1. Introduction

The well-known classical parabolic–parabolic Keller–Segel model reads as [Reference Keller and Segel24]

(1.1) \begin{equation} \begin{cases} \partial _t u=\Delta u-\alpha \nabla \cdot (u\nabla v), &\quad x\in \mathbb{R}^2,\quad t\gt 0,\\[5pt] \tau \partial _t v=\Delta v-\beta v+\gamma u, &\quad x\in \mathbb{R}^2,\quad t\gt 0,\\[5pt] \end{cases} \end{equation}

where $u=u(x,t)$ and $v=v(x,t)$ denote the cell density and the concentration of the chemical substance, respectively. $\alpha$ and $\gamma$ are positive constants. The constants $\tau$ and $\beta$ are non-negative. The system (1.1) can be regarded as one of the simplest models to describe the overall behaviour of cells under the influence of chemotaxis, that is the motion of cells partially orient their movement towards higher concentration of a certain chemical substance produced by cells themselves. A striking feature of the Keller–Segel system is that the behaviour of solutions is determined by the total mass of cells which remains constant over time, see [Reference Calvez and Corrias5, Reference Herrero and Velázquez16, Reference Mizoguchi31, Reference Nagai and Ogawa34] for instance. Namely, given a non-negative and suitable smooth initial data $u_0$ , any solution with $m=\|u_0\|_{L^1(\mathbb{R}^2)}\lt 8\pi/(\alpha \gamma )$ exists globally, while blow-up solution appears if $m\gt 8\pi/(\alpha \gamma )$ . Note that the main idea to prove the global existence is based on the following free energy functional,

\begin{align*} \mathcal{F}_{KS}=\int _{\mathbb{R}^2}u\log udx+\frac{\alpha }{2\gamma }\int _{\mathbb{R}^2}\left(|\nabla v|^2+\beta v^2\right)dx-\alpha \int _{\mathbb{R}^2}uvdx, \end{align*}

which is a monotonic non-increasing function with respect to time variable. In view of this fact, Calvez and Corrias use a minimisation principle for entropy functionals and Onofri’s inequality to derive a priori estimates under the sub-critical mass $m\lt 8\pi/(\alpha \gamma )$ , where the assumptions $u_0\log\! \big(1+|x|^2\big)\in L^1(\mathbb{R}^2)$ and $u_0\log u_0\in L^1(\mathbb{R}^2)$ are necessary [Reference Calvez and Corrias5], while these extra assumptions have been removed by applying a modified free energy functional with the Moser–Trudinger inequality in unbounded domain [Reference Mizoguchi31].

For the parabolic–elliptic Keller–Segel system (i.e. taking $\tau =0$ in (1.1)₂)

\begin{align*} 0=\Delta v-\beta v+\gamma u, \end{align*}

the above two-dimensional mass threshold phenomenon also exists. See [Reference Blanchet, Dolbeault and Perthame4, Reference Diaz, Nagai and Rakotoson8, Reference Nagai33] for the global well-posedness results and [Reference Biler and Nadzieja3, Reference Blanchet, Dolbeault and Perthame4] for the blow-up arguments. The main feature to prove the global existence of solutions in this simplified chemotaxis system over (1.1) is that $v$ could be expressed by the fundamental solution of the elliptic equation, then it leads to a single parabolic problem for $u$ . For example, if $\beta =0$ , an explicit expression for $v$ takes form like $v=\gamma K\ast u$ , so (1.1) $_1$ becomes

\begin{align*} \partial _t u=\Delta u-\alpha \gamma \nabla \cdot (u\nabla K\ast u), \quad x\in \mathbb{R}^2,\quad t\gt 0, \end{align*}

where $K=-(1/2\pi )\log |\cdot |$ . A direct application of the logarithmic Hardy–Littlewood–Sobolev inequality (see [Reference Beckner2]) on the corresponding free energy yields the global existence of solutions if $m\lt 8\pi/(\alpha \gamma )$ [Reference Blanchet, Dolbeault and Perthame4].

Compared with the one-population chemotaxis system (1.1), an interesting and complex question is to derive sharp conditions to recognise global existence and blow-up of solutions for the following multi-species chemotaxis model in $\mathbb{R}^2$ ,

(1.2) \begin{equation} \begin{cases} \begin{split} \partial _t u_i=&\Delta u_i-\sum \limits ^m_{j=1}\alpha _{i,j}\nabla \cdot (u_i\nabla v_j), &i\in \mathcal{I}=\{1,\cdots,n\}, \\ \tau _j\partial _t v_j=&\Delta v_j-\beta _jv_j+\sum \limits ^n_{i=1}\gamma _{i,j}u_i, &j\in \mathcal{J}=\{1,\cdots,m\}, \end{split} \end{cases} \end{equation}

where $\tau _j\geq 0$ , $j\in \mathcal{J}$ . This model was first proposed by Wolansky in [Reference Wolansky42] to describe the chemotactic movement of $n$ populations with respect to $m$ chemical substances. Here, $u_i=u_i(x,t)$ denotes the density of $i$ -th population, and $v_j=v_j(x,t)$ represents the concentration of $j$ -th chemical signal. The total number of species $|\mathcal I|=n\geq 1$ is assumed to be finite. $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times m}$ and $\boldsymbol{\gamma }=(\gamma _{i,j})_{n\times m}$ define a pair of $n\times m$ matrices for the sensitivity parameter and the production/consumption rate, respectively. $\beta _j\in \mathbb{R}$ , $j\in \mathcal{J}$ , presents the growth/degradation rate for chemical substance. We introduce $\boldsymbol{\beta }=(\beta _{j,l})_{m\times m}$ with $\beta _{j,l}=\beta _j\delta _{j,l}$ as a $m\times m$ diagonal matrix for convenience, where $\delta _{j,l}$ satisfies

\begin{align*} \delta _{j,l}=\begin{cases} 1,&\text{if}\quad j=l,\\[5pt] 0,&\text{if}\quad j\neq l. \end{cases} \end{align*}

It is very important to understand the multi-species chemotaxis in biology, and this phenomenon has been observed in numerous experiments. We take the following two examples. First one is that a system with two different species, reacting on one common chemical, appears in the cell sorting process during the early aggregation state of mound formation [Reference Weijer40]. And a two-species chemotaxis system with two chemicals has been proposed in [Reference Knútsdóttir, Pálsson and Edelstein-Keshet25] to imitate the breast cancer metastatic process. The readers could see [Reference Horstmann20, Reference Horstmann and Lucia21] for other biological motivations.

Just recently, some authors have started to look more closely at the parabolic–elliptic case of (1.2) (i.e. $\tau _j=0$ ) for $n$ -populations interacting via a self-produced chemical agent. Consider (1.2) with $|\mathcal{I}|=|\mathcal{J}|=n$ is subject to symmetric sensitivity coefficients matrix $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times n}$ with non-negative entries, zero matrix $\boldsymbol{\beta }$ and unit matrix $\boldsymbol{\gamma }$ , that is,

(1.3) \begin{equation} \begin{cases} \partial _t u_i=\Delta u_i-\sum \limits ^n_{j=1}\alpha _{i,j}\nabla \cdot (u_i\nabla v_j), \\[9pt] -\Delta v_j=u_j, \quad i,j \in \mathcal{I}=\{1,\cdots,n\}. \\[5pt] \end{cases} \end{equation}

Karmakar and Wolansky [Reference Karmakar and Wolansky22] had derived the global well-posedness of weak solutions with respect to time in the sub-critical regime

(1.4) \begin{equation} \begin{split} \Lambda _{\mathcal{K}}(\boldsymbol{{m}})\,:\!=\,8\pi \sum \limits _{i\in \mathcal{K}}m_i-\sum \limits _{i,k\in \mathcal{K}}\alpha _{i,k}m_im_k\gt 0, \quad \forall \,\,\emptyset \neq \mathcal{K} \subset \mathcal{I}, \end{split} \end{equation}

where $m_i=\|u_0\|_{L^1(\mathbb{R}^2)}$ . Furthermore, the borderline case of critical mass $\Lambda _{\mathcal{I}}(\boldsymbol{{m}})=0$ , and $\Lambda _{\mathcal{K}}(\boldsymbol{{m}})\gt 0$ , $\forall \,\emptyset \neq \mathcal{K} \subsetneq \mathcal{I}$ , has been considered in [Reference Karmakar and Wolansky23]. It shows that a free energy solution exists globally in time. According to analogous results about (1.1) mentioned above, it is expected that if the condition $\Lambda _{\mathcal{K}}(\boldsymbol{{m}})\geq 0$ for some $\emptyset \neq \mathcal{K} \subset \mathcal{I}$ is violated, a finite-time blow-up solution appears. Using the second-moment techniques in [Reference He and Tadmor15], some solutions of (1.3) blow-up in finite time provided $\Lambda _{\mathcal{I}}(\boldsymbol{{m}})\lt 0$ . While the basic strategy to prove global existence is the logarithmic Hardy–Littlewood–Sobolev inequality for system, see [Reference Shafrir and Wolansky38] for details. In particular, in the case of parabolic–elliptic system (1.2) with $|\mathcal{I}|=2$ , $|\mathcal{J}|=1$ , Espejo et al. [Reference Conca, Espejo and Vilches7, Reference Espejo, Vilches and Conca12] discovered a threshold curve to help us to determine whether the solutions of system are blow-up or global in time. See related works for parabolic–elliptic system (1.2) with $|\mathcal{I}|=|\mathcal{J}|=2$ in [Reference Hong, Wang and Wang18, Reference Hong, Wang, Yu and Zhang19]. Moreover, [Reference Espejo, Stevens and Suzuki9– Reference Espejo, Stevens and Velázquez11, Reference Espejo, Vilches and Conca13] could be refereed to characterise the simultaneous or non-simultaneous blow-up results in two-species model.

However, it should be noted that fewer papers have been considered on Cauchy problem of the fully parabolic multi-species (i.e. $\tau _j\gt 0$ in (1.2)) than the parabolic–elliptic case. For a two-dimensional bounded domain, the author and coauthors have researched the initial boundary problems of (1.2), and we have tried to find optimal conditions on the global existence or blow-up in [Reference Lin27–Reference Lin and Zhong30]. In this article, under a conflict-free situation given by Definition 1 $(ii)$ , a sufficient (or possibly optimal) condition on the global solvability of the Cauchy problem for parabolic–parabolic system (1.2) with arbitrary $|\mathcal{I}|=n\geq 1$ and $|\mathcal{J}|=m\geq 1$ has been obtained. For simplicity, we assume $\tau _j=1$ for all $j\in \mathcal{J}$ in (1.2) and consider

(1.5) \begin{equation} \begin{cases} \partial _t u_i=\Delta u_i-\sum \limits ^m_{j=1}\alpha _{i,j}\nabla \cdot (u_i\nabla v_j), &i\in \mathcal{I}, \\[5pt] \partial _t v_j=\Delta v_j-\beta _jv_j+\sum \limits ^n_{i=1}\gamma _{i,j}u_i, &j\in \mathcal{J}, \\[5pt] u_{i0}(x)=u_i(x,0), v_{j0}(x)=v_j(x,0),&i\in \mathcal{I},\quad j\in \mathcal{J}, \end{cases} \end{equation}

with initial data $\boldsymbol{{u}}_0=(u_{10},\cdots,u_{n0})$ and $ \boldsymbol{{v}}_0=(v_{10},\cdots,v_{m0})$ satisfying

(1.6) \begin{equation} \begin{cases} &u_{i0}(x)\in C^0(\mathbb{R}^2)\cap L^1(\mathbb{R}^2)\cap L^1\!\left(\mathbb{R}^2,\log \!\left(1+|x|^2\right)dx\right)\cap L^{\infty }(\mathbb{R}^2),\\[5pt] &u_{i0}\geq 0\quad \text{and}\quad u_{i0}\not \equiv 0,\quad i\in \mathcal{I},\\[5pt] &v_{j0}(x)\in W^{1,p}(\mathbb{R}^2)\cap W^{1,1}(\mathbb{R}^2)\quad \text{with some}\,\,p\gt 2,\quad j\in \mathcal{J}. \end{cases} \end{equation}

Before stating our main results, let us go over some notations and definitions in [Reference Wolansky42].

\begin{align*} \lambda _{i,k}\,:\!=\,\sum ^{m}_{j=1}\alpha _{i,j}\gamma _{k,j}=\boldsymbol{\alpha }^T_i\cdot \boldsymbol{\gamma }_k,\quad i,k\in \mathcal{I},\end{align*}

is the number to quantify the tendency of a population $i$ towards a population $k$ by taking an accounting of the action of all the agents, where $\boldsymbol{\alpha }_i=(\alpha _{i,1},\cdots,\alpha _{i,m})^T$ and $\boldsymbol{\gamma }_i=(\gamma _{i,1},\cdots,\gamma _{i,m})^T$ . The condition $\lambda _{i,k}\gt 0$ means that a population $i$ is attracted by a population $k$ ; otherwise, the population $i$ is repelled from the population $k$ if $\lambda _{i,k}\lt 0$ . Especially, a population is self-attracting (resp. self-repelling) if $\lambda _{i,i}\gt 0$ (resp. $\lambda _{i,i}\lt 0$ ). A pair $(i,k)$ of populations $i,k\in I$ is said to be in a conflict if $\lambda _{i,k}\times \lambda _{k,i}\lt 0$ . In general, $\boldsymbol{\lambda }=(\lambda _{i,k})_{i,k\in \mathcal{I}}$ is not symmetric. We assume that there exist nonzero constants $a_1,\cdots,a_n$ satisfying

(1.7) \begin{equation} a_i\lambda _{i,k}=a_{k}\lambda _{k,i},\quad i,k\in \mathcal{I}, \end{equation}

then $D_{\boldsymbol{{a}}}\boldsymbol{\lambda }$ is symmetric, where $D_{\boldsymbol{{a}}}=\text{Diag}\{a_1,\cdots,a_n\}$ . If $\boldsymbol{\lambda }$ is non-singular, there exists a $m\times m$ symmetric matrix $\boldsymbol{{B}}$ which transforms $\boldsymbol{\gamma }_i$ into $a_i\boldsymbol{\alpha }_i$ for all $i\in \mathcal{I}$ , i.e.

(1.8) \begin{equation} \boldsymbol{{B}}\boldsymbol{\gamma }_i=a_i\boldsymbol{\alpha }_i,\quad i\in \mathcal{I}. \end{equation}

In fact, denote

\begin{align*} \boldsymbol{\alpha }=(\alpha _{i,j})_{n\times m}=\left [ \begin{array}{c} \boldsymbol{\alpha }^T_1\\[5pt] \vdots \\[5pt] \boldsymbol{\alpha }^T_n \end{array} \right ],\,\, \boldsymbol{\gamma }=(\gamma _{i,j})_{n\times m}=\left [ \begin{array}{c} \boldsymbol{\gamma }^T_1\\[5pt] \vdots \\[5pt] \boldsymbol{\gamma }^T_n \end{array} \right ]. \end{align*}

First, we observe that the ranks $R(\boldsymbol{\alpha })=R(\boldsymbol{\gamma })=n$ due to $\boldsymbol{\lambda }=\boldsymbol{\alpha }\boldsymbol{\gamma }^T$ is non-singular. Then, there exists a solution $X=(x_{i,j})_{m\times m}$ to a linear system of equations $ \boldsymbol{\gamma }X=D_{\boldsymbol{{a}}}\boldsymbol{\alpha }$ since both the ranks of its coefficient matrix and augmented matrix equal to $n$ . Finally, the choice $\boldsymbol{{B}}=X^T$ fulfils (1.8). Moreover, using the symmetry of $D_{\boldsymbol{{a}}}\boldsymbol{\lambda }$ , one is able to show that $ \boldsymbol{\gamma }\boldsymbol{{B}}^T\boldsymbol{\gamma }^T=\boldsymbol{\gamma }\boldsymbol{{B}}\boldsymbol{\gamma }^T$ . This implies that $\boldsymbol{{B}}$ can be symmetric.

Now we give the following definitions throughout this paper.

The main result of this article is stated as follows.

Theorem 1.1. Let $\boldsymbol{\gamma }=(\gamma _{i,j})_{n\times m}$ , $\boldsymbol{\lambda }=(\lambda _{i,k})_{n\times n}$ , $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times m}$ with full column rank $R(\boldsymbol{\alpha })=m$ , and $\boldsymbol{\beta }=(\beta _{j,l})_{m\times m}$ with $\beta _{j,l}=\beta _j\delta _{j,l}$ , $\beta _{j}\in \mathbb{R}$ , $j,l\in \mathcal{J}$ . Assume that (1.5) is a conflict-free system with initial data $(\boldsymbol{{u}}_0,\boldsymbol{{v}}_0)$ satisfying (1.6). Suppose that there exist positive constants $a_1,\cdots,a_n$ and a positive definite matrix $\boldsymbol{{R}}=(r_{i,k})_{n\times n}$ with $r_{i,k}\geq 0$ , $i,k\in \mathcal{I}$ , such that

(1.9) \begin{equation} \boldsymbol{\alpha }^T\boldsymbol{{R}}^{-1}\boldsymbol{\alpha }\boldsymbol{\gamma }_i=a_i\boldsymbol{\alpha }_i,\quad \forall \,\,i\in \mathcal{I}. \end{equation}

Then for any initial data satisfying

(1.10) \begin{equation} \Lambda ^{\boldsymbol{{a}}}_{\mathcal{K}}(\boldsymbol{{m}})\,=\!:\, 8\pi \sum \limits _{i\in \mathcal{K}}a_im_i-\sum \limits _{i,k\in \mathcal{K}}a_ia_kr_{i,k}m_im_k\gt 0, \quad \forall \,\,\emptyset \neq \mathcal{K} \subset \mathcal{I}, \end{equation}

the corresponding initial boundary value problem (1.5) possesses a unique smooth global solution.

We would like to give an explanation for assumptions in Theorem 1.1. First, since $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times m}$ is required to be full column rank, it ensures that $\boldsymbol{{B}}=(b_{j,l})_{m\times m}=\boldsymbol{\alpha }^T\boldsymbol{{R}}^{-1}\boldsymbol{\alpha }$ is a positive definite matrix if $\boldsymbol{{R}}$ is chosen to be positive definition. Condition (1.9) can probably be viewed as one of necessary condition for the existence of energy functional for the conflict-free system (1.5) (see [Reference Horstmann20, Reference Lin and Zhong30, Reference Wolansky42]). In order to handle the whole domain case better in this paper, we use a modified free energy functional $\mathcal{F}$ as

\begin{align*} \begin{split} \mathcal{F}[\boldsymbol{{u}},\boldsymbol{{v}}]=&\sum \limits ^n_{i=1}a_i\int _{\mathbb{R}^2}(u_i+1)\log\! (u_i+1)dx+\frac{1}{2}\sum \limits ^{m}_{j=1}\sum \limits ^m_{l=1}b_{j,l}\int _{\mathbb{R}^2}\left (\nabla v_j\cdot \nabla v_l+\beta _{l}v_jv_l\right )dx\\[5pt] &-\sum \limits ^n_{i=1}\sum \limits ^{m}_{j=1}a_i\alpha _{i,j}\int _{\mathbb{R}^2}u_iv_jdx. \end{split} \end{align*}

Second, condition (1.10) can be regarded as an optimality condition to guarantee the global existence of solution to (1.5). This is because if $D_{\boldsymbol{{a}}}\boldsymbol{\lambda }$ is a positive definite matrix with $\lambda _{i,k}\geq 0$ , $\forall \,i,k\in \mathcal{I}$ , then (1.10) is actually equivalent to the following sub-critical mass condition obtained in [Reference Lin and Zhong30, Reference Wolansky42]

\begin{align*} 8\pi \sum \limits _{i\in \mathcal{K}}a_im_i-\sum \limits _{i,k\in \mathcal{K}}a_i\lambda _{i,k}m_im_k\gt 0, \quad \forall \,\,\emptyset \neq \mathcal{K} \subset \mathcal{I}. \end{align*}

Indeed, let $\boldsymbol{{R}}=(r_{i,k})_{n\times n}=D^{-1}_{\boldsymbol{{a}}}\boldsymbol{\lambda }^T$ . Then in terms of $\lambda _{i,k}=\boldsymbol{\alpha }^T_i\cdot \boldsymbol{\gamma }_k$ and $R(\boldsymbol{\alpha })=m$ , $\boldsymbol{{B}}=\boldsymbol{\alpha }^{T}{\boldsymbol{{R}}}^{-1}\boldsymbol{\alpha }$ is a positive definite matrix which satisfies $\boldsymbol{{B}}\boldsymbol{\gamma }_i=a_i\boldsymbol{\alpha }_i$ , $\forall \,\,i\in \mathcal{I}$ . Moreover, we obtain

\begin{align*} 8\pi &\sum \limits _{i\in \mathcal{K}}a_im_i-\sum \limits _{i,k\in \mathcal{K}}a_i\lambda _{i,k}m_im_k= 8\pi \sum \limits _{i\in \mathcal{K}}a_im_i-\sum \limits _{i,k\in \mathcal{K}}a_ia_kr_{i,k}m_im_k\gt 0,\quad \forall \,\,\emptyset \neq \mathcal{K} \subset \mathcal{I}, \end{align*}

from $r_{i,k}={\lambda _{k,i}}/a_i\geq 0$ , $i,k\in \mathcal{I}$ .

Theorem 1.1 gives a sharp criterion on the global existence of the general chemotaxis system (1.5). Hence, a large amount of known global existence results are particular cases in our paper. We give several typical examples here. It is obvious that the sub-critical mass condition (1.10) recovers the threshold condition, i.e. $m\lt 8\pi/(\alpha \gamma )$ , for global regularity of the Keller–Segel model (1.1). When $|\mathcal{I}|=2$ , $|\mathcal{J}|=1$ , consider a chemotaxis system involving two species that interact via one-single chemical signal [Reference Weijer40]

(1.11) \begin{equation} \left\{\begin{array}{l} \partial _t u_i=\Delta u_i-\alpha _{i,1}\nabla \cdot (u_i\nabla v_1),\quad i\in \mathcal{I}=\{1,2\}, \\[5pt] \partial _tv_{1}=\Delta v_1-v_1+\gamma _{1,1}u_1+\gamma _{2,1}u_2, \end{array}\right. \end{equation}

with $\alpha _{i,1}\gt 0$ , $\gamma _{i,1}=1$ , $i=1,2$ . Note that $\lambda _{i,k}=\alpha _{i,1}\gt 0$ , $i,k=1,2$ . Then, (1.9) can be satisfied if one takes $a_i=1/\alpha _{i,1}$ and chooses a positive definite matrix

\begin{align*} \boldsymbol{{R}}=\left [ \begin{array}{c@{\quad}c} \alpha ^2_{1,1}(1+\epsilon )&\alpha _{1,1}\alpha _{2,1}(1-\epsilon )\\[5pt] \alpha _{1,1}\alpha _{2,1}(1-\epsilon )&\alpha ^2_{2,1}(1+\epsilon ) \end{array} \right ] \end{align*}

for some small $\epsilon \in (0,1)$ . Then, (1.10) reads as

(1.12) \begin{equation} \begin{cases} 8\pi \gt \alpha _{1,1}(1+\epsilon )m_1,\\[5pt] 8\pi \gt \alpha _{2,1}(1+\epsilon )m_2,\\[5pt] 8\pi \!\left (\dfrac{m_1}{\alpha _{1,1}}+\dfrac{m_2}{\alpha _{2,1}}\right )\gt (m_1+m_2)^2+\epsilon (m_1-m_2)^2. \end{cases} \end{equation}

However, (1.12) can be simplified as $m_1\lt 8\pi/\alpha _{1,1}$ , $m_2\lt 8\pi/\alpha _{2,1}$ and $(m_1+m_2)^2\lt 8\pi \!\left (m_1/\alpha _{1,1}+m_2/\alpha _{2,1}\right )$ by letting $\epsilon \rightarrow 0$ , which coincides with global existence condition for (1.11) in a bounded domain ([Reference Lin27, Theorem 1.1]).

Now suppose that $|\mathcal{I}|=|\mathcal{J}|=n$ , $\boldsymbol{\beta }=\boldsymbol{0}$ , $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times n}$ with $\alpha _{i,j}\geq 0$ is positive definite, and $\boldsymbol{\gamma }$ is an unit matrix. Then, (1.5) becomes

(1.13) \begin{equation} \begin{cases} \partial _t u_i=\Delta u_i-\sum \limits ^n_{j=1}\alpha _{i,j}\nabla \cdot (u_i\nabla v_j), \\[8pt] \partial _tv_{j}=\Delta v_j+u_j, \quad i,j \in \mathcal{I}=\{1,\cdots,n\}. \\[5pt] \end{cases} \end{equation}

Taking $a_i=1$ and $\lambda _{i,k}=\alpha _{i,k}$ , $i,k\in \mathcal{I}$ , one can find that Cauchy problem (1.13) has a global solution if (1.4) is valid. This global result is similar to that of the parabolic–elliptic system (1.3) [Reference He and Tadmor15, Reference Karmakar and Wolansky22].

The idea to show the global existence is to derive an a prior estimate for modified total entropy

\begin{align*} \mathcal{S}[\boldsymbol{{u}}]=\sum \limits ^n_{i=1}\|(u_i+1)\log\! (u_i+1)\|_{L^1(\mathbb{R}^2)}. \end{align*}

For this purpose, we need to give a lower bound for $\mathcal{F}$ . In this situation, the last term consisting of $u_i$ and $v_j$ in $\mathcal{F}$ could be controlled by $\mathcal{S}$ and the last second term under (1.10). For the case of single variable, a common approach to achieve this goal is to use the well-known Moser–Trudinger inequality [Reference Moser32, Reference Trudinger39]

(1.14) \begin{equation} \frac{1}{2}\int _{\Omega }|\nabla \rho |^2dx-8\pi \log \!\left (\int _{\Omega }e^{\rho }dx\right )\geq -C,\quad \forall \,\,\rho \in H^1_0(\Omega ), \end{equation}

where $\Omega \subset \mathbb{R}^2$ is a bounded domain. For initial Neumann boundary value problem (1.1), Nagai et al. [Reference Nagai, Senba and Yoshida36] had used a version of (1.14) in the Sobolev space $W^{1,2}$ (see also [Reference Chang and Yang6]) to obtain the global existence of solution if $m\lt 8\pi/(\alpha \gamma )$ . Later, Mizoguchi [Reference Mizoguchi31] applies (1.14) to get a similar global result for Cauchy problem (1.1) in $\mathbb{R}^2$ . Hence, we expect the Moser–Trudinger inequality for vector is valid for our problem. Shafrir and Wolansky [Reference Shafrir and Wolansky38] had proved the following Moser–Trudinger inequality for system. For $\forall \,\,\rho _i\in H^1(\mathbb{S}^2)$ satisfying $\int _{\mathbb{S}^2}\rho _i=0,\,\,i\in \mathcal{I}$ , there exists a constant $C\gt 0$ such that

\begin{align*} \Phi _{\mathbb{S}^2}(\boldsymbol{\rho })= \frac{1}{2}\sum _{i,k\in \mathcal{I}}s_{i,k}\int _{\mathbb{S}^2}\nabla \rho _i\cdot \nabla \rho _k -\sum _{i\in \mathcal{I}}M_i\log \!\left (\frac{1}{4\pi }\int _{\mathbb{S}^2}\exp \!\left(\sum \limits _{k\in \mathcal{I}}s_{i,k}\rho _k\right)\right )\geq -C\end{align*}

if and only if

\begin{align*} \left \{ \begin{array}{l} \Lambda ^{\boldsymbol{{S}}}_{\mathcal{K}}(\boldsymbol{{M}})\geq 0,\quad \forall \,\emptyset \neq \mathcal{K}\subset \mathcal{I},\\[9pt] \text{if}\,\,\Lambda ^{\boldsymbol{{S}}}_{\mathcal{K}}(\boldsymbol{{M}})=0\,\,\text{for some}\,\,\mathcal{K},\,\,\text{then}\,\,s_{i,i}+\Lambda ^{\boldsymbol{{S}}}_{\mathcal{K}\setminus \{i\}}(\boldsymbol{{M}})\gt 0,\,\,\forall \,{i\in \mathcal{K}}, \end{array} \right . \end{align*}

where $\mathbb{S}^2\subset \mathbb{R}^3$ is the unit sphere, $\mathcal{I}=\{1,2,\cdots,n\}$ , $\boldsymbol{{M}}\,:\!=\,\{M_1,\cdots,M_n\}\in (\mathbb{R}_{+})^n$ , $\boldsymbol{{S}}\,:\!=\,(s_{i,k})_{n\times n}$ is a positive definite matrix with $s_{i,k}\geq 0$ , $i,k\in \mathcal{I}$ , and $\Lambda ^{\boldsymbol{{S}}}_{\mathcal{K}}$ is given by

\begin{align*} \Lambda ^{\boldsymbol{{S}}}_{\mathcal{K}}(\boldsymbol{{M}})=8\pi \sum \limits _{i\in \mathcal{K}}M_i-\sum \limits _{i,k\in \mathcal{K}}s_{i,k}M_iM_k,\quad \forall \,\,\emptyset \neq \mathcal{K}\subset \mathcal{I}. \end{align*}

We will firstly transform the above Moser–Trudinger inequality for system to $\mathbb{R}^2$ via the stereographic projection and next use it to show that $\mathcal{S}$ is bounded under the assumption (1.10). Then, one invokes the Moser iterative to obtain the global existence of solutions to (1.5). Finally, we should point out that such idea has been used to establish the global well-posedness of solutions to initial Neumann boundary value problem for multi-species and chemicals in a two-dimensional bounded domain [Reference Lin and Zhong30].

Our second object is to show certain conflict-free parabolic system admits a global solution for any initial data. More precisely,

Theorem 1.2. Let $\boldsymbol{\alpha }=(\alpha _{i,j})_{n\times m}$ , $\boldsymbol{\gamma }=(\gamma _{i,j})_{n\times m}$ , $\boldsymbol{\lambda }=(\lambda _{i,k})_{n\times n}$ and $\boldsymbol{\beta }=(\beta _{j,l})_{m\times m}$ with $\beta _{j,l}=\beta _j\delta _{j,l}$ , $\beta _{j}\in \mathbb{R}$ , $j,l\in \mathcal{J}$ . Assume that (1.5) is a conflict-free system with initial data $(\boldsymbol{{u}}_0,\boldsymbol{{v}}_0)$ satisfying (1.6). Suppose that there exist positive constants $a_1,\cdots,a_n$ and a positive definite matrix $\boldsymbol{{B}}=(b_{j,l})_{m\times m}$ such that

\begin{align*} \boldsymbol{{B}}\boldsymbol{\gamma }_i=-a_i\boldsymbol{\alpha }_i,\quad \forall \,\,i\in \mathcal{I}, \end{align*}

then Cauchy problem (1.5) possesses a unique smooth global solution.

Compared with the proof of Theorem 1.1, the main difference to prove Theorem 1.2 is to derive a prior estimates for the modified total entropy $\mathcal{S}$ through the following modified free energy functional

\begin{align*} \mathcal{G}[\boldsymbol{{u}},\boldsymbol{{v}}]=&\sum \limits ^n_{i=1}a_i\int _{\mathbb{R}^2}(u_i+1)\log\! (u_i+1)dx+\frac{1}{2}\sum \limits ^{m}_{j=1}\sum \limits ^m_{l=1}b_{j,l}\int _{\mathbb{R}^2}\left (\nabla v_j\cdot \nabla v_l+\beta _{l}v_jv_l\right )dx. \end{align*}

This paper is organised as follows. In Section 2, we would like to establish the local existence of smooth solutions and present some basic inequalities. Theorem 1.1 will be proved in Section 3. The existence of modified free energy functional $\mathcal{F}$ will be shown firstly, then the stepwise bounds on the total entropy, $L^2$ and $L^{\infty }$ norms under the condition (1.10) will end the proof of this theorem. In Section 4, we will prove Theorem 1.2 by making use of $\mathcal{G}$ . In appendix, the proof of Lemma 2.1 will be contained.

We introduce some notations which will be used later. Let $|\mathcal I|=n\geq 1$ denote the total number of species, and $|\mathcal{J}|=m\geq 1$ represent the total number of chemical substances. $|\Omega |$ is the Lebesgue measure of $\Omega \subset \mathbb{R}^2$ . For $\alpha _{i,j}$ , $\gamma _{i,j}$ , $b_{i,j}$ , $r_{i,k}$ , $t_{i,j}$ , $a_i$ , $\beta _{j}\in \mathbb{R}$ , $i,k\in \mathcal{I}$ , $j\in \mathcal{J}$ , we define

\begin{align*} \alpha ^* & =\max \limits _{i\in \mathcal{I},j\in \mathcal{J}}\{|\alpha _{i,j}|\}, \,\,\gamma ^*=\max \limits _{i\in \mathcal{I},j\in \mathcal{J}}\{|\gamma _{i,j}|\},\,\,b^*=\max \limits _{i\in \mathcal{I},j\in \mathcal{J}}\{|b_{i,j}|\},\,\,r^*=\max \limits _{i,k\in \mathcal{I}}\{|r_{i,k}|\},\\ t^* & =\max \limits _{i\in \mathcal{I},j\in \mathcal{J}}\{|t_{i,j}|\},\,\,a^*=\max \limits _{i\in \mathcal{I}}\{|a_i|\}, \,\,\beta ^*=\max \limits _{j\in \mathcal{J}}\{|\beta _j|\},\,\,\beta _*=\min \limits _{j\in \mathcal{J}}\{\beta _j\}\,\,\text{if}\,\,\beta _j\gt 0. \end{align*}