Ill-posedness of the hyperbolic Keller-Segel model in Besov spaces

Abstract

In this paper, we give a new construction of $u_0\in B^\sigma _{p,\infty }$ such that the corresponding solution to the hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^\sigma _{p,\infty }(\mathbb {R}^d)$ with $d\ge 1$ and $1\le p\le \infty$, which implies the ill-posedness for this equation in $B^\sigma _{p,\infty }$. Our result generalizes the recent work in Zhang et al. (J Differ Equ 334:451-489, 2022) where the case $d=1$ and $p=2$ was considered.

1 Introduction

Chemotaxis is the active motion of organisms influenced by chemical gradients. The most prominent model for this process goes back to Patlak, Keller and Segel [6,7,8, 16] which takes the form of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+ \nabla \cdot (D_{1}(u,S)\nabla u-\chi (u,S)\nabla S)=0,\\ \tau S_{t}=D_{2} \Delta S+k(u,S), \end{array}\right. } \end{aligned}$$

(1.1)

here u(x, t) represents the cell density at position $x \in {\mathbb {R}}^{d}$, time $t > 0$, and S(x, t) is the concentration of a chemical signal. The motility $D_{1}(u, S)$ and the chemotactic sensitivity $\chi (u, S)$ rely on the cell density and on the signal concentration. The term k(u, S) depicts production and decay or consumption of the signal and $D_{2}$ is the diffusion constant for S. The parameter $\tau$ illustrates that movement of the species and dynamics of the signal have different characteristic time scales. The Keller-Segel model has been applied to many different problems, ranging from bacteria chemotaxis to cancer growth or the immune response.

Dolak and Schmeiser [5] derived a convection equation with a small diffusion term as higher order correction from a kinetic model for chemotaxis. Inspired by this, Dolak and Schmeiser proposed the following parabolic-type Keller-Segel equations with small diffusivity:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u=- \nabla \cdot (u(1-u)\nabla S-\epsilon \nabla u), &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d},\\ -\Delta S=u - S, &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d}. \\ \end{array}\right. } \end{aligned}$$

(1.2)

Burger, Dolak and Schmeiser [3] studied the asymptotic behavior of solutions of the chemotaxis model (1.2) in multiple spatial dimensions. Of particular interest is the practically relevant case of small diffusivity, where (as in the one-dimensional case) the cell densities form plateau-like solutions for large time. Some other results related to (1.2) can be found in [18,19,20].

Nie and Yuan [14] considered the Cauchy problem for multidimensional chemotaxis system

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} \partial _{t}u-\Delta u=\nabla \cdot (uv), &{}{}\text{ in }\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d},\\ \partial _{t}v-\nabla u=0, &{}{}\text{ in }\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d}, \,\ \\ (u,v)|_{t=0}=(u_{0},v_{0}), &{}{}\text{ in }\quad {\mathbb {R}}^{d}, \end{array} \end{array}\right. } \end{aligned}$$

(1.3)

they proved that (1.3) is well-posed in $\dot{B}_{p,\sigma }^{\frac{d}{p}-2}\times (\dot{B}_{p,\sigma }^{\frac{d}{p}-1})^{d}$ when $p<2d$ and is ill-posed when $p>2d$. Later, Nie and Yuan [15] also obtaind that (1.3) is ill-posed in $\dot{B}_{p,1}^{\frac{d}{p}-2}\times (\dot{B}_{p,1}^{\frac{d}{p}-1})^{d}$ when $p=2d$. Almost in the same time, Xiao and Fei [21] proved that (1.3) is ill-posed in $\dot{B}_{p,\sigma }^{\frac{d}{p}-2}\times (\dot{B}_{p,\sigma }^{\frac{d}{p}-1})^{d}$ when $p=2d, \sigma >2$. Recently, Li, Yu and Zhu [13] proved that (1.3) is ill-posed in $\dot{B}_{p,r}^{\frac{d}{p}-2}\times (\dot{B}_{p,r}^{\frac{d}{p}-1})^{d}$ when $1\le r<d$.

In this paper, we consider the Cauchy problem for following hyperbolic Keller-Segel equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u=- \nabla \cdot (u(1-u)\nabla S), &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d},\\ -\Delta S=u - S , &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d}, \\ u(x,0)=u_{0}(x) , &{}\text {in}\quad {\mathbb {R}}^{d}. \end{array}\right. } \end{aligned}$$

(1.4)

The unknown scale functions u(x, t) and S(x, t) denote the cell density and the concentration of chemical substance, respectively. Dolak and Schmeiser [4] firstly established the existence and unique of global smooth solution to one dimensional version of (1.2) with suitable conditions on the initial data. On a time scale characteristic for the convective effects, they also proved that the corresponding sequence of solutions $u^\epsilon$ converges to the weak entropy solution u to (1.1) as $\epsilon \rightarrow 0$. Laterly, Burger, Difrancesco and Dolak [2] obtained the unique local-in-time solution to (1.2) with the initial data belonging to $L^1(\mathbb {R}^d)\cap L^\infty (\mathbb {R}^d)$. Perthame and Dalibard [17] proved the existence of an entropy solution to (1.4) by passing to the limit in a sequence of solutions to the parabolic approximation. Lee and Liu [9] proved the sub-threshold for finite time shock formation to solutions of (1.1) in one-dimension.

Recently, Zhou, Zhang and Mu [23] obtained the existence and uniqueness of solution of (1.4) in ${{B}}_{p,r}^{s}({\mathbb {R}}^{d})$ when $1\le p,r\le \infty ,s>1+\frac{d}{p}$. Later, Zhang, Mu and Zhou [22] proved that (1.1) is ill-posed in ${{B}}_{2,\infty }^{s}({\mathbb {R}})$ with $s>\frac{3}{2}$ and (1.4) is local well-posed in ${{B}}_{p,1}^{s}({\mathbb {R}}^{d})$ when $1\le p<\infty , s=1+\frac{d}{p}$. However, their initial data seems to be valid only for $p=2$ when proving the ill-posedness in ${B}_{p,\infty }^{s}$. Motivated by the recent works in [11, 12], we aim to extend the ill-posedness result in [22] to more general case, i.e, $1\le p\le \infty$ and $d\ge 1$. The main result of the paper is the following theorem:

Theorem 1.1

Let $d\ge 1$. Assume that

$$\begin{aligned} s>1+\frac{d}{p}\quad \text {with}\quad 1\le p\le \infty , \end{aligned}$$

then there exists $u_{0}\in {B}_{p,\infty }^{s}({\mathbb {R}}^{d})$ and a positive constant $\epsilon _{0}$ such that the data-to-solution map $u_{0}\mapsto u$ of the Cauchy problem (1.1) satisfies

$$\begin{aligned} \limsup _{t\rightarrow 0^+}\Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s}}\ge \epsilon _{0}. \end{aligned}$$

Remark 1.1

Theorem 1.1 demonstrates the ill-posedness of (1.4) in ${B}_{p,\infty }^{s}({\mathbb {R}}^{d})$. Precisely speaking, we can construct $u_{0}\in {B}_{p,\infty }^{s}({\mathbb {R}}^{d})$ such that the corresponding solutions of the Keller-Segel equation do not converge to $u_{0}$ in the metric of ${B}_{p,\infty }^{s}({\mathbb {R}}^{d})$ as $t\rightarrow 0^+$.

Remark 1.2

We should mention that the key decomposition technique and the special initial data used in [22] can not be applied to the present case $p\ne 2$ any more. To overcome these difficulties, we construct a new initial data which is completely different from [22]. In particular, by utilizing the commutator estimate and some basic analysis, we make the proof more simple.

The rest of the paper is organized as follows. In Sect. 2, we introduce some basic definitions and key lemmas. In Sect. 3, we present the proof of Theorem 1.1.

2 Preliminaries

Notation  The notation $A\lesssim B$ (resp., $A \gtrsim B$) means that there exists a harmless positive constant c such that $A \le cB$ (resp., $A \ge cB$). Given a Banach space X, we denote its norm by $\Vert \cdot \Vert _{X}$. For a Banach space X and for any $0<T\le \infty$, we use standard notation $L^{p}(0,T;X)$ to denote the quasi-Banach space of Bochner measurable functions f from (0, T) to X endowed with the norm

$$\begin{aligned} \Vert f\Vert _{{L^{p}_{T}}X}:= {\left\{ \begin{array}{ll} \left(\displaystyle \int \limits _0^T \Vert f(\cdot ,t)\Vert ^{p}_{X}\,dt\right)^{\frac{1}{p}},&{}\hbox {if}\,\, 1\le p<\infty ,\\ \sup _{0\le t \le T}\Vert f(\cdot ,t)\Vert _X,&{}\hbox {if}\,\, p=\infty . \end{array}\right. } \end{aligned}$$

Let us recall that for all $f\in {\mathcal {S}}'$, the Fourier transform ${\widehat{f}}$, is defined by

$$\begin{aligned} ({\mathcal {F}} f)(\xi )={\widehat{f}}(\xi )=\int \limits _{\mathbb {R}^d}e^{-ix\cdot \xi }f(x)\text {d}x \quad \text{ for } \text{ any }\, \xi \in \mathbb {R}^d. \end{aligned}$$

The inverse Fourier transform of any g is given by

$$\begin{aligned} ({\mathcal {F}}^{-1} g)(x)=\frac{1}{(2 \pi )^d} \int \limits _{\mathbb {R}^d} e^{i x \cdot \xi }g(\xi ) \textrm{d}\xi . \end{aligned}$$

Next, we recall some facts on the Littlewood-Paley theory which can be found in [1].

Let $\varphi \in C_c^{\infty }({\mathbb {R}}^d)$ and $\chi \in C_c^{\infty }({\mathbb {R}}^d)$ be a radial positive function such that

$$\begin{aligned}&\textrm{supp}\ \varphi \subset \{\xi \in {\mathbb {R}}^d:\frac{3}{4}\le |\xi |\le \frac{8}{3}\}, \,\, \,\, \textrm{supp}\ \chi \subset \{\xi \in {\mathbb {R}}^d:|\xi |\le \frac{4}{3}\},\quad \\ {}&\chi (\xi )+ \sum _{j\ge 0}\varphi (2^{-j}\xi )=1\ \text {for any}\ \xi \in {\mathbb {R}}^d, \\ {}&|i-j|\ge 2\Rightarrow \textrm{supp}\ \varphi (2^{-i}\cdot )\cap \textrm{supp}\ \varphi (2^{-j}\cdot )=\varnothing , \\ {}&j\ge 1\Rightarrow \textrm{supp}\ \varphi (2^{-j}\cdot )\cap \textrm{supp}\ \chi (x)=\varnothing ,\\&\varphi (\xi )\equiv 1\quad \text {for}\quad \frac{4}{3}\le |\xi |\le \frac{3}{2}. \end{aligned}$$

We can define the nonhomogeneous localization operators as follows.

$$\begin{aligned}&\Delta _ju=0,\,\,j\le -1;\,\,\,\, \Delta _ju=\chi (D)u,\,\,j=-1;\,\,\,\, \Delta _ju=\varphi (2^{-j}D)u,\,\,j\ge 0, \end{aligned}$$

where the pseudo-differential operator $f(D):u\rightarrow {\mathcal {F}}^{-1}(f {\mathcal {F}}u)$.

Let us now define the Besov spaces as follows.

Definition 2.1

( [1]) Let $s\in {\mathbb {R}}$ and $(p,r)\in [1, \infty ]^2$. The nonhomogeneous Besov space $B^{s}_{p,r}(\mathbb {R}^d)$ is defined by

$$\begin{aligned} B^{s}_{p,r}(\mathbb {R}):=\Big \{f\in {\mathcal {S}}'(\mathbb {R}^d):\,\Vert f\Vert _{B^{s}_{p,r}(\mathbb {R}^d)}<\infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} {\Vert f\Vert _{B^{s}_{p,r}(\mathbb {R}^d)}=} {\left\{ \begin{array}{ll} \left( \sum _{j\ge -1}2^{sjr}\Vert \Delta _jf\Vert ^r_{L^p(\mathbb {R}^d)}\right) ^{\frac{1}{r}}, &{}\hbox {if}\,\, 1\le r<\infty ,\\ \sup _{j\ge -1}2^{sj}\Vert \Delta _jf\Vert _{L^p(\mathbb {R}^d)}, &{}\hbox {if}\,\,r=\infty . \end{array}\right. } \end{aligned}$$

Remark 2.1

It should be emphasized that $B^{s}_{p,\infty }(\mathbb {R}^d)$ with $s>\frac{d}{p}$ is a Banach algebra and $B^s_{p,\infty }(\mathbb {R}^d)\hookrightarrow B^t_{p,\infty }(\mathbb {R}^d)$ with $s>t$. These facts will be often used implicity.

Finally, we recall some lemmas which be used later.

Lemma 2.1

(Bernstein’s inequality, [1]) Let ${\mathcal {C}}$ be an annulus and ${\mathcal {B}}$ be a ball. There exists a constant C such that for any nonnegative integer k, any couple $(p,q)\in [1,\infty ]^{2}$ with $1\le p\le q$, and any $L^{p}$ function u we have

$$\begin{aligned}&\sup _{|\alpha |=k}\Vert \partial ^{\alpha }u\Vert _{L^{q}({\mathbb {R}}^{d})}\le C^{k+1}\lambda ^{k+d(\frac{1}{p}-\frac{1}{q})}\Vert u\Vert _{L^{p}({\mathbb {R}}^{d})}, \,\, \textrm{supp} {{\widehat{u}}}\subset \lambda {\mathcal {B}}, \\ {}&C^{-(k+1)}\lambda ^{k}\Vert u\Vert _{L^{p}({\mathbb {R}}^{d})}\le \sup _{|\alpha |=k}\Vert \partial ^{\alpha }u\Vert _{L^{p}({\mathbb {R}}^{d})}\le C^{(k+1)}\lambda ^{k}\Vert u\Vert _{L^{p}({\mathbb {R}}^{d})},\,\, \textrm{supp}{{\widehat{u}}}\subset \lambda {\mathcal {C}}. \end{aligned}$$

Lemma 2.2

( [1]) A smooth function $f:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ is said to be an $S^{m}$-multiplier: if $\forall \alpha \in {\mathbb {N}}^{d}$, there exists a constant $C_{\alpha }>0$ such that

$$\begin{aligned} |\partial ^{\alpha }f(\xi )|\le C_{\alpha }(1+|\xi |)^{m-\alpha },\,\,\xi \in {\mathbb {R}}^{d}. \end{aligned}$$

If f is a $S^{m}$-multiplier, then the operator f(D) is continuous from ${B}_{p,r}^{s}$ to ${B}_{p,r}^{s-m}$ for all $s\in {\mathbb {R}}$ and $1\le p,r\le \infty$.

Lemma 2.3

( [1]) For $1\le p\le \infty$ and $s>0$, there exists a constant C,depending continuously on p and s, we have

$$\begin{aligned}&\big \Vert 2^{js}\Vert [\Delta _{j},v]\cdot \nabla f\Vert _{L^{p}}\big \Vert _{\ell ^{\infty }}\le C(\Vert \nabla v\Vert _{L^{\infty }}\Vert f\Vert _{{B}_{p,\infty }^{s}}+\Vert \nabla f\Vert _{L^{\infty }}\Vert \nabla v\Vert _{{ B}_{p,\infty }^{s-1}}), \end{aligned}$$

where $[\Delta _{j},v]\cdot \nabla f=\Delta _{j}(v\cdot \nabla f)-v\cdot \Delta _{j}\nabla f$.

3 Proof of Theorem 1.1

For convenience of computation, we rewrite (1.1) as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+(1-2u)\nabla S\cdot \nabla u+u(1-u)\Delta S=0, &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d}, \\ S=(1-\Delta )^{-1}u, &{}\text {in}\quad {\mathbb {R}}^{+}\times {\mathbb {R}}^{d}, \\ u(x,0)=u_{0}(x) , &{}\text {in}\quad {\mathbb {R}}^{d}. \end{array}\right. } \end{aligned}$$

(3.1)

Let ${\widehat{\phi }}\in C_{0}^{\infty }({\mathbb {R}})$ be an even, real-valued and nonnegative function which satisfies

$$\begin{aligned} {{\widehat{\phi }}(\xi )=} {\left\{ \begin{array}{ll} 1,&{} \hbox {if}\,\, |\xi |\le \frac{1}{4^{d}},\\ 0,&{} \hbox {if}\,\, |\xi |\ge \frac{1}{2^{d}}. \end{array}\right. } \end{aligned}$$

Remark 3.1

By the Fourier-Plancherel formula, we have $\phi (x)={\mathcal {F}}^{-1}({\widehat{\phi }}(\xi ))$. It is easy to check that

$$\begin{aligned}&\phi (0)=\frac{1}{2\pi }\int \limits _{\mathbb {R}}{\widehat{\phi }}(\xi )\textrm{d}\xi >0\quad \text {and}\quad \phi '(0)=\frac{1}{2\pi }\int \limits _{\mathbb {R}}\widehat{\phi '}(\xi )\textrm{d}\xi =0. \end{aligned}$$

Lemma 3.1

Define the function $f_{n}(x)$ by

$$\begin{aligned} f_{n}(x)=\phi (x_{1})\sin \left(\frac{17}{12}2^{n}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d}),\quad n\ge 3. \end{aligned}$$

Then

$$\begin{aligned} {\Delta _{j}(f_{n})=} {\left\{ \begin{array}{ll} f_{j}, &{}\hbox {if}\,\, j=n,\\ 0, &{}\hbox {if}\,\, j\ne n. \end{array}\right. } \end{aligned}$$

Proof

Notice that

$$\begin{aligned} \textrm{supp} \ {\widehat{f}}_{n}&\subset \Big \{\xi \in \mathbb {R}^d: \ \frac{17}{12}2^{n}-\frac{1}{2}\le |\xi |\le \frac{17}{12}2^{n}+\frac{1}{2}\Big \}, \end{aligned}$$

using the definition of $\Delta _{j}$ enables us to get the desired result. For more details see [10]. $\square$

Proposition 3.1

Define the initial data $u_0(x)$ as

$$\begin{aligned}&S_{0}(x):=\sum _{n=3}^{\infty }2^{-n(s+2)}f_{n}(x),\\&u_{0}(x):=(1-\Delta )S_{0}(x). \end{aligned}$$

If $s>1+\frac{d}{p}$, we have

$$\begin{aligned} \Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}\le C. \end{aligned}$$

Proof

By Lemma 3.1, we have

$$\begin{aligned} \Delta _{j}S_{0}=2^{-j(s+2)}f_{j}(x). \end{aligned}$$

(3.2)

Combining Lemma 2.2 and (3.2) yields

$$\begin{aligned} \Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}&\le C\Vert S_{0}\Vert _{{B}_{p,\infty }^{s+2}}=\sup _{j\ge 0}2^{(s+2) j}\Vert \Delta _{j}S_0\Vert _{L^p}\le C. \end{aligned}$$

We complete the proof of Proposition 3.1. $\square$

Using Proposition 3.1 and Theorem 1.1 in [23], we can obtain that there exists a short time $T>0$ that (3.1) has a unique solution $u\in L^{\infty }([0,T);{B}_{p,\infty }^{s})\cap Lip([0,T);{B}_{p,\infty }^{s-1})$ for $s>1+\frac{d}{p}$. Moreover, it holds

$$\begin{aligned} \Vert u(t)\Vert _{L^{\infty }_{T}({B}_{p,\infty }^{s})}\le C\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}. \end{aligned}$$

(3.3)

Proposition 3.2

Let $s-1>\frac{d}{p}$ and $\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}\lesssim 1$. Assume that $u\in L^{\infty }(0,T;{B}_{p,\infty }^{s}({\mathbb {R}}^{d}))$ be the solution of (1.4), then we have

$$\begin{aligned} \Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s-1}} \lesssim t. \end{aligned}$$

Proof

Using the Newton-Leibniz formula, Minkowski’s inequality, Remark 2.1, Lemma 2.2 and Proposition 3.1, we have

$$\begin{aligned} \Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s-1}}&\le \int \limits _{0}^{t} \Vert (1-2u)\nabla S\cdot \nabla u\Vert _{{B}_{p,\infty }^{s-1}}\textrm{d}\tau +\int \limits _{0}^{t} \Vert u(1-u)\Delta S\Vert _{{ B}_{p,\infty }^{s-1}}\textrm{d}\tau \\ {}&\lesssim t\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s-2}}\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s}}+t\Vert u\Vert _{L_{t}^{\infty }{ B}_{p,\infty }^{s-2}}\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s-1}}\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s}}\\ {}&\quad +t\Vert u\Vert _{L_{t}^{\infty }{ B}_{p,\infty }^{s-1}}^{3}+t\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s-1}}^{2} \\ {}&\lesssim t\left(\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s}}^{3}+\Vert u\Vert _{L_{t}^{\infty }{B}_{p,\infty }^{s}}^{2}\right) \\ {}&\lesssim t\left(\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}^{3}+\Vert u_0\Vert _{{B}_{p,\infty }^{s}}^{2}\right)\\&\lesssim t, \end{aligned}$$

where we have used (3.3).

We complete the proof of Proposition 3.2. $\square$

Proposition 3.3

Let $s-1>\frac{d}{p}$ and $\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}\lesssim 1$. Assume that $u\in L^{\infty }(0,T;{B}_{p,\infty }^{s}({\mathbb {R}}^{d}))$ be the solution of (1.1), then we have

$$\begin{aligned} \Vert h(t,u_{0})\Vert _{{B}_{p,\infty }^{s-2}}\lesssim t^{2}, \end{aligned}$$

where we denote

$$\begin{aligned} h(t,u_{0}):=u-u_{0}+tv_{0} \end{aligned}$$

and

$$\begin{aligned} v_{0}:=\nabla \cdot (u_0(1-u_0)\nabla S_0)=(1-2u_{0})\nabla S_{0}\cdot \nabla u_{0}+u_{0}(1-u_{0})\Delta S_{0}. \end{aligned}$$

Proof

Using the Newton-Leibniz formula, Minkowski’s inequality, Remark 2.1, Lemma 2.2 and (3.3), we have

$$\begin{aligned} \Vert h(t,u_{0})\Vert _{{B}_{p,\infty }^{s-2}}&\le \int \limits _0^t \Vert \partial _{\tau }u+v_{0}\Vert _{{B}_{p,\infty }^{s-2}}\textrm{d}\tau \\ {}&\le \int \limits _0^t \Vert \nabla \cdot (u_0(1-u_0)\nabla S_0)-\nabla \cdot (u(1-u)\nabla S)\Vert _{{B}_{p,\infty }^{s-2}}\textrm{d}\tau \\ {}&\lesssim \int \limits _0^t \Vert u_0(1-u_0)\nabla S_0-u(1-u)\nabla S\Vert _{{B}_{p,\infty }^{s-1}}\textrm{d}\tau \\&\lesssim \int \limits _0^t \Vert u(\tau )-u_{0}\Vert _{{B}_{p,\infty }^{s-1}}\textrm{d}\tau \\&\lesssim t^2, \end{aligned}$$

where we have used Proposition 3.2 in the last step.

We complete the proof of Proposition 3.3. $\square$

Now we present the proof of Theorem 1.1.

Proof of Theorem 1.1.

Notice that $u(t)-u_0=h(t,u_{0})-tv_0,$ then

$$\begin{aligned} \nonumber \Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s}}&\ge 2^{js}\Vert \Delta _{j}(h(t,u_{0})-tv_{0})\Vert _{L^{p}} \nonumber \\ {}&\ge 2^{js}t\Vert \Delta _{j}v_{0}\Vert _{L^{p}}-2^{js}\Vert \Delta _{j}h(t,u_{0})\Vert _{L^{p}} \nonumber \\ {}&\ge 2^{js}t\Vert \Delta _{j}((1-2u_{0})\nabla S_{0}\cdot \nabla u_{0})\Vert _{L^{p}} -2^{js}t\Vert \Delta _{j}(u_{0}^{2}\Delta S_{0})\Vert _{L^{p}} \nonumber \\ {}&\quad -2^{js}t\Vert \Delta _{j}(u_{0}\partial _{x}^{2} S_{0})\Vert _{L^{p}} -2^{js}\Vert \Delta _{j}h(t,u_{0})\Vert _{L^{p}}. \end{aligned}$$

(3.4)

It is not difficult to deduce that

$$\begin{aligned}&2^{js}t\Vert \Delta _{j}(u_{0}^{2}\Delta S_{0})\Vert _{L^{p}}\lesssim t\Vert u_{0}^{2}\Delta S_{0}\Vert _{{B}_{p,\infty }^{s}} \lesssim t\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}^{3}\lesssim t,\\&2^{js}t\Vert \Delta _{j}(u_{0}\Delta S_{0})\Vert _{L^{p}}\lesssim t\Vert u_{0}\Delta S_{0}\Vert _{{B}_{p,\infty }^{s}} \lesssim t\Vert u_{0}\Vert _{{B}_{p,\infty }^{s}}^{2}\lesssim t ,\\&2^{js}\Vert \Delta _{j}h(t,u_{0})\Vert _{L^{p}}\le 2^{2j}\Vert h(t,u_{0})\Vert _{{B}_{p,\infty }^{s-2}} \lesssim t^{2}2^{2j}. \end{aligned}$$

Gathering the above estimates together with (3.4) yields

$$\begin{aligned} \nonumber \Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s}}&\ge 2^{js}\Vert \Delta _{j}(h+tv_{0})\Vert _{L^{2}} \nonumber \\ {}&\ge 2^{js}t\Vert \Delta _{j}((1-2u_{0})\nabla S_{0}\cdot \nabla u_{0})\Vert _{L^{p}} -Ct-Ct^{2}2^{2j} \nonumber \\ {}&\ge 2^{js}t\Vert (1-2u_{0})\nabla S_{0}\cdot \Delta _{j}\nabla u_{0}\Vert _{L^{p}}\nonumber \\ {}&\quad -2^{js}t\Vert [\Delta _{j},(1-2u_{0})\nabla S_{0}]\cdot \nabla u_{0}\Vert _{L^{p}}-Ct-Ct^{2}2^{2j}. \end{aligned}$$

(3.5)

On the one hand, by Lemma 2.3, we deduce

$$\begin{aligned} 2^{js}\Vert [\Delta _{j},(1-2u_{0})\nabla S_{0}]\cdot \nabla u_{0}\Vert _{L^{p}}\le C. \end{aligned}$$

(3.6)

On the other hand, we have

$$\begin{aligned} 2^{js}\Vert (1-2u_{0})(\nabla S_{0}\cdot \Delta _{j}\nabla u_{0})\Vert _{L^{p}}&= 2^{js}\left\Vert(1-2u_{0})\sum _{i=1}^{d}\partial _{x_{i}}S_{0}\Delta _{j}\partial _{x_{i}}u_{0}\right\Vert_{L^{p}}\ge J-K, \end{aligned}$$

(3.7)

where

$$\begin{aligned}&J:=2^{js}\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\Delta _{j}\partial _{x_{1}}u_{0}\right\Vert_{L^{p}},\\&K:=2^{js}\sum _{i=2}^{d}\left\Vert(1-2u_{0})\partial _{x_{i}}S_{0}\Delta _{j}\partial _{x_{i}}u_{0}\right\Vert_{L^{p}}. \end{aligned}$$

By Lemma 3.1, we infer

$$\begin{aligned} J&=2^{-2j}\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\partial _{x_{1}}(1-\Delta )f_{j}\right\Vert_{L^{p}} \ge 2^{-2j}\left(J_{1}-J_{2}-J_{3}\right), \end{aligned}$$

(3.8)

where

$$\begin{aligned}&J_{1}:=\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\partial ^3_{x_{1}}f_{j}\right\Vert_{L^{p}},\\&J_{2}:=\sum _{i=2}^d\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\partial _{x_{1}}\partial ^2_{x_i}f_{j}\right\Vert_{L^{p}},\\&J_{3}:=\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\partial _{x_{1}}f_{j}\right\Vert_{L^{p}}. \end{aligned}$$

We have

$$\begin{aligned} \partial ^3_{x_{1}}f_{j}(x)=-\left(\frac{17}{12}\right)^32^{3j}\phi (x_{1})\cos \left(\frac{17}{12}2^{j}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})+R, \end{aligned}$$

where

$$\begin{aligned} R&=\frac{17}{4}2^{j}\phi ^{\prime \prime }(x_{1})\cos \left(\frac{17}{12}2^{j}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})\\&\quad -3\left(\frac{17}{12}\right)^22^{2j}\phi ^{\prime }(x_{1})\sin \left(\frac{17}{12}2^{j}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})\\&\quad +\phi ^{\prime \prime \prime }(x_{1})\sin \left(\frac{17}{12}2^{j}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d}). \end{aligned}$$

Obviously, $\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}R\right\Vert_{L^{p}}\le C2^{2j}$, then

$$\begin{aligned} J_{1}&\ge \left(\frac{17}{12}\right)^32^{3j}\left\Vert(1-2u_{0})\partial _{x_{1}}S_{0}\phi (x_{1})\cos \left(\frac{17}{12}2^{j}x_{1}\right)\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})\right\Vert_{L^{p}}-C2^{2j}. \end{aligned}$$

By the construction of $f_n$, it is not difficult to deduce that

$$\begin{aligned} S_{0}(0)=\sum _{n=3}^{\infty }2^{-n(s+2)}f_{n}(0)=0. \end{aligned}$$

By easy computations, we have

$$\begin{aligned} \Delta S_{0}(x)&=\partial _{x_{1}}^{2}S_{0}(x)+\sum _{i=2}^{d}\partial _{x_{i}}^{2}S_{0}(x)\\&=\sum _{n=3}^{\infty }2^{-n(s+2)}\partial _{x_{1}}^{2}f_{n}+\sum _{i=2}^{d}\sum _{n=3}^{\infty }2^{-n(s+2)}\partial _{x_{i}}^{2} f_{n}. \end{aligned}$$

Noticing that the construction of $f_n$ again and using the fact $\phi '(0)=0$ from Remark 3.1, then we obtain

$$\begin{aligned} \Delta S_{0}(0)=0, \end{aligned}$$

which implies that

$$\begin{aligned} u_{0}(0)=S_{0}(0)-\Delta S_{0}(0)=0. \end{aligned}$$

Since $(1-2u_{0})\partial _{x_{1}}S_{0}\phi (x_{1})\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})$ is a real-valued and continuous function on $\mathbb {R}$, then there exists some $\delta >0$ such that for any $x\in B_{\delta }(0):=\{x\in {\mathbb {R}}^{d}:\,|x|\le \delta \}$

$$\begin{aligned}&\left|[(1-2u_{0})\partial _{x_{1}}S_{0}\phi (x_{1})\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})](x)\right|\nonumber \\ \ge&\, \frac{1}{2}\left|[(1-2u_{0})\partial _{x_{1}}S_{0}\phi (x_{1})\phi (x_{2})\cdot \cdot \cdot \phi (x_{d})](0)\right|\nonumber \\ =&\,\frac{1}{2}\phi ^{d}(0)|\partial _{x_{1}}S_{0}(0)|\nonumber \\ =&\,\frac{17}{24}\phi ^{2d}(0)\sum \limits ^{\infty }_{n=3}2^{-n(s+1)}=:c_0>0. \end{aligned}$$

Thus we have for j large enough

$$\begin{aligned} J_{1}&\ge c_02^{3j}\left\Vert\cos \left(\frac{17}{12}2^{j}x_{1}\right)\right\Vert_{L^{p}(B_{\delta }(0))}-C2^{2j} \ge {\tilde{c}}_02^{3j}. \end{aligned}$$

By direct computations, we can verify that

$$\begin{aligned} J_{2}+J_{3}\le C2^{j}. \end{aligned}$$

Thus, we have

$$\begin{aligned} J\ge C2^{j}. \end{aligned}$$

(3.9)

Similarly, we also have

$$\begin{aligned} K\le C. \end{aligned}$$

(3.10)

Combining (3.9) and (3.10), we have

$$\begin{aligned} 2^{js}t\Vert (1-2u_{0})\nabla S_{0}\cdot \Delta _{j}\nabla u_{0}\Vert _{L^{p}}&\ge C2^{j}t. \end{aligned}$$

(3.11)

Inserting (3.11) and (3.6) into (3.5), we deduce that for large j

$$\begin{aligned} \Vert u(t)-u_{0}\Vert _{{B}_{p,\infty }^{s}}\ge C2^{j}t-Ct-C2^{2j}t^{2}\ge C2^{j}t-C2^{2j}t^{2}. \end{aligned}$$

Thus, picking $t2^{j}\approx \epsilon _0$ with small $\epsilon _0$, we have

$$\begin{aligned} \Vert u(t)-u_0\Vert _{B^{s}_{p,\infty }}\ge C\epsilon _0-C\epsilon _0^2\ge c_1\epsilon _0. \end{aligned}$$

This completes the proof of Theorem 1.1. $\square$