The Keller-Segel system of parabolic-parabolic type in homogeneous Besov spaces framework

Abstract

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat equations.

Introduction

Consider the Keller-Segel system of parabolic-parabolic type in ${\mathbb{R}}^n$, $n\ge 2$;

$$\{\begin{array}{cccc}\hfill {\partial }_{t}u& =\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),\hfill & \hfill x& \in {\mathbb{R}}^n,t>0,\hfill \\ \hfill {\partial }_{t}v& =\mathrm{\Delta }v+u,\hfill & \hfill x& \in {\mathbb{R}}^n,t>0,\hfill \\ \hfill u(0,x)& =u_0(x),\hfill & \hfill x& \in {\mathbb{R}}^n,\hfill \\ \hfill v(0,x)& =v_0(x),\hfill & \hfill x& \in {\mathbb{R}}^n,\hfill \end{array}$$

where $u=u(t,x)$ and $v=v(t,x)$ stand for the density of amoebae and the concentration of the chemo-attractant, respectively, while $u_0=u_0(x)$ and $v_0=v_0(x)$ are the given initial data.

The aim of this article is to show the existence and uniqueness of local strong solutions of (1.1) for arbitrary initial data $(u_0,v_0)\in {\dot{B}}_{p_1,q}^{-2+n/p_1}({\mathbb{R}}^n)\times {\dot{B}}_{p_2,q}^{n/p_2}({\mathbb{R}}^n)$ for some $1\le p_1,p_2<\infty$ and $1\le q<\infty$. Here, ${\dot{B}}_{p,q}^s({\mathbb{R}}^n)$ is the homogeneous Besov space. We also prove the existence and uniqueness of global strong solutions for small $(u_0,v_0)\in {\dot{B}}_{p_1,q}^{-2+n/p_1}({\mathbb{R}}^n)\times {\dot{B}}_{p_2,q}^{n/p_2}({\mathbb{R}}^n)$, including the case of $q=\infty$. If a pair of function $(u,v)$ solves (1.1), then $(u_{\lambda }(t,x),v_{\lambda }(t,x)):=({\lambda }^{2}u({\lambda }^{2}t,\lambda x),v({\lambda }^{2}t,\lambda x))$ becomes also the solution of (1.1) for all $\lambda >0$. In addition, the Banach space $X\times Y$ with norms ${\Vert \cdot \Vert }_X$ and ${\Vert \cdot \Vert }_Y$ is said to be scaling invariant to (1.1) if the conditions ${\Vert u_{\lambda }(0,\cdot )\Vert }_X={\Vert u_0\Vert }_X$ and ${\Vert v_{\lambda }(0,\cdot )\Vert }_Y={\Vert v_0\Vert }_Y$ are satisfied for all $\lambda >0$. Notice that ${\dot{B}}_{p_1,q}^{-2+n/p_1}({\mathbb{R}}^n)\times {\dot{B}}_{p_2,q}^{n/p_2}({\mathbb{R}}^n)$ is one of scaling invariant spaces to (1.1). The notion of scaling invariant spaces is traced back to the Fujita-Kato principle for the Navier-Stokes system:

$$\{\begin{array}{cccc}\hfill {\partial }_{t}u& =\mathrm{\Delta }u-(u\cdot \mathrm{\nabla })u-\mathrm{\nabla }\pi ,\hfill & \hfill x& \in {\mathbb{R}}^n,t>0,\hfill \\ \hfill \mathrm{\nabla }\cdot u& =0,\hfill & \hfill x& \in {\mathbb{R}}^n,t>0,\hfill \\ \hfill u(0,x)& =u_0(x),\hfill & \hfill x& \in {\mathbb{R}}^n.\hfill \end{array}$$

Fujita and Kato [11] chose ${\dot{H}}^{1/2,2}({\mathbb{R}}^3)$ as the scaling invariant space to (1.2). After their work, Kato [15] and Giga [12] showed the existence of the strong solution of (1.2) for $L^n({\mathbb{R}}^n)$ and Cannone [8] also showed for ${\dot{B}}_{3,\infty }^{-1+3/p}({\mathbb{R}}^3)$ with $3<p\le 6$. For further results in other function spaces, see, e.g., Taylor [27], Giga-Miyakawa [13], Kozono-Yamazaki [22] and Koch-Tataru [16]. For the Keller-Segel system, Biler [2] constructed self-similar solutions in ${\mathbb{R}}^n$, and then Kozono-Sugiyama [20], [21] showed the existence of strong solutions $(u,v)$ of (1.1) in the class

$$u\in C([0,\infty );H^{-2+n/p,p}({\mathbb{R}}^n))\cap C((0,\infty );H^{2,p}({\mathbb{R}}^n))\cap C^1((0,\infty );L^p({\mathbb{R}}^n)),v\in C([0,\infty );H^{n/p,p}({\mathbb{R}}^n))\cap C^1((0,\infty );L^p({\mathbb{R}}^n))$$

for small $(u_0,v_0)\in H^{-2+n/p,p}({\mathbb{R}}^n)\times H^{n/p,p}({\mathbb{R}}^n)$ with $\max \{1,n/4\}<p<n/2$ and

$$u\in BC([0,\infty );L^1({\mathbb{R}}^n))\cap C((0,\infty );H^{2,p}({\mathbb{R}}^n))\cap C^1((0,\infty );L^p({\mathbb{R}}^n)),v\in C([0,\infty );L^1({\mathbb{R}}^n))\cap C((0,\infty );H^{n/p,p}({\mathbb{R}}^n))\cap C^1((0,\infty );L^p({\mathbb{R}}^n))(n/3<\exists p<n/2)$$

for small $(u_0,v_0)\in L^{n/2,\infty }({\mathbb{R}}^n)\times \mathrm{BMO}$ with additional conditions, respectively. Notice that they obtained the strong solutions by showing the regularity of mild solutions.

Compared with previous results, we shall construct a strong solution $(u,v)$ in the class

$${\partial }_{t}u,\mathrm{\Delta }u\in L^{{\alpha }_1,q}((0,T);{\dot{B}}_{r,1}^{s_1}({\mathbb{R}}^n)),{\partial }_{t}v,\mathrm{\Delta }v\in L^{{\alpha }_2,q}((0,T);{\dot{B}}_{r,1}^{s_2}({\mathbb{R}}^n))$$

with $2/{\alpha }_1+n/r=4+s_1$, $2/{\alpha }_2+n/r=2+s_2$ and $1\le q<\infty$ for sufficiently small $0<T<\infty$. To this end, we follow the approach by Kozono-Shimizu [19], i.e., our method relies on the maximal Lorentz regularity theorem for the Keller-Segel system in the homogeneous Besov spaces framework. The crucial point of the approach by Kozono-Shimizu [19] stems from the analysis of the heat equation:

$$\{\begin{array}{cccc}\hfill {\partial }_{t}u-\mathrm{\Delta }u& =f\hfill & \hfill & \text{in }(0,T)\times {\mathbb{R}}^n,\hfill \\ \hfill u(0)& =u_0\hfill & \hfill & \text{in }{\mathbb{R}}^n.\hfill \end{array}$$

The heat equation (1.3) is divided into two parts, namely

$$\{\begin{array}{cccc}\hfill {\partial }_{t}u-\mathrm{\Delta }u& =0\hfill & \hfill & \text{in }(0,T)\times {\mathbb{R}}^n,\hfill \\ \hfill u(0)& =u_0\hfill & \hfill & \text{in }{\mathbb{R}}^n\hfill \end{array}$$

and

$$\{\begin{array}{cccc}\hfill {\partial }_{t}u-\mathrm{\Delta }u& =f\hfill & \hfill & \text{in }(0,T)\times {\mathbb{R}}^n,\hfill \\ \hfill u(0)& =0\hfill & \hfill & \text{in }{\mathbb{R}}^n.\hfill \end{array}$$

For (1.4), by the estimate of the heat semigroup ${\{e^{t\mathrm{\Delta }}\}}_{t\ge 0}$ (cf. Kozono-Ogawa-Taniuchi [17, Lemma 2.2]) and the theory of real interpolation, they obtain

$${\Vert \mathrm{\Delta }{e}^{t\mathrm{\Delta }}{u}_0\Vert }_{L^{\alpha ,q}((0,\infty );{\dot{B}}_{r,1}^s)}\le C{\Vert u_0\Vert }_{{\dot{B}}_{p,q}^{\sigma }},\text{for}\frac{1}{\alpha }=\frac{n}{2}(\frac{1}{p}-\frac{1}{r})+\frac{1}{2}(s+2-\sigma ).$$

On the other hand, for (1.5), by the maximal $L^p$ regularity for the Laplacian (cf. Hieber [14, Corollary 1.6.10]) and real interpolation, they also prove

$${\Vert {\partial }_{t}u\Vert }_{L^{\alpha ,q}((0,T);{\dot{B}}_{r,\beta }^s)}+{\Vert \mathrm{\Delta }u\Vert }_{L^{\alpha ,q}((0,T);{\dot{B}}_{r,\beta }^s)}\le C{\Vert f\Vert }_{L^{\alpha ,q}((0,T);{\dot{B}}_{r,\beta }^s)}.$$

Combining the estimates (1.6) and (1.7), they derive the maximal Lorentz regularity of (1.3), which is employed to prove the existence of the strong solution of (1.2) for initial data $u_0\in {\dot{B}}_{p,q}^{-1+n/p}({\mathbb{R}}^n)$. Here, notice that, the class of solutions to (1.2) is determined from the estimate (1.6). In our situation, however, the Keller-Segel system (1.1) consists of two parabolic-type PDEs so that the suitable choice of function spaces for the initial data ${\dot{B}}_{p_1,q}^{-2+n/p_1}({\mathbb{R}}^n)\times {\dot{B}}_{p_2,q}^{n/p_2}({\mathbb{R}}^n)$ becomes more delicate due to the nonlinear term $-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v)$. Particularly, in the case of $q=\infty$, the initial data $v_0$ belong to ${\dot{B}}_{p_2,\infty }^{n/p_2}({\mathbb{R}}^n)$ with $n/p_2>0$, and hence we cannot directly apply the argument of [19] to show the existence of the local strong solution to (1.1) because the number $n/p_2$ expects to be negative if we follow the approach due to [19].

Let us give a few comments on our results. First, our results are based on the maximal Lorentz regularity estimate for the Laplacian, which is established by Kozono and Shimizu [19]. As far as the author knows, this paper is the first article, which succeeds to apply the maximal Lorentz regularity of the heat equation to a nonlinear parabolic-type PDE excluded the Navier-Stokes system. Namely, arguments used in this paper will work not only for the Keller-Segel system but for other parabolic-type PDEs, especially, for coupling systems of parabolic-type PDEs. The advantage of the maximal Lorentz regularity approach is that we can obtain strong solutions directly, which means that it is not necessary to construct mild solutions — the existence proof of strong solutions becomes more elegant and short.

The main result of this paper reads as follows.

Theorem 1.1

• Let $1\le p_1,p_2<\infty$ and $1\le q<\infty$. Assume that

  $$-\frac{1}{n}<\frac{1}{p_1}-\frac{1}{p_2}<\frac{2}{n}.$$

  For every $u_0\in {\dot{B}}_{p_1,q}^{-2+n/p_1}$ and $v_0\in {\dot{B}}_{p_2,q}^{n/p_2}$, there exist $0<T<\infty$ and a unique solution $(u,v)$ on $(0,T)\times {\mathbb{R}}^n$ of (1.1) in the class

  $${\partial }_{t}u,\mathit{\Delta }u\in L^{{\alpha }_1,q}((0,T);{\dot{B}}_{r,1}^{s_1}),{\partial }_{t}v,\mathit{\Delta }v\in L^{{\alpha }_2,q}((0,T);{\dot{B}}_{r,1}^{s_2})$$

  for some $1<r<\infty ,s_1,s_2\in \mathbb{R}$ and $1<{\alpha }_1<{\alpha }_2<\infty$ such that

  $$\frac{2}{{\alpha }_1}+\frac{n}{r}=4+s_1,\frac{2}{{\alpha }_2}+\frac{n}{r}=2+s_2.$$

  In addition, it holds that

  $$u\in L^{{\alpha }_2,q}((0,T);{\dot{B}}_{r,1}^{s_2}),v\in L^{{\alpha }_2^{⁎},q}((0,T);{\dot{B}}_{r_2^{⁎},1}^{s_2^{⁎}})$$

  for some $r\le r_2^{⁎}\le \infty ,s_2^{⁎}\in \mathbb{R}$ and ${\alpha }_2<{\alpha }_2^{⁎}<\infty$ satisfying

  $$\frac{2}{{\alpha }_2^{⁎}}+\frac{n}{r_2^{⁎}}=s_2^{⁎}.$$

• Let $1\le p_1,p_2<\infty$ satisfy (1.8). Suppose that $1\le q\le \infty$. There exists $\epsilon =\epsilon (n,p_1,p_2,q)>0$ such that if $u_0\in {\dot{B}}_{p_1,q}^{-2+n/p_1}$ and $v_0\in {\dot{B}}_{p_2,q}^{n/p_2}$ satisfy

  $${\Vert u_0\Vert }_{{\dot{B}}_{p_1,q}^{-2+n/p_1}}+{\Vert v_0\Vert }_{{\dot{B}}_{p_2,q}^{n/p_2}}\le \epsilon ,$$

  then we may take $T=\infty$ in (1.9) and (1.10). Concerning the uniqueness for $q=\infty$, there exists a constant $\eta =\eta (n,p_1,p_2,r,s_1,s_2)>0$ such that if $(u_1,v_1)$ and $(u_2,v_2)$ are the solution of (1.1) in the class (1.9) with

  $$\sum _{i=1}^2[\underset{t\to \infty }{\lim \sup }\left\{t\mu {(\tau \in (0,\infty )|{\Vert {\partial }_t{u}_i(\tau )\Vert }_{{\dot{B}}_{r,1}^{s_1}}+{\Vert \mathit{\Delta }{u}_i(\tau )\Vert }_{{\dot{B}}_{r,1}^{s_1}}>t)}^{\frac{1}{{\alpha }_1}}\right\}+\underset{t\to \infty }{\lim \sup }\left\{t\mu {(\tau \in (0,\infty )|{\Vert {\partial }_t{v}_i(\tau )\Vert }_{{\dot{B}}_{r,1}^{s_2}}+{\Vert \mathit{\Delta }{v}_i(\tau )\Vert }_{{\dot{B}}_{r,1}^{s_2}}>t)}^{\frac{1}{{\alpha }_2}}\right\}]\le \eta ,$$

  then it holds that $u_1\equiv u_2$ and $v_1\equiv v_2$.

Remark 1.2

• We have not yet obtained a local strong solution for arbitrary large initial data in the case $q=\infty$ due to the reasons mentioned above.

• Compared with the previous studies due to [20], [21], the initial data belong to the homogeneous Besov spaces. In particular, the initial data $u_0$ of the density of amoebae can be taken as a singular data, e.g., the Dirac measure $\epsilon \delta (x)$ with small coefficient $\epsilon >0$ in 2D case, since $\delta \in {\dot{B}}_{p,\infty }^{-n+n/p}({\mathbb{R}}^n)$.

• In 2D case, it is well-known that there exist global solutions of (1.1) if $u_0\in L^1({\mathbb{R}}^2)$ satisfies the condition ${\int }_{{\mathbb{R}}^2}{u}_{0}dx<8\pi$, without taking $v_0$ small. We should refer to [7], [25] for the result. Their method is mainly based on the a priori estimate of (1.1) by skillful technique. On the other hand, we need smallness assumption on both $u_0\in {\dot{B}}_{p_1,q}^{-2+2/p_1}({\mathbb{R}}^2)$ and $v_0\in {\dot{B}}_{p_2,q}^{2/p_2}({\mathbb{R}}^2)$. Although our result does not yield the threshold initial mass of $u_0$ like 8π, we are able to obtain more general class ${\dot{B}}_{p_1,q}^{-2+2/p_1}({\mathbb{R}}^2)$ of $u_0$ which ensures the existence of global solutions. For instance, by taking $p_1=1$ and $q=\infty$, we see that the class ${\dot{B}}_{1,\infty }^0({\mathbb{R}}^2)$ of $u_0$ with smallness assumption is larger than that in $L^1({\mathbb{R}}^2)$, i.e., $L^1({\mathbb{R}}^2)\subset {\dot{B}}_{1,\infty }^0({\mathbb{R}}^2)$.

Remark 1.3

• We should notice that it is also known results on the existence of global solution of the Keller-Segel system for Neumann problems in bounded domains Ω. Establishment a pioneer work to deal with such a problem might be in the paper [9] by Cao, who obtained global classical solutions of the Neumann problems under the condition that ${\Vert u_0\Vert }_{L^{n/2}(\mathrm{\Omega })}$ and ${\Vert \mathrm{\nabla }{v}_0\Vert }_{L^n(\mathrm{\Omega })}$ are small. Winkler [36] recently treated 1D case and showed the existence of global classical solutions, even if $u_0$ is Radon measures on $\bar{\mathrm{\Omega }}$, whenever $v_0\in L^2(\mathrm{\Omega })$ holds. Besides, the global existence and blow-up phenomena of solutions for Neumann problems have been fully in the series of papers of Winkler [29], [30], [32], [33].

• Moreover, we can expect that one obtain farther-reaching results by considering additional structures of the Keller-Segel system, e.g., with considering logistic sources. For contributions to the asymptotic behavior, blow-up phenomena and instantaneous regularization of the solution of the logistic Keller-Segel system, we refer to [23], [24], [31], [34], [35], [37]. We also note that Biler et al. [3], [4], [5], [6] obtained various results for other settings in the Keller-Segel system.

The rest of this paper is organized as follows. In Section 2, we will recall the notations of functional spaces and the basic propositions. Section 3 shows the maximal Lorentz regularity theorem for the heat equations based on the argument due to Kozono-Shimizu [19]. Section 4 deals with estimating the nonlinear term $-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v)$. Finally, combining the results obtained in Sections 3 and 4, we show our main result, Theorem 1.1, in Section 5.