\Vert _{{\widetilde{L}}^{\infty }_{t}(\dot{B}_{p,1}^{\frac{d}{p}-2})}^{h} \nonumber \\&+\Vert a\Vert _{L^{1}_{t}(\dot{B}_{p,1}^{\frac{d}{p}})}^{h} +\Vert q\Vert ^{h} _{{\widetilde{L}}^{\infty }_{t}(\dot{B}_{p,1}^{\frac{d}{p}-1})} +\Vert q\Vert ^{h} _{L^{1}_{t}(\dot{B}_{p,1}^{\frac{d}{p}-1})}. \end{aligned}$$

(7)

The main purpose of this paper is to establish the optimal time-decay rates of strong solutions constructed in Theorem 1 in $\mathbb {R}^{d}$ ($d\ge 2$). To do this, let’s rewrite (4) as the nonlinear perturbation form of constant equilibrium state (1, 0), looking at the nonlinearities as source terms. Consequently, in terms of (a, q) with $\varrho =a+1$, system (4) becomes

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}a -\Delta a-\Lambda q=f, \\ \partial _{t}q+\Lambda a=0 \end{array} \right. \end{aligned}$$

(8)

with $f=-\mathrm {div}\left( a \nabla \Lambda ^{-1} q\right)$. The main result is stated as follows.

Theorem 2

Let $d\ge 2$ and p fulfill (6). Denote by $(\varrho ,q)$ the global solution constructed in Theorem 1. Let the real number $\sigma _{1}$ fulfill

$$\begin{aligned} 1-\frac{d}{2}<\sigma _{1}\le \sigma _{0} \ \ \hbox {with} \ \ \sigma _{0}=\frac{2d}{p}-\frac{d}{2}+1. \end{aligned}$$

(9)

There exists a small positive constant $c=c(p,d)$ such that if

$$\begin{aligned} \mathcal {D}_{p,0}\triangleq \left\| (a_{0},q_{0})\right\| ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}\le c, \end{aligned}$$

(10)

then we have for $t\rightarrow \infty$,

$$\begin{aligned} \Vert \Lambda ^{\sigma }a\Vert _{L^{p}}\lesssim & {} \big (\mathcal {D}_{p,0}+\Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}\big ) t^{-\frac{\sigma _{1}+\sigma }{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{p})} \ \ \hbox {if} \ \ -{\widetilde{\sigma }}_{1}<\sigma \le \frac{d}{p}, \\ \Vert \Lambda ^{\sigma } q\Vert _{L^{p}}\lesssim & {} \big (\mathcal {D}_{p,0}+\Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}\big )\langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}-\frac{d}{2}(\frac{1}{p}-\frac{1}{p})} \ \ \hbox {if} \ \ -{\widetilde{\sigma }}_{1}<\sigma \le \frac{d}{p}-1, \end{aligned}$$

where ${\widetilde{\sigma }}_{1}=\sigma _{1}+d\big (\frac{1}{2}-\frac{1}{p}\big )$ and $\Lambda ^{\sigma }f\triangleq \mathcal {F}^{-1}\left( |\xi |^{\sigma }\mathcal {F}f\right)$ for all $\sigma \in \mathbb {R}$.

Remark 1

In contrast with [1, 2], there are apparent innovative aspects in Theorem 2. Precisely, the result [1] studied the case of $\sigma _{1}=\sigma _{0}-1=\frac{d}{2}$ and $p=2$ for simplicity and [2] only considered the case of $\sigma _{1}=\sigma _{0}-1=\frac{2d}{p}-\frac{d}{2}$ which is obviously a general value in our $\big (1-\frac{d}{2}, \frac{2d}{p}-\frac{d}{2}+1\big ]$ for p satisfying (6) . In the present paper, we try to establish the time-weighted energy inequality in more general $L^{p}$ framework. More importantly, the regularity assumption of low frequencies (see (9)–(10)) is new, which enables us to enjoy larger freedom on the choice of $\sigma _{1}$.

Remark 2

We emphasize that the decay index $\alpha =\sigma _{1}+\frac{d}{2}+\frac{1}{2}-\varepsilon$ coming from the second and third terms of functional $\mathcal {D}_{p}(t)$ are optimal. In fact, the decay rates for the high frequencies cannot exceed those of the low–low interactions such as $\mathrm {div}(a^{\ell }\nabla \Lambda ^{-1}q^{\ell })$:

$$\begin{aligned}&\Vert \mathrm {div}\,(a^{\ell }\nabla \Lambda ^{-1} q^{\ell })\Vert ^{h}_{{\widetilde{L}}_{t}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \Vert a^{\ell }\nabla \Lambda ^{-1} q^{\ell }\Vert _{{\widetilde{L}}_{t}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})}\\\lesssim & {} \Vert a^{\ell }\Vert _{L_{t}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})} \Vert q^{\ell }\Vert _{{\widetilde{L}}_{t}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})} +\Vert q^{\ell }\Vert _{{\widetilde{L}}_{t}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})} \Vert q^{\ell }\Vert _{L_{t}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})}\\\lesssim & {} \langle \tau \rangle ^{-(\sigma _{1}+\frac{d}{2}+\frac{1}{2}-\varepsilon )} \big (\langle \tau \rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}}\Vert a^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})}\big ) \big (\langle \tau \rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }\Vert q^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1-2\varepsilon }_{2,1})}\big )\\&+\langle \tau \rangle ^{-(\sigma _{1}+\frac{d}{2}+\frac{1}{2}-\varepsilon )} \big (\langle \tau \rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}}\Vert a^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})}\big ) \big (\langle \tau \rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }\Vert a^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1-2\varepsilon }_{2,1})}\big ). \end{aligned}$$

In particular, if $\sigma _{1}=\sigma _{0}$, then the value of $\alpha$ becomes $\frac{2d}{p}+\frac{3}{2}-\varepsilon$ in (22). If $\sigma _{1}=\sigma _{0}-1=\frac{2d}{p}-\frac{d}{2}$, then the value of $\alpha$ becomes $\frac{2d}{p}+\frac{1}{2}-\varepsilon$. Obviously, such optimal exponent has not yet been observed in [1, 2].

Furthermore, we have more decay estimates of $\dot{B}^{-\sigma _{1}}_{2,\infty }$-$L^{r}$ type.

Corollary 1

Under the additional assumption (9) and (10), the corresponding solution a satisfies

$$\begin{aligned} \Vert \Lambda ^{\sigma }a\Vert _{L^{r}}\lesssim \big (\mathcal {D}_{p,0}+\Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}\big ) t^{-\frac{\sigma _{1}+\sigma }{2}-\frac{d}{2}(\frac{1}{2}-\frac{1}{r})} \end{aligned}$$

for all $p\le r\le \infty$ and $\sigma \in \mathbb {R}$ fulfilling $-{\widetilde{\sigma }}_{1}<\sigma +d\big (\frac{1}{p}-\frac{1}{r}\big )\le \frac{d}{p}$, and q satisfies

$$\begin{aligned} \Vert \Lambda ^{\sigma } q\Vert _{L^{r}} \lesssim \big (\mathcal {D}_{p,0}+\Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}\big )\langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}-\frac{d}{2}(\frac{1}{p}-\frac{1}{r})} \end{aligned}$$

for all $p\le r\le \infty$ and $\sigma \in \mathbb {R}$ fulfilling $-{\widetilde{\sigma }}_{1}<\sigma +d\big (\frac{1}{p}-\frac{1}{r}\big )\le \frac{d}{p}-1$.

Remark 3

The proof of Corollary 1 is omitted for brevity, and the readers can refer [24] for similar argument. Moreover, the optimal exponents for high frequencies allow the derivative indices of a to be somewhat relaxed, which makes a further improvement in comparison with the results of [1, 2].

Finally, we end the section with the strategies for proving Theorem 2. Precisely, we shall proceed the proof in three steps, according to three terms of the time-weighted functional $\mathcal {D}_{p}(t)$. By the spectral analysis, Hao in [22] pointed out that the Green function for the linearized form of (8) just behaves like a heat equation at low frequency. Consequently, we use the low-frequency time-decay properties of the semi-group defined by the left-hand side of (8), and standard Duhamel principle to bound those nonlinear terms. The purpose of the second and third steps is to track the optimal time-decay exponents for high frequencies with the general assumptions (9)–(10). In the second step, we introduce the auxiliary function

$$\begin{aligned} b=a-\Lambda ^{-1} q \end{aligned}$$

(11)

to present the optimal time-decay estimates of the high frequencies part of the solution. The last step is dedicated to establish gain of regularity and decay altogether for the high frequencies of a, which strongly depends on the parabolic maximal regularity for the heat semi-group (see Proposition 7). In order to highlight the new observation on the decay exponents, we show the proof in detail, which may be of interest for further efforts.

The rest of the paper unfolds as follows. In Sect. 2, we provide a short review of Littlewood–Paley decomposition, Besov spaces and related analysis tools. Section 3 is devoted to the proof of Theorem 2. In the last section, we give a conclusion that describes the highlights of our works.

2 Preliminary

Throughout the paper, $C>0$ stands for a generic “constant.” For brevity, we write $f\lesssim g$ instead of $f\le Cg$. The notation $f\approx g$ means that $f\lesssim g$ and $g\lesssim f$. For any Banach space X and $f,\, g\in X$, we agree that $\Vert (f,g)\Vert _{X}\triangleq \Vert f\Vert _{X}+\Vert g\Vert _{X}$. For all $T>0$ and $\theta \in [1,+\infty ]$, we denote by $L_{T}^{\theta }(X) \triangleq L^{\theta }([0,T];X)$ the set of measurable functions $f:[0,T]\rightarrow X$ such that $t\mapsto \Vert f(t)\Vert _{X}$ are in $L^{\theta }(0,T)$.

To make the paper self-contained, we briefly recall Littlewood–Paley decomposition, Besov spaces and analysis tools. The reader is referred to Chap. 2 and Chap. 3 of [25] for more details. Firstly, let’s introduce the homogeneous Littlewood–Paley decomposition. For that purpose, we fix some smooth radial non increasing function $\chi$ with $\mathrm {Supp}\,\chi \subset B\left( 0,\frac{4}{3}\right)$ and $\chi \equiv 1$ on $B\left( 0,\frac{3}{4}\right)$, then set $\varphi (\xi ) =\chi (\xi /2)-\chi (\xi )$ so that

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\varphi (2^{-k}\cdot ) =1\ \ \hbox {in}\ \ \mathbb {R}^{d}\setminus \{0\} \ \ \hbox {and}\ \ \mathrm {Supp}\,\varphi \subset \big \{\xi \in \mathbb {R}^{d}:3/4\le |\xi |\le 8/3\big \}. \end{aligned}$$

The homogeneous dyadic blocks $\dot{\Delta }_{k}$ are defined by

$$\begin{aligned} \dot{\Delta }_{k}f\triangleq \varphi (2^{-k}D)f=\mathcal {F}^{-1}(\varphi (2^{-k}\cdot )\mathcal {F}f)=2^{kd}h(2^{k}\cdot )*f\ \ \hbox {with}\ \ h\triangleq \mathcal {F}^{-1}\varphi . \end{aligned}$$

Formally, we have the homogeneous decomposition as follows

$$\begin{aligned} f=\sum _{k\in \mathbb {Z}}\dot{\Delta }_{k}f, \end{aligned}$$

(12)

for any tempered distribution $f\in \mathcal {S}'(\mathbb {R}^{d})$. As it holds only for modulo polynomials, it is convenient to consider the subspace of those tempered distributions f such that

$$\begin{aligned} \lim _{k\rightarrow -\infty }\Vert \dot{S}_{k}f\Vert _{L^{\infty } }=0, \end{aligned}$$

(13)

where $\dot{S}_{k}f$ stands for the low-frequency cutoff defined by $\dot{S}_{k}f\triangleq \chi (2^{-k}D)f$. Indeed, if (13) is fulfilled, then (12) holds in $\mathcal {S}'(\mathbb {R}^{d})$. For convenience, we denote by $\mathcal {S}'_{0}(\mathbb {R}^{d})$ the subspace of tempered distributions satisfying (13).

Based on the Littlewood–Paley decomposition, Besov spaces and related analysis tools will come into play in our paper.

Definition 1

For $s\in \mathbb {R}$ and $1\le p,\, r\le \infty ,$ the homogeneous Besov spaces $\dot{B}^{s}_{p,r}$ is defined by

$$\begin{aligned} \dot{B}^{s}_{p,r}\triangleq \big \{f\in \mathcal {S}'_{0}:\Vert f\Vert _{\dot{B}^{s}_{p,r}}<+\infty \big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{s}_{p,r}}\triangleq \Vert (2^{js}\Vert \dot{\Delta }_{k}f\Vert _{L^p})\Vert _{\ell ^{r}(\mathbb {Z})}. \end{aligned}$$

(14)

On the other hand, a class of mixed space-time Besov spaces are also used, which was initiated by J.-Y. Chemin and N. Lerner [26] (see also [27] for the particular case of Sobolev spaces).

Definition 2

For $T>0, \, s\in \mathbb {R}$, $1\le r,\,\theta \le \infty$, the homogeneous Chemin–Lerner space ${\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})$ is defined by

$$\begin{aligned} {\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})\triangleq \big \{f\in L^{\theta }(0,T;\mathcal {S}'_{0}):\Vert f\Vert _{{\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})}<+\infty \big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})}\triangleq \Vert (2^{ks}\Vert \dot{\Delta }_{k} f\Vert _{L^{\theta }_{T}(L^{p})})\Vert _{\ell ^{r}(\mathbb {Z})}. \end{aligned}$$

(15)

For notational simplicity, index T will be omitted if $T=+\infty$. We also use the following functional space:

$$\begin{aligned} \widetilde{\mathcal {C}}_{b}(\mathbb {R_{+}};\dot{B}_{p,r}^{s})\triangleq \big \{f \in \mathcal {C}(\mathbb {R_{+}};\dot{B}_{p,r}^{s}) \ \hbox {s.t} \ \Vert f\Vert _{{\widetilde{L}}^{\infty }(\dot{B}_{p,r}^{s})}<+\infty \big \} . \end{aligned}$$

The above norm (15) may be linked with those of the standard spaces $L_{T}^{\theta } (\dot{B}_{p,r}^{s})$ by means of Minkowski’s inequality.

Remark 4

It holds that

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})}\le \Vert f\Vert _{L^{\theta }_{T}(\dot{B}^{s}_{p,r})}\ \ \hbox {if} \ \ r\ge \theta ; \ \ \ \Vert f\Vert _{{\widetilde{L}}^{\theta }_{T}(\dot{B}^{s}_{p,r})}\ge \Vert f\Vert _{L^{\theta }_{T}(\dot{B}^{s}_{p,r})}\ \ \hbox {if}\ \ r\le \theta . \end{aligned}$$

Restricting the above norms (14) and (15) to the low or high frequencies parts of distributions will be fundamental in our approach. For example, we fix some integer $k_{0}$ (the value of which will follow from the proofs of our main results) and put¹

$$\begin{aligned}&\Vert f\Vert _{\dot{B}_{p,1}^{s}}^{\ell } \triangleq \sum \limits _{k\le k_{0}}2^{ks}\Vert \dot{\Delta }_{k}f\Vert _{L^{p}} \ \hbox {and} \ \Vert f\Vert _{\dot{B}_{p,1}^{s}}^{h}\triangleq \sum \limits _{k\ge k_{0}-1}2^{js}\Vert \dot{\Delta }_{k}f\Vert _{L^{p}}, \end{aligned}$$

(16)

$$\begin{aligned}&\Vert f\Vert _{{\widetilde{L}}_{T}^{\infty } (\dot{B}_{p,1}^{s})}^{\ell } \triangleq \sum \limits _{k\le k_{0}}2^{ks}\Vert \dot{\Delta }_{k}f\Vert _{L_{T}^{\infty } (L^{p})} \ \hbox {and} \nonumber \\&\quad \Vert f\Vert _{{\widetilde{L}}_{T}^{\infty } (\dot{B}_{p,1}^{s})}^{h}\triangleq \sum \limits _{k\ge k_{0}-1}2^{js} \Vert \dot{\Delta }_{k}f\Vert _{L_{T}^{\infty } (L^{p})}. \end{aligned}$$

(17)

We often use the following classical properties (see [24, 25, 28]):

• Scaling invariance: For any $s\in \mathbb {R}$ and $(p,r)\in [1,\infty ]^{2}$, there exists a constant $C=C(s,p,r,d)$ such that for all $\lambda >0$ and $f\in \dot{B}_{p,r}^{s}$, we have

  $$\begin{aligned} C^{-1}\lambda ^{s-\frac{d}{p}}\Vert f\Vert _{\dot{B}_{p,r}^{s}} \le \Vert f(s \,\cdot )\Vert _{\dot{B}_{p,r}^{s}}\le C\lambda ^{s-\frac{d}{p}}\Vert f\Vert _{\dot{B}_{p,r}^{s}}. \end{aligned}$$

• Completeness: $\dot{B}^{s}_{p,r}$ is a Banach space whenever $s<\frac{d}{p}$ or $s\le \frac{d}{p}$ and $r=1$.

• Interpolation: The following inequalities are satisfied for $1\le p,\,r_{1},\,r_{2},\,r\le \infty$, $s_{1}\ne s_{2}$ and $\theta \in (0,1)$:

  $$\begin{aligned} \Vert f\Vert _{\dot{B}_{p,r}^{\theta s_{1}+(1-\theta )s_{2}}}\lesssim \Vert f\Vert _{\dot{B}_{p,r_{1}}^{s_{1}}}^{\theta } \Vert f\Vert _{\dot{B}_{p,r_2}^{s_{2}}}^{1-\theta } \ \ \ \hbox {with} \ \ \ \frac{1}{r}=\frac{\theta }{r_{1}}+\frac{1-\theta }{r_{2}}. \end{aligned}$$

• Action of Fourier multipliers: If F is a smooth homogeneous of degree m function on $\mathbb {R}^{d}\backslash \{0\}$ and F(D) maps $\mathcal {S}'_{0}$ to itself, then

  $$\begin{aligned} F(D):\dot{B}_{p,r}^{s}\rightarrow \dot{B}_{p,r}^{s-m}. \end{aligned}$$

  In particular, the gradient operator maps $\dot{B}_{p,r}^{s}$ in $\dot{B}_{p,r}^{s-1}$.

The embedding properties will be used several times throughout the paper.

Proposition 3

([24, 25, 28]) (Embedding for Besov spaces on $\mathbb {R}^{d}$)

• For any $p\in [1,\infty ]$, we have the continuous embedding $\dot{B}^{0}_{p,1}\hookrightarrow L^{p}\hookrightarrow \dot{B}^{0}_{p,\infty }.$

• If $s\in \mathbb {R}$, $1\le p_{1}\le p_{2}\le \infty$ and $1\le r_{1}\le r_{2}\le \infty ,$ then $\dot{B}^{s}_{p_1,r_1}\hookrightarrow \dot{B}^{s-d \,(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}$.

• The space $\dot{B}^{\frac{d}{p}}_{p,1}$ is continuously embedded in the set of bounded continuous functions (going to zero at infinity if, additionally, $p<\infty$).

The following product estimates in Besov spaces play a fundamental role in our analysis of the nonlinear terms.

Proposition 4

([24, 25, 28, 29]) Let $s>0$ and $1\le p,\, r\le \infty$. Then, $\dot{B}^{s}_{p,r}\cap L^{\infty }$ is an algebra and

$$\begin{aligned} \Vert fg\Vert _{\dot{B}^{s}_{p,r}}\lesssim \Vert f\Vert _{L^{\infty }}\Vert g\Vert _{\dot{B}^{s}_{p,r}}+\Vert f\Vert _{L^{\infty }}\Vert g\Vert _{\dot{B}^{s}_{p,r}}. \end{aligned}$$

Let the real numbers $s_{1},$ $s_{2}$, $p_1$ and $p_2$ be such that

$$\begin{aligned} s_{1}+s_{2}>0,\quad s_{1}\le \frac{d}{p_{1}}, \quad s_{2}\le \frac{d}{p_{2}},\quad s_{1}\ge s_{2},\quad \frac{1}{p_{1}}+\frac{1}{p_{2}}\le 1. \end{aligned}$$

Then, we have

$$\begin{aligned} \Vert fg\Vert _{\dot{B}^{s_{2}}_{q,1}}\lesssim \Vert f\Vert _{\dot{B}^{s_{1}}_{p_{1},1}}\Vert g\Vert _{\dot{B}^{s_{2}}_{p_{2},1}}\quad \hbox {with}\quad \frac{1}{q}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{s_{1}}{d}. \end{aligned}$$

Additionally, for exponents $s>0$ and $1\le p_{1},\,p_{2},\,q\le \infty$ satisfying

$$\begin{aligned} \frac{d}{p_{1}}+\frac{d}{p_{2}}-d\le s \le \min \left( \frac{d}{p_{1}},\frac{d}{p_{2}}\right) \quad \hbox {and}\quad \frac{1}{q}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{s}{d}, \end{aligned}$$

we have

$$\begin{aligned} \Vert fg\Vert _{\dot{B}^{-s}_{q,\infty }}\lesssim \Vert f\Vert _{\dot{B}^{s}_{p_{1},1}}\Vert g\Vert _{\dot{B}^{-s}_{p_{2},\infty }}. \end{aligned}$$

To handle the case of $p>d$ in the proof of Theorem 2, just resorting to Proposition 4 does not allow to get suitable bounds for the low-frequency part of some nonlinear terms, so we need to utilize the following non-classical product estimates.

Proposition 5

([24, 28, 29]) Let $k_{0}\in \mathbb {Z},$ and denote $z^{\ell }\triangleq \dot{S}_{k_{0}}z$, $z^{h}\triangleq z-z^{\ell }$ and, for any $s\in \mathbb {R}$,

$$\begin{aligned} \Vert z\Vert _{\dot{B}^{s}_{2,\infty }}^{\ell }\triangleq \sup _{k\le k_{0}}2^{ks} \Vert \dot{\Delta }_{k} z\Vert _{L^2}. \end{aligned}$$

There exists a universal integer $N_{0}$ such that for any $2\le p\le 4$ and $s>0,$ we have

$$\begin{aligned}&\Vert f g^{h}\Vert _{\dot{B}^{-\delta _{0}}_{2,\infty }}^{\ell }\le C \big (\Vert f\Vert _{\dot{B}^{s}_{p,1}}+\Vert \dot{S}_{k_{0}+N_{0}}f\Vert _{L^{{p}^{*}}}\big )\Vert g^{h}\Vert _{\dot{B}^{-s}_{p,\infty }}, \end{aligned}$$

(18)

$$\begin{aligned}&\Vert f^{h} g\Vert _{\dot{B}^{-\delta _{0}}_{2,\infty }}^{\ell } \le C \big (\Vert f^{h}\Vert _{\dot{B}^{s}_{p,1}}+\Vert \dot{S}_{k_{0}+N_{0}}f^{h}\Vert _{L^{p^{*}}}\big )\Vert g\Vert _{\dot{B}^{-s}_{p,\infty }} \end{aligned}$$

(19)

with $\delta _{0}\triangleq \frac{2d}{p}-\frac{d}{2}$ and $\frac{1}{p^{*}}\triangleq \frac{1}{2}-\frac{1}{p},$ and C depending only on $k_{0}$, d and s.

Let us here recall the following classical Bernstein inequality:

$$\begin{aligned} \Vert D^{k}f\Vert _{L^{b}} \le C^{1+k} \lambda ^{k+d(\frac{1}{a}-\frac{1}{b})}\Vert f\Vert _{L^{a}} \end{aligned}$$

(20)

that holds for all function f such that $\mathrm {Supp}\,\mathcal {F}f\subset \left\{ \xi \in \mathbb {R}^{d}: |\xi |\le R\lambda \right\}$ for some $R>0$ and $\lambda >0$, if $k\in \mathbb {N}$ and $1\le a\le b\le \infty$.

More generally, if we assume f to satisfy $\mathrm {Supp}\,\mathcal {F}f\subset \big \{\xi \in \mathbb {R}^{d}:R_{1}\lambda \le$ $|\xi |\le R_{2}\lambda \big \}$ for some $0<R_{1}<R_{2}$ and $\lambda >0$, then for any smooth homogeneous of degree m function A on $\mathbb {R}^d\setminus \{0\}$ and $1\le a\le \infty ,$ we have (see, e.g., Lemma 2.2 in [25]):

$$\begin{aligned} \left\| A(D)f\right\| _{L^{a}}\lesssim \lambda ^{m}\Vert f\Vert _{L^{a}}. \end{aligned}$$

(21)

An obvious consequence of (20) and (21) is that $\Vert D^{k}f\Vert _{\dot{B}^{s}_{p, r}}\thickapprox \Vert f\Vert _{\dot{B}^{s+k}_{p, r}}$ for all $k\in \mathbb {N}$.

We also need the following nonlinear generalization of (21):

Proposition 6

([29]) If $\mathrm {Supp}\,\mathcal {F}f\subset \big \{\xi \in \mathbb {R}^{d}:R_{1}\lambda \le |\xi |\le R_{2}\lambda \big \}$, then there exists c depending only on d, $R_{1}$ and $R_{2}$ so that for all $1<p<\infty$,

$$\begin{aligned} c\lambda ^{2}\left( \frac{p-1}{p}\right) \int _{\mathbb {R}^{d}}|f|^{p}dx\le (p-1)\int _{\mathbb {R}^{d}}|\nabla f|^{2}|f|^{p-2}dx =-\int _{\mathbb {R}^{d}} \Delta f|f|^{p-2}f dx. \end{aligned}$$

Let us finally recall the following parabolic regularity estimate for the heat equation to end this section.

Proposition 7

([24, 25, 28, 29]) Let $s\in \mathbb {R}$, $(p,r)\in \left[ 1,\infty \right] ^{2}$ and $1\le \rho _{2}\le \rho _{1}\le \infty$. Let u satisfy

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _{t}u-\mu \Delta u=f,\\ u|_{t=0}=u_{0}. \end{array} \right. \end{aligned}$$

Then, for all $T>0$, the following a priori estimate is fulfilled:

$$\begin{aligned} \mu ^{\frac{1}{\rho _1}}\Vert u\Vert _{{{\widetilde{L}}}_{T}^{\rho _1}(\dot{B}^{s+\frac{2}{\rho _1}}_{p,r})}\lesssim \Vert u_{0}\Vert _{\dot{B}^{s}_{p,r}}+\mu ^{\frac{1}{\rho _{2}}-1}\Vert f\Vert _{{{\widetilde{L}}}^{\rho _{2}}_{T}(\dot{B}^{s-2+\frac{2}{\rho _{2}}}_{p,r})}. \end{aligned}$$

3 The Proof of Theorem 2

We here prove Theorem 2 taking for granted the global-in-time existence result of Theorem 1. To do this, we give the following proposition.

Proposition 8

Let $d\ge 2$ and p and $\sigma _{1}$ fulfill (6) and (9), respectively. Moreover, the initial data satisfy the assumption of Theorem 1 and (10). Let (a, q) be the global to system (8) and define

$$\begin{aligned}&\mathcal {D}_{p}(t) \triangleq \sup _{\sigma \in \big [\varepsilon -\sigma _{1},\frac{d}{2}+1\big ]}\Big \Vert \langle \tau \rangle ^{\frac{\sigma _{1}+\sigma }{2}}(a,q)\Big \Vert _{L^{\infty }_{t} (\dot{B}_{2,1}^{\sigma })}^{\ell }+\left\| \langle \tau \rangle ^{\alpha } (a,\nabla q)\right\| _{{\widetilde{L}}^{\infty }_{t}(\dot{B}_{p,1}^{\frac{d}{p}-2})}^{h}\nonumber \\&\quad +\left\| \tau ^{\alpha } \nabla a\right\| ^{h} _{{\widetilde{L}}^{\infty }_{t}(\dot{B}_{p,1}^{\frac{d}{p}-1})} \end{aligned}$$

(22)

with $\langle t\rangle =\sqrt{1+t^{2}}$ and $\alpha \triangleq \sigma _{1}+\frac{d}{2}+\frac{1}{2}-\varepsilon$ for sufficiently small $\varepsilon >0$, then we have

$$\begin{aligned} \mathcal {D}_{p}(t)\lesssim \mathcal {D}_{p,0}+\Vert (a_{0},\nabla q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}} \ \ \hbox {for all} \ t\ge 0. \end{aligned}$$

(23)

Remark 5

If replacing (10) by the slightly stronger assumption

$$\begin{aligned} \Vert (a_{0},q_{0})\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,1}}\le c, \end{aligned}$$

then we can take $\varepsilon =0$ in both $\alpha$ and $\mathcal {D}_{p}(t)$.

Next, let us prove Proposition 8, based on which we further establish the time-decay estimates of strong solutions. The proof is divided into three steps, due to the three terms of the time-weighted functional $\mathcal {D}_{p}(t)$. In what follows, we shall use repeatedly elementary inequality for $0\le s_{1}\le s_{2}$ with $s_{2}>1$:

$$\begin{aligned} \int _{0}^{t}\langle t-\tau \rangle ^{-s_{1}}\langle \tau \rangle ^{-s_{2}}d\tau \lesssim \langle t\rangle ^{-s_{1}}. \end{aligned}$$

(24)

3.1 First Step: Bounds for the Low Frequencies

Let $\left( E(t)\right) _{t\ge 0}$ be the semi-group associated with the left-hand side of (8). It follows from the standard Duhamel formula that

$$\begin{aligned} \left( \begin{array}{c} a\\ q \end{array} \right) =E(t) \left( \begin{array}{c} a_{0}\\ q_{0}\\ \end{array} \right) +\int _{0}^{t}E\left( t-\tau \right) \left( \begin{array}{c} f(\tau )\\ 0 \end{array} \right) d\tau . \end{aligned}$$

(25)

Let us state smoothing estimate of the low frequencies of the linearized solution $\left( a_{L},q_{L}\right) \triangleq E(t)\left( a_{0},q_{0}\right)$, which behaves like that of heat kernel.

Lemma 1

Let $\left( a_{L},q_{L}\right)$ be the solution to the linear system

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}a_{L} -\Delta a_{L}-\Lambda q_{L}=0, \\ \partial _{t}q_{L}+\Lambda a_{L}=0 \end{array} \right. \end{aligned}$$

with the initial data

$$\begin{aligned} (a_{L},q_{L})|_{t=0}=(a_{0},q_{0}). \end{aligned}$$

Then, for any $k_{0}\in \mathbb {Z}$, there exists a positive constant $c_{0}=c_{0}\left( k_{0}\right)$ such that

$$\begin{aligned} \Vert (a_{L,k},q_{L,k})(t)\Vert _{L^{2}}\lesssim e^{-c_{0}2^{2k}t} \Vert (a_{0,k},q_{0,k})\Vert _{L^{2}} \end{aligned}$$

for $t\ge 0$ and $k\le k_{0}$, where we set $z_{k}=\dot{\Delta }_{k}z$ for any $z\in \mathcal {S}'(\mathbb {R}^{d})$.

The proof of Lemma 1 follows from essentially the similar argument to the related works of [24, 28], and is omitted for brevity. Based on the key Lemma 1, we can perform the similar procedure as in [29] to get for $\sigma _{1}+\sigma >0$,

$$\begin{aligned} \sup _{t\ge 0}\left\langle t\right\rangle ^{\frac{\sigma _{1}+\sigma }{2}}\Vert E(t)U_{0}\Vert ^{\ell }_{\dot{B}^{\sigma }_{2,1}}\lesssim \Vert U_{0}\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}, \end{aligned}$$

(26)

where we denote $\langle t\rangle \triangleq \sqrt{1+t^{2}}$ and $U_{0}\triangleq (a_{0},q_{0})$. Furthermore, it follows from Duhamel’s formula that

$$\begin{aligned} \Big \Vert \int _{0}^{t}E(t-\tau )f(\tau )d\tau \Big \Vert ^{\ell }_{\dot{B}^{s}_{2,1}}\lesssim \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert f(\tau )\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau . \end{aligned}$$

We claim that if p satisfies (6), then we get for all $t\ge 0$,

$$\begin{aligned} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert f(\tau )\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau \lesssim \langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\left( \mathcal {D}^{2}_{p}(t)+\mathcal {E}^{2}_{p}(t)\right) , \end{aligned}$$

(27)

where $\mathcal {E}_{p}(t)$ and $\mathcal {D}_{p}(t)$ have been defined by (7) and (22), respectively.

In order to prove (27), we decompose f as follows:

$$\begin{aligned}&f=\mathrm {div}\,(a\nabla \Lambda ^{-1} q)=\mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{\ell })\nonumber \\&\quad +\mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{h} )+\mathrm {div}\,(a^{h}\nabla \Lambda ^{-1}q). \end{aligned}$$

(28)

To bound the first second terms, we present a non-classical product inequality:

$$\begin{aligned}&\Vert FG\Vert _{\dot{B}^{\frac{d}{p}-\frac{d}{2}-\delta _{1}}_{2,\infty }} \lesssim \Vert F\Vert _{\dot{B}^{\frac{d}{p}-\frac{d}{2}-\delta _{1}}_{p,1}}\Vert G\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}, \end{aligned}$$

(29)

$$\begin{aligned}&\Vert FG\Vert _{\dot{B}^{-\delta _{1}}_{2,\infty }} \lesssim \Vert F\Vert _{\dot{B}^{-\delta _{1}}_{2,1}} \Vert G\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}} \end{aligned}$$

(30)

for all $-\frac{d}{2}< \delta _{1}\le \delta _{0}=\frac{2d}{p}-\frac{d}{2}$.

Indeed, if $\delta _{1}=\delta _{0}=\frac{2d}{p}-\frac{d}{2}$, then $\frac{d}{p}-\frac{d}{2}-\delta _{1}=-\frac{d}{p}$. It follows from the last term of Proposition 4 with $s=\frac{d}{p}$, $p_{1}=p$ and $q=p_{2}=2$ that

$$\begin{aligned} \Vert FG\Vert _{\dot{B}^{-\frac{d}{p}}_{2,\infty }} \lesssim \Vert F\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}}\Vert G\Vert _{\dot{B}^{\frac{d}{p}}_{2,\infty }}, \end{aligned}$$

namely

$$\begin{aligned} \Vert FG\Vert _{\dot{B}^{\frac{d}{p}-\frac{d}{2}-\delta _{1}}_{2,\infty }} \lesssim \Vert F\Vert _{\dot{B}^{\frac{d}{p}-\frac{d}{2}-\delta _{1}}_{p,1}}\Vert G\Vert _{\dot{B}^{\frac{d}{p}}_{2,\infty }}. \end{aligned}$$

Noticing that the embedding $\dot{B}^{\frac{d}{p}}_{2,1}\hookrightarrow \dot{B}^{\frac{d}{p}}_{2,\infty }$, we can get (29). If $-\frac{d}{2}<\delta _{1}<\delta _{0}$, then (29) follows from the second term of Proposition 4 with $\frac{d}{p}\triangleq s_{1}\ge s_{2}=\frac{d}{p}-\frac{d}{2}-\delta _{1}, \ \ q=p_{2}=2 \ \hbox {and} \ \ p_{1}=p$ satisfying $s_{1}+s_{2}>0$. As for (30), if $p=2$, then it coincides with (29). If $2<p\le d^{*}=\frac{2d}{d-2}$, then (30) follows from the second term of Proposition 4 with $\frac{d}{2}\triangleq s_{1}\ge s_{2}\triangleq -\delta _{1}$, $q=p_{1}=p_{2}=2$ satisfying $s_{1}+s_{2}>0$.

In addition, we shall utilize repeatedly that, according to the embedding, the definition of $\mathcal {D}_{p}(t)$ and the fact that $-\sigma _{1}<1-\sigma _{1}<\frac{d}{2}<\frac{d}{2}+1$,

$$\begin{aligned} \Vert (a,q)^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}\lesssim \langle \tau \rangle ^{-\frac{\sigma _{1}}{2}-\frac{d}{4}}\mathcal {D}_{p}(\tau ), \ \ \ \ \ \Vert (a,q)^{\ell } \Vert _{\dot{B}^{1-\sigma _{1}}_{2,1}}\lesssim \langle \tau \rangle ^{-\frac{1}{2}}\mathcal {D}_{p}(\tau ) \end{aligned}$$

(31)

and also that, due to $-\sigma _{1}<\frac{d}{2}-1\le \frac{d}{p}<\frac{d}{2}+1$ (note that $p\le \frac{2d}{d-2}$) and $\frac{d}{p}-\frac{d}{2}-\sigma _{1}<\frac{d}{p}-1$,

$$\begin{aligned} \Vert a^{\ell }\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}\lesssim \langle \tau \rangle ^{-\frac{\sigma _{1}}{2}-\frac{d}{2p}}\mathcal {D}_{p}(\tau ), \ \ \ \ \ \Vert q^{h}\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}\lesssim \langle \tau \rangle ^{-\alpha }\mathcal {D}_{p}(\tau ). \end{aligned}$$

(32)

To handle the term with $\mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{\ell })$, we write that, owing to Bernstein inequality and (30),

$$\begin{aligned} \Vert \mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{\ell })\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}\lesssim & {} \Vert a^{\ell } \nabla \Lambda ^{-1} q^{\ell }\Vert ^{\ell }_{\dot{B}^{1-\sigma _{1}}_{2,\infty }}\lesssim \Vert a^{\ell } \nabla \Lambda ^{-1} q^{\ell }\Vert _{\dot{B}^{1-\sigma _{1}}_{2,\infty }}\\\lesssim & {} \Vert a^{\ell }\Vert _{\dot{B}^{1-\sigma _{1}}_{2,1}}\Vert \nabla \Lambda ^{-1} q^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}, \end{aligned}$$

where we used that the relation $1-\frac{d}{2}<\sigma _{1}\le \sigma _{0}\triangleq \frac{2d}{p}-\frac{d}{2}+1$ implies that $-\frac{d}{2}<\sigma _{1}-1\le \sigma _{0}-1=\frac{2d}{p}-\frac{d}{2}=\delta _{0}$. As we know, $\nabla \Lambda ^{-1}$ is a homogeneous Fourier multiplier of degree 0, and then we have $\Vert \nabla \Lambda ^{-1} q^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}\lesssim \Vert q^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}$. Consequently, it follows from (31) that

$$\begin{aligned}&\int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\Vert \mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{\ell })\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert a^{\ell }\Vert _{\dot{B}^{1-\sigma _{1}}_{2,1}}\Vert \nabla \Lambda ^{-1} q^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert a^{\ell }\Vert _{\dot{B}^{1-\sigma _{1}}_{2,1}}\Vert q^{\ell }\Vert _{\dot{B}^{\frac{d}{2}}_{2,1}}d\tau \\\lesssim & {} \mathcal {D}^{2}_{p}(t)\int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\langle \tau \rangle ^{-\frac{\sigma _{1}}{2}-\frac{d}{4}-\frac{1}{2}} d\tau . \end{aligned}$$

As $\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}>1$ and $\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}\ge \frac{\sigma _{1}+\sigma }{2}$ for $\sigma _{1}$ fulfilling (9) and $\sigma \le \frac{d}{2}+1$, inequality (24) ensures that

$$\begin{aligned} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert \mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{\ell })\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau \lesssim \langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\mathcal {D}^{2}_{p}(t). \end{aligned}$$

It follows from (29) with $\delta _{1}=\delta _{0}=\frac{2d}{p}-\frac{d}{2}$ that

$$\begin{aligned} \Vert FG\Vert _{\dot{B}^{-\frac{d}{p}}_{2,\infty }} \lesssim \Vert F\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}} \Vert G\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}. \end{aligned}$$

(33)

Next, let us look at the term with $\mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{h})$. Thanks to $-\frac{d}{p}\le \frac{d}{2}-\frac{2d}{p}=1-\sigma _{0}\le 1-\sigma _{1}<\frac{d}{2}$ for all $p\ge 2$, we use the embedding $\dot{B}^{-\frac{d}{p}}_{2,\infty }\hookrightarrow \dot{B}^{1-\sigma _{1}}_{2,\infty }$ for the low frequencies together with Bernstein inequality and (33), and then get

$$\begin{aligned} \Vert \mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{h})\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}\lesssim & {} \Vert a^{\ell } \nabla \Lambda ^{-1} q^{h}\Vert ^{\ell }_{\dot{B}^{1-\sigma _{1}}_{2,\infty }}\lesssim \Vert a^{\ell } \nabla \Lambda ^{-1} q^{h}\Vert ^{\ell }_{\dot{B}^{-\frac{d}{p}}_{2,\infty }}\\\lesssim & {} \Vert a^{\ell } \nabla \Lambda ^{-1} q^{h}\Vert _{\dot{B}^{-\frac{d}{p}}_{2,\infty }}\lesssim \Vert \nabla \Lambda ^{-1}q^{h}\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}}\Vert a^{\ell }\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}\\\lesssim & {} \Vert q^{h}\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}}\Vert a^{\ell }\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}, \end{aligned}$$

where we noticed that $\nabla \Lambda ^{-1}$ is a homogeneous Fourier multiplier of degree 0. As the relation $-\frac{d}{p}<\frac{d}{p}-1$ ensures the embedding $\dot{B}^{\frac{d}{p}-1}_{p,1}\hookrightarrow \dot{B}^{-\frac{d}{p}}_{p,1}$ for the high frequencies, we have $\Vert q^{h}\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}}\lesssim \Vert q^{h}\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}$. Hence, we conclude that

$$\begin{aligned}&\int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\Vert \mathrm {div}\,(a^{\ell } \nabla \Lambda ^{-1} q^{h})\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert q^{h}\Vert _{\dot{B}^{-\frac{d}{p}}_{p,1}}\Vert a^{\ell }\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert q^{h}\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}\Vert a^{\ell }\Vert _{\dot{B}^{\frac{d}{p}}_{2,1}}d\tau . \end{aligned}$$

Using the relations $\frac{\sigma _{1}}{2}+\frac{d}{2p}+\alpha>\frac{\sigma _{1}}{2}+\frac{d}{2p}+1\ge \frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}>1$ ($\alpha >1$ and $p\le \frac{2d}{d-2}$) and $\frac{\sigma _{1}}{2}+\frac{d}{2p}+\alpha \ge \frac{\sigma _{1}+\sigma }{2}$ for all $\sigma \le \frac{d}{2}+1$ together with (32) and (24), we end up with

$$\begin{aligned} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert \mathrm {div}\,(a^{\ell }q^{h})\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau\lesssim & {} \mathcal {D}^{2}_{p}(t)\int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\langle \tau \rangle ^{-\frac{\sigma _{1}}{2}-\frac{d}{2p}-\alpha } d\tau \\\lesssim & {} \langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\mathcal {D}^{2}_{p}(t). \end{aligned}$$

Let us finally bound the term with $\mathrm {div}\,(a^{h}\nabla \Lambda ^{-1}q)$. We can develop the following inequality

$$\begin{aligned} \Vert F^{h}G\Vert ^{\ell }_{\dot{B}^{-\delta _{0}}_{2,\infty }} \lesssim \Vert F^{h}\Vert _{\dot{B}^{\frac{d}{p}}_{p,1}} \Vert G\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}} \ \ \hbox {with} \ \ \delta _{0}=\frac{2d}{p}-\frac{d}{2}. \end{aligned}$$

(34)

Indeed, in the case $2\le p\le d$, the interested reader is referred to [28] for the proof of (34). Here, let us only present the proof of (34) in the case $p>d$. Applying (19) with $s=1-\frac{d}{p}>0$ yields

$$\begin{aligned} \Vert F^{h}G\Vert ^{\ell }_{\dot{B}^{-\delta _{0}}_{2,\infty }} \lesssim \big (\Vert F^{h}\Vert _{\dot{B}^{1-\frac{d}{p}}_{p,1}} +\sum _{k=k_{0}}^{k_{0}+N_{0}-1}\Vert \dot{\Delta }_{k}F^{h}\Vert _{L^{p^{*}}}\big )\Vert G\Vert _{\dot{B}_{p,1}^{\frac{d}{p}-1}}. \end{aligned}$$

As $p^{*}\ge p$, we get from Bernstein inequality that

$$\begin{aligned} \Vert \dot{\Delta }_{k}F^{h}\Vert _{L^{p^{*}}}\lesssim \Vert \dot{\Delta }_{k}F^{h}\Vert _{L^{p}} \ \ \hbox {for} \ \ k_{0}\le k<k_{0}+N_{0}. \end{aligned}$$

Hence, taking advantage of the relation $1-\frac{d}{p}\le \frac{d}{p}$, we have for $p>d$,

$$\begin{aligned} \Vert F^{h}G\Vert ^{\ell }_{\dot{B}^{-\delta _{0}}_{2,\infty }} \lesssim \Vert F^{h}\Vert _{\dot{B}_{p,1}^{1-\frac{d}{p}}}\Vert G\Vert _{\dot{B}_{p,1}^{\frac{d}{p}-1}} \lesssim \Vert F^{h}\Vert _{\dot{B}_{p,1}^{\frac{d}{p}}}\Vert G\Vert _{\dot{B}_{p,1}^{\frac{d}{p}-1}}. \end{aligned}$$

Note that $-\delta _{0}=1-\sigma _{0}\le 1-\sigma _{1}$, then we take advantage of (34) and get

$$\begin{aligned}&\int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert \mathrm {div}\,(a^{h}\nabla \Lambda ^{-1}q)\Vert ^{\ell }_{\dot{B}^{-\sigma _{1}}_{2,\infty }}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert a^{h}\nabla \Lambda ^{-1}q\Vert ^{\ell }_{\dot{B}^{1-\sigma _{1}}_{2,\infty }}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert a^{h}\nabla \Lambda ^{-1}q\Vert ^{\ell }_{\dot{B}^{1-\sigma _{0}}_{2,\infty }}d\tau \\\lesssim & {} \int _{0}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}} \Vert a^{h}\Vert _{\dot{B}^{\frac{d}{p}}_{p,1}}\Vert q\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}d\tau \\= & {} \left( \int _{0}^{1}+\int _{1}^{t}\right) (\cdots )d\tau \triangleq I_{1}+I_{2}. \end{aligned}$$

It follows from the definitions of $\mathcal {E}_{p}(t)$ that $I_{1}\lesssim \langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\mathcal {E}^{2}_{p}(1)$. We note that, thanks to the relations $-\sigma _{1}<\frac{d}{2}-1<\frac{d}{2}+1$ and $\alpha \ge \frac{\sigma _{1}}{2}+\frac{d}{4}-\frac{1}{2}$ for small enough $\varepsilon >0$,

$$\begin{aligned} \Vert q\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}\lesssim \Vert q\Vert ^{\ell }_{\dot{B}^{\frac{d}{2}-1}_{2,1}}+\Vert q\Vert ^{h}_{\dot{B}^{\frac{d}{p}-1}_{p,1}}\lesssim \langle \tau \rangle ^{-\frac{\sigma _{1}}{2}-\frac{d}{4}+\frac{1}{2}}\mathcal {D}_{p}(\tau ) \end{aligned}$$

together with (24), we thus get, if $t\ge 1$,

$$\begin{aligned} I_{2}\lesssim & {} \big (\sup _{\tau \in [1,t]}\tau ^{\alpha }\Vert a\Vert ^{h}_{\dot{B}^{\frac{d}{p}}_{p,1}}\big ) \big (\sup _{\tau \in [1,t]}\langle \tau \rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}-\frac{1}{2}}\Vert q\Vert _{\dot{B}^{\frac{d}{p}-1}_{p,1}}\big )\\&\times \int _{1}^{t}\langle t-\tau \rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\langle \tau \rangle ^{-\alpha -\frac{\sigma _{1}}{2}-\frac{d}{4}+\frac{1}{2}}d\tau \lesssim \langle t\rangle ^{-\frac{\sigma _{1}+\sigma }{2}}\mathcal {D}^{2}_{p}(t), \end{aligned}$$

since the relations $\sigma _{1}>1-\frac{d}{2}$ and $\alpha >1$ lead to $\alpha +\frac{\sigma _{1}}{2}+\frac{d}{4}-\frac{1}{2}>1$ and $\alpha +\frac{\sigma _{1}}{2}+\frac{d}{4}-\frac{1}{2}\ge \frac{\sigma _{1}+\sigma }{2}$ for all $\sigma \le \frac{d}{2}+1$.

Putting together all the above estimates lead to the (27). Then, combining with (26) for bounding the term of (25) pertaining to the data, we end up with

$$\begin{aligned} \langle t\rangle ^{\frac{\sigma _{1}+\sigma }{2}}\left\| (a,q)(t)\right\| ^{\ell }_{\dot{B}^{\sigma }_{2,1}} \lesssim \mathcal {D}_{p,0}+\mathcal {D}^{2}_{p}(t)+\mathcal {E}^{2}_{p}(t) \ \ \hbox {for all} \ \ t\ge 0, \end{aligned}$$

(35)

provided that $-\sigma _{1}<\sigma \le \frac{d}{2}+1$.

3.2 Decay Estimates for the High Frequencies of $(a,\Lambda q)$

In this section, we shall utilize the $L^{p}$ energy method in terms of the auxiliary function b defined in (11), and handle the second term of functional $\mathcal {D}_{p}(t)$. First of all, let us recall the following key lemma (see [29] for more details).

Lemma 2

Let $X:[0,T]\rightarrow \mathbb {R}_{+}$ be a continuous function such that $X^{p}$ is differentiable for some $p\ge 1$ and fulfills

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}X^{p}+QX^{p}\le KX^{p-1} \end{aligned}$$

for some constant $Q\ge 0$ and measurable function $K:[0,T]\rightarrow \mathbb {R}_{+}$.

Define $X_{\varepsilon }=\left( X^{p}+\varepsilon ^{p}\right) ^{\frac{1}{p}}$ for $\varepsilon >0$. Then, it holds that

$$\begin{aligned} \frac{d}{dt}X_{\varepsilon }+Q X_{\varepsilon }\le K+Q\varepsilon . \end{aligned}$$

In terms of the useful auxiliary function (11), it follows from (8) that

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}b-\Delta b=f+b+\Lambda ^{-1}q, \\ \partial _{t}\Lambda q+\Lambda q=-\Lambda ^{2} b \end{array} \right. \end{aligned}$$

(36)

with $f=-\mathrm {div}\left( a \nabla \Lambda ^{-1} q\right)$.

Let us observe that applying $\dot{\Delta }_{k}$ to the first equation of (36) gives for all $k\in \mathbb {Z}$,

$$\begin{aligned} \partial _{t}b_{k}-\Delta b_{k}=f_{k}+b_{k}+\Lambda ^{-1}q_{k} \ \hbox {with} \ b_{k}\triangleq \dot{\Delta }_{k}b, \ q_{k}\triangleq \dot{\Delta }_{k}q \ \hbox {and} \ f_{k}\triangleq \dot{\Delta }_{k}f. \end{aligned}$$

(37)

Multiplying both sides of (37) by $|b_{k}|^{p-2}b_{k}$ and integrating over $\mathbb {R}^{d}$ yields,

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\Vert b_{k}\Vert ^{p}_{L^{p}} -\int \Delta b_{k}|b_{k}|^{p-2} b_{k} dx =\int \left( f_{k}+b_{k}+\Lambda ^{-1} q_{k}\right) |b_{k}|^{p-2}b_{k}dx. \end{aligned}$$

The key observation is that the second term of the l.h.s., although not spectrally localized, may be handled from Proposition 6. Therefore, we end up (for some constant $c_{p}$ depending only p) with

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\Vert b_{k}\Vert ^{p}_{L^{p}}+c_{p} 2^{2k}\Vert b_{k}\Vert ^{p}_{L^{p}} \le \left( \Vert f_{k}\Vert _{L^{p}}+\Vert b_{k}\Vert _{L^{p}}+\Vert \Lambda ^{-1}q_{k}\Vert _{L^{p}}\right) \Vert b_{k}\Vert ^{p-1}_{L^{p}}. \end{aligned}$$

Furthermore, using the notation $\Vert \cdot \Vert _{\varepsilon ,L^{p}}\triangleq (\Vert \cdot \Vert ^{p}_{\varepsilon ,L^{p}}+\varepsilon ^{p})^{\frac{1}{p}}$, it follows from Lemma 2 that for all $\varepsilon >0$,

$$\begin{aligned} \frac{d}{dt}\Vert b_{k}\Vert _{\varepsilon ,L^{p}}+c_{p}2^{2k}\Vert b_{k}\Vert _{\varepsilon ,L^{p}} \le \Vert f_{k}\Vert _{L^{p}}+\Vert b_{k}\Vert _{L^{p}}+\Vert \Lambda ^{-1}q_{k}\Vert _{L^{p}}+c_{p}2^{2k}\varepsilon . \end{aligned}$$

(38)

Next, applying the operator $\dot{\Delta }_{k}$ to the second equation of (36) gives

$$\begin{aligned} \partial _{t}\Lambda q_{k}+\Lambda q_{k}=-\Lambda ^{2} b_{k}. \end{aligned}$$

Multiplying by $|\Lambda q_{k}|^{p-2}\Lambda q_{k}$ and integrating on $\mathbb {R}^{d}$, we arrive at

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\Vert \Lambda q_{k}\Vert ^{p}_{L^{p}}+\Vert \Lambda q_{k}\Vert ^{p}_{L^{p}}=\int -\Lambda ^{2} b_{k} |\Lambda q_{k}|^{p-2}\Lambda q_{k} dx. \end{aligned}$$

Taking advantage of Hölder and Bernstein inequalities leads to

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\Vert \Lambda q_{k}\Vert ^{p}_{L^{p}}+\Vert \Lambda q_{k}\Vert ^{p}_{L^{p}}\le C2^{2k}\Vert b_{k}\Vert _{L^{p}}\Vert \Lambda q_{k}\Vert ^{p-1}_{L^{p}}. \end{aligned}$$

Once again using Lemma 2 yields

$$\begin{aligned} \frac{d}{dt}\Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}}+\Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}} \le C 2^{2k}\Vert b_{k}\Vert _{L^{p}}+\varepsilon . \end{aligned}$$

(39)

Adding up (38) to (39) (multiplied by $\gamma c_{p}$ for some $\gamma >0$) yields

$$\begin{aligned}&\frac{d}{dt}\left( \Vert b_{k}\Vert _{\varepsilon ,L^{p}}+\gamma c_{p} \Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}}\right) +c_{p}2^{2k}\Vert b_{k}\Vert _{\varepsilon ,L^{p}}+\gamma c_{p}\Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}} \nonumber \\\le & {} \Vert f_{k}\Vert _{L^{p}}+\Vert b_{k}\Vert _{L^{p}}+\Vert \Lambda ^{-1} \,q_{k}\Vert _{L^{p}} +C\gamma c_{p}2^{2k}\Vert b_{k}\Vert _{L^{p}}+(c_{p}2^{2k}+\gamma c_{p})\varepsilon . \end{aligned}$$

Now, as $\Lambda ^{-1}$ is a homogeneous Fourier multiplier of degree $-1$, we get

$$\begin{aligned} \Vert \Lambda ^{-1} \,q_{k}\Vert _{L^{p}}\lesssim 2^{-2k}\Vert \Lambda q_{k}\Vert _{L^{p}} \lesssim 2^{-2k_{0}}\Vert \Lambda q_{k}\Vert _{L^{p}}\ \ \hbox {for all} \ \ k\ge k_{0}-1. \end{aligned}$$

Choosing $k_{0}$ suitably large and $\gamma$ small enough, we notice that the last three terms of the above inequality may be absorbed by the left-hand side. Hence, we conclude that there exist some $k_{0}\in \mathbb {Z}$ and $c_{0} > 0$ so that for all $k\ge k_{0}-1$,

$$\begin{aligned}&\frac{d}{dt}\left( \Vert b_{k}\Vert _{\varepsilon ,L^{p}}+\gamma c_{p} \Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}}\right) +c_{0}\left( \Vert b_{k}\Vert _{\varepsilon ,L^{p}}+\gamma c_{p}\Vert \Lambda q_{k}\Vert _{\varepsilon ,L^{p}}\right) \\\le & {} \Vert f_{k}\Vert _{L^{p}}+(c_{p}2^{2k}+\gamma c_{p})\varepsilon . \end{aligned}$$

Integrating in time and having $\varepsilon$ tend to 0, we obtain (taking smaller $c_{0}$ if needed)

$$\begin{aligned} e^{c_{0}t}\left\| (b_{k},\Lambda q_{k})(t)\right\| _{L^{p}}\lesssim \left\| (b_{k},\Lambda q_{k})(0)\right\| _{L^{p}}+ \int _{0}^{t}e^{c_{0}\tau }\left\| f_{k}(\tau )\right\| _{L^{p}}d\tau . \end{aligned}$$

It is obvious that $(a_{k}, \Lambda q_{k})$ fulfills a similar inequality, for we arrive at

$$\begin{aligned} a=b+\Lambda ^{-1}q, \end{aligned}$$

which leads for $k\ge k_{0}-1$ to

$$\begin{aligned} \Vert a_{k}-b_{k}\Vert _{L^{p}}\lesssim 2^{-2k_{0}}\Vert \Lambda \,q_{k}\Vert _{L^{p}}\lesssim 2^{-2k_{0}}\Vert \Lambda \, q_{k}\Vert _{L^{p}}. \end{aligned}$$

Hence, there exists a constant $c_{0}>0$ such that for all $k\ge k_{0}-1$ and $t\ge 0$, we can get

$$\begin{aligned} \left\| (a_{k},\Lambda \, q_{k})(t)\right\| _{L^{p}}\lesssim e^{-c_{0}t}\left\| (a_{k},\Lambda \, q_{k})(0)\right\| _{L^{p}}+ \int _{0}^{t}e^{-c_{0}(t-\tau )}\left\| f_{k}(\tau )\right\| _{L^{p}}d\tau . \end{aligned}$$

(40)

Now, multiplying both sides of (40) by $\left\langle t\right\rangle ^{\alpha }2^{k(\frac{d}{p}-2)}$, taking the supremum on [0, T], and summing up on $k\ge k_{0}-1$ yields

$$\begin{aligned}&\Vert \langle t \rangle ^{\alpha }(a,\Lambda q)\Vert ^{h}_{\tilde{L}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})}\lesssim \Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}\nonumber \\&+\sum _{k\ge k_{0}-1}\sup _{t \in [0,T]}\Big (\langle t\rangle ^{\alpha }\int _{0}^{t}e^{-c_{0}(t-\tau )}2^{k(\frac{d}{p}-2)}\Vert f_{k}(\tau )\Vert _{L^{p}}d\tau \Big ). \end{aligned}$$

(41)

In order to handle the sum, we first see that

$$\begin{aligned}&\sum _{k\ge k_{0}-1}\sup _{t\in [0,2]}\Big (\langle t\rangle ^{\alpha }\int _{0}^{t}e^{-c_{0}(t-\tau )}2^{k(\frac{d}{p}-2)}\Vert f_{k}(\tau )\Vert _{L^{p}}d\tau \Big )\\\lesssim & {} \int _{0}^{2}\sum _{k\ge k_{0}-1}2^{k(\frac{d}{p}-2)}\Vert f_{k}(\tau )\Vert _{L^{p}}d\tau . \end{aligned}$$

Using Proposition 4 to treat $\left\| f_{k}(\tau )\right\| _{L^{p}}$, we eventually have

$$\begin{aligned} \int _{0}^{2}\sum _{k\ge k_{0}-1}2^{k(\frac{d}{p}-2)}\left\| f_{k}(\tau )\right\| _{L^{p}}d\tau \lesssim \int _{0}^{2} \left\| a\right\| _{\dot{B}^{\frac{d}{p}}_{p,1}}\left\| q\right\| _{\dot{B}^{\frac{d}{p}-1}_{p,1}}d\tau \lesssim \mathcal {E}^{2}_{p}(2). \end{aligned}$$

(42)

Secondly, let us handle the supremum for $2\le t\le T$ in the last term of (41). To do this, one can split the integral on [0, t] into integrals on [0, 1] and [1, t]. The [0, 1] part of the integral can be bounded by C$\mathcal {E}^{2}_{p}(1)$ exactly as the supremum on [0, 2] treated before, see [24, 28, 29] for more details. In order to bound the [1, t] part of the integral for $2 \le t \le T$, we start from

$$\begin{aligned}&\sum _{k\ge k_{0}-1}\sup _{t\in [2,T]}\Big (\langle t\rangle ^{\alpha }\int _{1}^{t}e^{-c_{0}(t-\tau )}2^{k(\frac{d}{p}-2)}\Vert f_{k}(\tau )\Vert _{L^{p}}d\tau \Big )\nonumber \\\lesssim & {} \sum _{k\ge k_{0}-1} 2^{k(\frac{d}{p}-2)}\sup _{t\in [1,T]} t^{\alpha } \Vert f_{k}(t)\Vert _{L^{p}}. \end{aligned}$$

(43)

To deal with the right-hand side of (43), we get from (24) that

$$\begin{aligned} \sum _{k\ge k_{0}-1} 2^{k(\frac{d}{p}-2)}\sup _{t\in [1,T]} t^{\alpha } \Vert f_{k}(t)\Vert _{L^{p}} \lesssim \Vert t^{\alpha }\mathrm {div}\,(a\nabla \Lambda ^{-1} q)\Vert ^{h}_{\tilde{L}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})}. \end{aligned}$$

Regarding the term with $\mathrm {div}\,(a\nabla \Lambda ^{-1} q)$, its decomposition is the same as (28). Product laws in Proposition 4 adapted to tilde spaces ensure that

$$\begin{aligned}&\Vert t^{\alpha }\mathrm {div}\,(a^{h}\nabla \Lambda ^{-1} q)\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \Vert t^{\alpha }a^{h}\nabla \Lambda ^{-1} q\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-1}_{p,1})}\\\lesssim & {} \Vert t^{\alpha }a\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}}_{p,1})} \Vert q\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-1}_{p,1})}\lesssim \mathcal {D}_{p}(T)\mathcal {E}_{p}(T),\\&\Vert t^{\alpha }\mathrm {div}\,(a^{\ell }\nabla \Lambda ^{-1} q^{h})\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \Vert t^{\alpha }a^{\ell }\nabla \Lambda ^{-1} q^{h}\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-1}_{p,1})}\\\lesssim & {} \Vert a\Vert ^{\ell }_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}}_{p,1})} \Vert t^{\alpha }q\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-1}_{p,1})} \lesssim \mathcal {E}_{p}(T)\mathcal {D}_{p}(T). \end{aligned}$$

Bernstein inequality and embedding imply that

$$\begin{aligned}&\Vert t^{\alpha }\mathrm {div}\,(a^{\ell }\nabla \Lambda ^{-1} q^{\ell })\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \Vert t^{\alpha }a^{\ell }\nabla \Lambda ^{-1} q^{\ell }\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})}\\\lesssim & {} \Vert t^{\frac{\sigma _{1}}{2}+\frac{d}{4}}a^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})} \Vert t^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }q^{\ell }\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})}\nonumber \\&+\Vert t^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }q^{\ell }\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})} \Vert t^{\frac{\sigma _{1}}{2}+\frac{d}{4}}a^{\ell }\Vert _{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}}_{2,1})}. \end{aligned}$$

According to the definition of $\mathcal {D}_{p}(t)$ and tilde norms, we arrive at

$$\begin{aligned} \Vert t^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }(a,q)^{\ell }\Vert _{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1}_{2,1})}\lesssim \Vert \langle t\rangle ^{\frac{\sigma _{1}}{2}+\frac{d}{4}+\frac{1}{2}-\varepsilon }(a,q)\Vert ^{\ell }_{L_{T}^{\infty }(\dot{B}^{\frac{d}{2}+1-2\varepsilon }_{2,1})}\lesssim \mathcal {D}_{p}(T). \end{aligned}$$

Therefore, we deduce that

$$\begin{aligned} \Vert t^{\alpha }\mathrm {div}\,(a^{\ell }\nabla \Lambda ^{-1} q^{\ell })\Vert ^{h}_{{\widetilde{L}}_{T}^{\infty }(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \mathcal {D}^{2}_{p}(T). \end{aligned}$$

Consequently, putting all estimates together, we conclude that

$$\begin{aligned} \sum _{k\ge k_{0}-1} 2^{k\,(\frac{d}{p}-2)}\sup _{t\in [1,T]} t^{\alpha } \Vert f_{k}(t)\Vert _{L^{p}} \lesssim \mathcal {E}_{p}(T)\mathcal {D}_{p}(T)+\mathcal {D}^{2}_{p}(T). \end{aligned}$$

(44)

Plugging (44) in (43), and remembering (41) and (42), we end up with

$$\begin{aligned} \Vert \langle t\rangle ^{\alpha }(a,\Lambda q)\Vert ^{h}_{{\widetilde{L}}^{\infty }_{T}(\dot{B}^{\frac{d}{p}-2}_{p,1})} \lesssim \Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}+\mathcal {E}^{2}_{p}(T)+\mathcal {D}^{2}_{p}(T). \end{aligned}$$

(45)

3.3 Decay and Gain of Regularity for the High Frequencies of a

In order to estimate the last term in $\mathcal {D}_{p}(t)$, we shall take full advantage of the parabolic smoothing effect. To do this, we observe that a satisfies

$$\begin{aligned} \partial _{t}a-\Delta a=\Lambda q-\mathrm {div}\,(a\nabla \Lambda ^{-1}q). \end{aligned}$$

(46)

To achieve the desired estimates, we reformulate (46) as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}\left( t^{\alpha }\Delta a\right) -\Delta \left( t^{\alpha }\Delta a\right) =\alpha t^{\alpha -1}\Delta a-t^{\alpha }\Lambda ^{3}q-t^{\alpha }\Delta \mathrm {div}\,(a\nabla \Lambda ^{-1}q),\\ t\Delta a|_{t=0}=0. \end{array} \right. \end{aligned}$$

With the aid of Proposition 7 and Bernstein inequality, we get for $k\ge k_{0}-1$,

$$\begin{aligned} \Vert \tau ^{\alpha } \Lambda a\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})}\lesssim & {} \Vert \tau ^{\alpha -1} a \Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})} +\Vert \tau ^{\alpha } \Lambda q\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})}\\&+\Vert \tau ^{\alpha }\mathrm {div}\,(a\nabla \Lambda ^{-1}q)\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})}. \end{aligned}$$

Note that $\alpha >1$, we get that, owing to (45),

$$\begin{aligned} \Vert \tau ^{\alpha -1} a\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})} +\Vert \tau ^{\alpha } \Lambda q\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})}\lesssim & {} \Vert \langle \tau \rangle ^{\alpha } (a,\Lambda q)\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})}\\\lesssim & {} \Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}+\mathcal {E}^{2}_{p}(t)+\mathcal {D}^{2}_{p}(t). \end{aligned}$$

Bounding the norm $\Vert \tau ^{\alpha }\mathrm {div}\,(a\nabla \Lambda ^{-1}q)\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-2}_{p,1})}$ is exactly same as Step 2, one can conclude that

$$\begin{aligned} \Vert \tau ^{\alpha } \Lambda a\Vert ^{h}_{{\widetilde{L}}^{\infty }_{t}(\dot{B}^{\frac{d}{p}-1}_{p,1})} \lesssim \Vert (a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}} +\mathcal {D}_{p}(t)\mathcal {E}_{p}(t) +\mathcal {E}^{2}_{p}(t)+\mathcal {D}^{2}_{p}(t). \end{aligned}$$

(47)

Finally, adding up (47) to (35) and (45) yields for all $T\ge 0$,

$$\begin{aligned} \mathcal {D}_{p}(T)\lesssim \mathcal {D}_{p,0} +\Vert ( a_{0},\Lambda q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}} +\mathcal {E}^{2}_{p}(T)+\mathcal {D}^{2}_{p}(T). \end{aligned}$$

As Theorem 1 ensures that $\mathcal {E}_{p}(t)\lesssim \mathcal {E}_{p,0}\ll 1$, one can conclude that (23) is fulfilled for all time if $\mathcal {D}_{p,0}$ and $\mathcal {E}_{p,0}$ are small enough. This completes the proof of Proposition 8.

In what follows, let us give the proof of Theorem 2. Due to the embedding $\dot{B}_{2,1}^{\sigma +d(\frac{1}{2}-\frac{1}{p})}\hookrightarrow \dot{B}^{\sigma }_{p,1}$ for all $p\ge 2$, we get

$$\begin{aligned} \sup _{t\in [0,T]} t^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}\Vert \Lambda ^{\sigma }a\Vert _{\dot{B}^{0}_{p,1}}\lesssim & {} \Vert t^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}a\Vert ^{\ell }_{L^{\infty }_{T}(\dot{B}^{\sigma +d(\frac{1}{2}-\frac{1}{p})}_{2,1})} \\&+\Vert t^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}a\Vert ^{h}_{L^{\infty }_{T}(\dot{B}^{\sigma }_{p,1})}. \end{aligned}$$

Set $-\tilde{\sigma }_{1}<\sigma \le \frac{d}{p}$, and then $-\sigma _{1}<\sigma +d(\frac{1}{2}-\frac{1}{p})\le \frac{d}{2}$. Hence, by taking advantage of (22) and (23), we observe that

$$\begin{aligned} \Vert t^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}a\Vert ^{\ell }_{L^{\infty }_{T}(\dot{B}^{\sigma +d(\frac{1}{2}-\frac{1}{p})}_{2,1})}\lesssim & {} \Vert \langle t\rangle ^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p}) }a\Vert ^{\ell }_{L^{\infty }_{T}(\dot{B}^{\sigma +d(\frac{1}{2}-\frac{1}{p})}_{2,1})}\\\lesssim & {} \mathcal {D}_{p,0}+\Vert (a_{0},\nabla q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}, \end{aligned}$$

since $\tilde{\sigma }_{1}+\sigma =\sigma _{1}+\sigma +d\big (\frac{1}{2}-\frac{1}{p}\big )$. On the other hand, if $\varepsilon >0$ is sufficiently small, then we have $\frac{\sigma _{1}}{2}+\frac{d}{2}+\frac{1}{2}-\varepsilon >1+\frac{d}{4}-\varepsilon \ge \frac{d}{4}\ge \frac{d}{2}(\frac{1}{2}-\frac{1}{p})+\frac{\sigma }{2}$ for all $\sigma \le \frac{d}{p}$, which implies that $\alpha \ge \frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})$. Consequently, we deduce that

$$\begin{aligned} \Vert t^{\frac{\sigma _{1}+\sigma }{2}+\frac{d}{2}(\frac{1}{2}-\frac{1}{p})}a\Vert ^{h}_{L^{\infty }_{T}(\dot{B}^{\sigma }_{p,1})} \lesssim \Vert t^{\alpha }a\Vert ^{h}_{\tilde{L}^{\infty }_{T}(\dot{B}^{\frac{d}{p}}_{p,1})} \lesssim \mathcal {D}_{p,0}+\Vert (a_{0},\nabla q_{0})\Vert ^{h}_{\dot{B}^{\frac{d}{p}-2}_{p,1}}. \end{aligned}$$

Hence, using $\dot{B}_{p,1}^{0}\hookrightarrow L^{p}$ yields the desired result for a. Bounding the large-time behavior of q works almost the same. The proof of Theorem 2 is complete.

4 Conclusion

The Keller–Segel model is an effective mathematical model (derived by Keller and Segel), which can be used to describe the phenomenon of chemotaxis in biological sciences. In the present paper, we formulate an additional smallness assumption of the low frequencies of the initial data and then establish the time-weighted energy inequality in the critical Besov spaces, which leads to the optimal time-decay estimates of global strong solutions to the Cauchy problem of chemotaxis model. In comparison with [1, 2], there are apparent innovative aspects in our paper. On the one hand, we propose a new regularity assumption, which allows us to enjoy larger freedom on the choice of regularity index of the low frequencies. On the other hand, we emphasize that the decay index $\alpha$ coming from the second and third terms of functional $\mathcal {D}_{p}(t)$ are optimal, which allows the derivative indices of a to be somewhat relaxed. In short, our works improve the previous results.