Abstract
This paper deals with a two-species attraction–repulsion chemotaxis system
under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\Omega \subseteq {\mathbb {R}}^{n}\), where \(\tau \in \{0,1\},\xi _{i},\chi _{i}>0\) and \(f_{i}(u,w)(i=1,2)\) satisfy
with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\). It is proved that in any space dimension \(n\ge 1\), the above system possesses a unique global and uniformly bounded classical solution regardless of \(\tau =0\) or \(\tau =1\) under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states.
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1 Introduction
To describe some complex biological processes (such as cell sorting process [30]) in two species, we consider the following two-species attraction–repulsion chemotaxis system with competitive and nonlocal kinetic terms
where \(\tau \in \{0,1\},\xi _{i},\chi _{i}>0(i=1,2)\), \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) is a smoothly bounded domain and \(f_{1},f_{2}\) satisfy
with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\). Here the movements of the two populations are described by random diffusion (i.e. \(\Delta u,\Delta w\)), chemoattractant (i.e. \(-\xi _{1}\nabla \cdot (u\nabla v),-\xi _{2}\nabla \cdot (w\nabla z)\)) and chemorepellent (i.e. \(+\chi _{1}\nabla \cdot (u\nabla z),+\chi _{2}\nabla \cdot (w\nabla v)\)). Moreover, in view of the classical Lotka–Volterra [26] and nonlocal dynamics, we assume that both populations reproduce and compete (or cooperate) themselves, and mutually compete (or cooperate) with the other. u(x, t) and w(x, t) represent the density of two populations, and v(x, t), z(x, t) stand for the concentrations of the chemical substances. \(\xi _{1},\xi _{2}\) refer to the chemoattraction sensitivity coefficients and \(\chi _{1},\chi _{2}\) stand for the chemorepulsion sensitivity coefficients. The biological significance of these parameters \(a_{i},b_{i}(i=0,1,\ldots \cdot \cdot ,4)\) is explained in [28]. The initial data \((u_{0},\tau v_{0},w_{0},\tau z_{0})\) is nonnegative and satisfies
Chemotaxis, a directed movement of biological cells or organisms in response to the concentration gradient of a chemical signal, is well known to play a significant role in a wide range of biological applications, such as pattern formation [5], embryonic development [18], wound healing [32] and blood vessel formation [7], etc. Notably, Keller and Segel [16] introduced the pioneering works of the chemotaxis which describes the aggregation of cellular slime molds Dictyostelium discoideum
where \(\xi \in {\mathbb {R}}\) and \(\tau \in \{0,1\}\). Here u(x, t) and v(x, t) represent the density of Dictyostelium discotylum and the concentration of chemical signals, respectively. Many scholars have studied deeply the system (1.4) in the past decades. For instance, when \(\xi =\tau =1\), all solutions of system (1.4) are globally bounded in one-dimensional domains [29]. If the initial mass \(\int _{\Omega }u_{0}dx\) of cells is small (i.e. \(\int _{\Omega }u_{0}dx<4\pi \)), then the solution of system (1.4) also is global and bounded in two-dimensional domains [9], whereas the initial mass of cells is large enough (i.e. \(\int _{\Omega }u_{0}dx>4\pi \) and \(\int _{\Omega }u_{0}dx\in \{4K\pi ,K\in {\mathbb {N}}\}\)) then the aggregation occurs either in finite or infinite time [33]. Considering the volume-filling effect [31], Winkler [41] proved that the aggregation of cells happens in finite time. Moreover, Tao and Winkler [35] obtained that solutions of the above system are global and uniformly bounded. Of course, there are a large amount of results that include the global existence, boundedness and blow-up behavior of solutions to variants of the above system, we can refer to [6, 10,11,12,13, 17, 27, 36, 43,44,45].
To further understand system (1.1), let us mention some previous advances. When the system (1.1) has no repulsive mechanisms (i.e. \(\chi _{1}=\chi _{2}=0\)) and nonlocal dynamics (i.e. \(a_{3}=a_{4}=b_{3}=b_{4}=0\)), system (1.1) becomes the following two-species and two-stimuli chemotaxis system with Lotka–Volterra competitive terms
where \(\tau \in \{0,1\}\) and \(\alpha ,\beta ,\gamma ,\delta ,\xi _{i},\mu _{i},a_{i}>0(i=1,2)\). For the parabolic-elliptic-parabolic-elliptic case (i.e. \(\tau =0\)), if the production efficiency of the signals v, z is the same as the consumption (i.e. \(\alpha =\beta =\delta =\gamma =1\)), Zheng and Mu [51] obtained that all solutions of system (1.5) are globally bounded in two-dimensional domains. When \(a_{1},a_{2}\in (0,1)\), Zheng et al. [52] showed that the two species can maintain constant coexistence stable state. Furthermore, in two-dimensional domains, when \(\mu _{1}=\mu _{2}=0\) in (1.5), Yu et al. [46] proved that aggregations of species occurs if the initial masses \(\int _{\Omega }u_{0}dx\) and \(\int _{\Omega }w_{0}dx\) are large sufficiently (i.e. \(\int _{\Omega }u_{0}dx\cdot \int _{\Omega }u_{0}dx-2\pi (\frac{\int _{\Omega }u_{0}dx}{\xi _{1}\beta }+\frac{\int _{\Omega }w_{0}dx}{\xi _{2}\delta })>0\)) and that globally bounded solutions exist if the initial masses \(\int _{\Omega }u_{0}dx\) and \(\int _{\Omega }w_{0}dx\) are small enough (i.e. \(\max \{\int _{\Omega }u_{0}dx,\int _{\Omega }w_{0}dx\}<4\pi \)) and \(\xi _{1}=\xi _{2}=\alpha =\beta =\delta =\gamma =1\). On the other hand, due to existence of Lotka–Volterra competition, Tu et al. [38] derived the coexistence stable state of two species under the weak competition (i.e. \(a_{1},a_{2}\in (0,1)\)) and semi-trivial equilibrium under strong competition (\(a_{1}\ge 1>a_{2}>0\)). Recently, the previous results of [38] was improved by Wang and Mu in [39]. For the fully parabolic case (i.e. \(\tau =1\)), relying on the maximal Sobolev regularity, Zheng and Mu [51] derived that when the chemotactic sensitivities are small enough as related to the Lotka–Volterra competitive terms in the sense that \(\frac{\xi _{2}}{\mu _{1}}<\theta _{0}\) and \(\frac{\xi _{1}}{\mu _{2}}<\theta _{0}\) for some \(\theta _{0}>0\), then system (1.5) possesses a globally bounded classical solution in any space dimension \(n\ge 1\). For more related contents, we can refer to [19, 20, 22, 47,48,49, 53]. However, to the best of our knowledge the literature does not provide any qualitative analysis on the solution behavior when attraction–repulsion chemotaxis as well as nonlocal competitive Lokta-Volterra competitive terms involving both species are present.
So far several special cases for system (1.1) are studied by some scholars. For example, when \(\tau =0,n=2\) and system (1.1) has no kinetic terms (i.e. \(f_{1}=f_{2}\equiv 0\)), the nonradial solutions blow up in finite time [21]. Considering the small initial masses and repulsive mechanisms (i.e. \(\chi _{1},\chi _{2}>0\)), the global boundedness of solutions is established under the conditions that \(n=2\) and \(\max \{\int _{\Omega }u_{0}dx,\int _{\Omega }w_{0}dx\}<\frac{4}{(\chi _{1}+\chi _{2}+\xi _{1}+\xi _{2})C_{GN}}\) or \(\chi _{1}=\chi _{2}=\xi _{1}=\xi _{2}>0\) in [21]. Similarly, due to the effect of repulsive mechanisms (i.e. \(\chi _{1},\chi _{2}>0\)), Liu and Dai [23] proved the global boundedness of solutions to system (1.1) if \(\min \{\chi _{1},\chi _{2}\}>\xi _{1}+\xi _{2}\) holds. Without nonlocal terms in (1.1) (i.e. \(a_{3}=a_{4}=b_{3}=b_{4}=0\)), Zheng and Hu [50] obtained the constant equilibrium and semi-trivial equilibrium of two species. In addition, when \(f_{1}\) and \(f_{2}\) satisfy (1.2), Zheng et al. [54] studied the fully parabolic two-species chemotaxis system with indirect signal production, and derived that the global boundedness and stability of the constant steady state under some suitable assumptions. Recently, when \(f_{1}\) and \(f_{2}\) satisfy (1.2) with \(a_{0},a_{1},b_{0},b_{2}>0,a_{2},a_{3},a_{4},b_{1},b_{3},b_{4}\in {\mathbb {R}}\), the boundedness and stabilization of global solutions for system (1.1) were derived by Hu et al. in [14]. However, the authors only considered the constant coexistence stable state under the locally intraspecific competition and globally interspecific cooperation (i.e. \(a_{1},a_{4},b_{2},b_{3}>0,a_{2},b_{1}<0\)) in [14].
As analyzed above, the boundedness of global solutions to system (1.1) still remains open under locally competitive and nonlocal kinetic terms. For the fully competitive case (i.e. \(a_{i},b_{i}>0(i=1,2),a_{j},b_{j}<0(j=3,4)\)), the large time behavior of global solutions to system (1.1) is also an unsolved question. Hence, this paper gives an affirmative respond. For simplicity, we introduce a notation that is \((a)_{+}:=\max \{0,a\}\) for all \(a\in {\mathbb {R}}\). Our main results are stated as follows.
Firstly, we consider the boundedness of solutions for (1.1) under the case \(\tau =0\).
Theorem 1.1
Let \(\tau =0,\xi _{i},\chi _{i}>0(i=1,2)\) and \(\Omega \subseteq {\mathbb {R}}^{n}\) be a smoothly bounded domain. Assume that \(f_{1},f_{2}\) satisfy (1.2) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\), and the initial data \((u_{0},w_{0})\) satisfies (1.3). If the following conditions hold:
-
\(n=1\) and
$$\begin{aligned} a_{1}>\bigg ((a_{3})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |, b_{2}>\bigg ((b_{4})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |; \end{aligned}$$(1.6) -
\(n\ge 2\), (1.6) and
$$\begin{aligned} \min \big \{\chi _{1}+a_{1},\chi _{2}+b_{2}\big \}>\xi _{1}+\xi _{2} \end{aligned}$$(1.7)or
$$\begin{aligned} \xi _{1}<\frac{na_{2}}{n-2},\xi _{2}<\frac{nb_{1}}{n-2}. \end{aligned}$$(1.8)Then system (1.1) admits a unique global classical solution (u, v, w, z), which is uniformly bounded in \(\Omega \times (0,\infty )\) in the sense that there exists \(C>0\) independent of t such that
$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z\Vert _{L^{\infty }(\Omega )}\le C\;\;for\;all\;t>0. \end{aligned}$$
Remark 1.1
When \(\tau =0\) and \(\chi _{1}=\chi _{2}=a_{3}=a_{4}=b_{3}=b_{4}=0\), the results of Theorem 1.1 cover those of [38, 50]. When \(n\ge 2\) and \(a_{2},b_{1}>0\), Theorem 1.1 not only includes the conditions \(\min \big \{\chi _{1}+a_{1},\chi _{2}+b_{2}\big \}>\xi _{1}+\xi _{2}\) of [14], but also expands the conditions \(\xi _{1}<\frac{na_{2}}{n-2}\) and \(\xi _{2}<\frac{nb_{1}}{n-2}\). Moreover, compared with the results in [21], we can remove the small initial condition \(\max \big \{\Vert u_{0}\Vert _{L^{1}(\Omega )},\Vert w_{0}\Vert _{L^{1}(\Omega )}\big \}<\frac{4}{(\chi _{1}+\chi _{2}+\xi _{1}+\xi _{2})C_{GN}}\) in this paper.
Next, we study the global stability of bounded solution for (1.1) when \(\tau =0\). When weak interspecific competition occurs, i.e.,
holds with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\), then the unique positive constant steady state \((u_{*},v_{*},w_{*},z_{*})\) can be recorded as
Theorem 1.2
Let the conditions in Theorem 1.1 and (1.9) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\) hold. Assume that system (1.1) admits a unique global classical solution (u, v, w, z) with the property
for all \(t\ge 1,\) where \(K>0\) and \(\vartheta \in (0,1)\). Suppose that there exist \(\theta _{1},\theta _{2}\in (0,1)\) such that
and
Then for some fixed time \(t_{1}>0\), there exist \(C>0\) and \(\lambda _{1}>0\) such that
for all \(t>t_{1}\), where \((u_{*},v_{*},w_{*},z_{*})\) is given in (1.10).
Now, we consider the strong competition case, i.e.,
with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\), then the unique semi-trivial equilibrium \((u_{\star },0,0,z_{\star })\) is
Theorem 1.3
Let the conditions in Theorem 1.1 and (1.15) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\) hold. Assume that system (1.1) admits a unique global classical solution (u, v, w, z) with the property (1.11). Suppose that there exist \(\theta _{3},\theta _{4}\in (0,1)\) such that
and
Then the unique global solution (u, v, w, z) has the following properties:
-
(i)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}\), then for some fixed time \(t_{2}>0\) there exist \(C>0\) and \(\lambda _{2}>0\) such that
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le Ce^{-\lambda _{2}t} \end{aligned} \end{aligned}$$(1.19)for all \(t>t_{2}\), where \((u_{\star },z_{\star })\) is given by (1.16).
-
(ii)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}=\frac{a_{0}}{b_{0}}\), then for some fixed time \(t_{3}>0\) there exist \(C>0\) and \(\lambda _{3}>0\) satisfying
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C(t-t_{3})^{-\lambda _{3}} \end{aligned}\nonumber \\ \end{aligned}$$(1.20)for all \(t>t_{3}\), where \((u_{\star },z_{\star })\) is given by (1.16).
On the other hand, we are concerned with the fully parabolic case, i.e. \(\tau =1\).
Theorem 1.4
Let \(\tau =1,\xi _{i},\chi _{i}>0(i=1,2)\) and \(\Omega \subseteq {\mathbb {R}}^{n}\) be a smoothly bounded domain. Assume that \(f_{1},f_{2}\) satisfy (1.2) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\) and the initial data \((u_{0},v_{0},w_{0},z_{0})\) satisfies (1.3). Moreover, suppose that
-
\(n=1,2\), the condition (1.6);
-
\(n=3\), the condition
$$\begin{aligned} \left\{ \begin{aligned} a_{1}&>\max \bigg \{\bigg ((a_{3})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |,\frac{7\xi _{1}^{2}}{b_{1}}+2b_{1},\frac{7\chi _{1}^{2}}{b_{1}}+2b_{1}+28\bigg (\frac{a_{2}^{2}}{b_{1}}+b_{1}\bigg )\bigg \},\\ b_{2}&>\max \bigg \{\bigg ((b_{4})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |,\frac{7\chi _{2}^{2}}{a_{2}}+2a_{2},\frac{7\xi _{2}^{2}}{a_{2}}+2a_{2}+28\bigg (\frac{b_{1}^{2}}{a_{2}}+a_{2}\bigg )\bigg \}; \end{aligned} \right. \nonumber \\ \end{aligned}$$(1.21) -
\(n\ge 4\), the condition
$$\begin{aligned} \left\{ \begin{aligned} a_{1}&>\max \bigg \{\bigg ((a_{3})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |,\xi _{1}+\chi _{1}+C_{S}(\chi _{1}+\xi _{2})\bigg \},\\ b_{2}&>\max \bigg \{\bigg ((b_{4})_{+}+\frac{(a_{4})_{+}+(b_{3})_{+}}{2}\bigg )|\Omega |,\xi _{2}+\chi _{2}+C_{S}(\chi _{2}+\xi _{1})\bigg \} \end{aligned} \right. \end{aligned}$$(1.22)are satisfied, where \(C_{S}\) is given in Lemma 2.7. Then system (1.1) admits a unique global classical solution (u, v, w, z), which is uniformly bounded in \(\Omega \times (0,\infty )\) in the sense that there exists \(C>0\) independent of t such that
$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z\Vert _{L^{\infty }(\Omega )}\le C\;\;for\;all\;t>0. \end{aligned}$$
Remark 1.2
When \(\tau =1\) and \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\), Theorem 1.4 includes the result of [14] under the effect of locally interspecific competition. If \(n=2\) and \(\chi _{1}=\chi _{2}=a_{3}=a_{4}=b_{3}=b_{4}=0\), compared with the results in [22], Theorem 1.4 only requires the conditions \(a_{1},b_{2}>0\), which improves the conditions \(a_{1}>\xi _{1}+\xi _{2}C_{S}\) and \(b_{2}>\xi _{2}+\xi _{1}C_{S}\) in [22]. Moreover, the small initial condition in [23] can also be removed in this paper.
Finally, we study the global stability of bounded solutions for (1.1) when \(\tau =1\).
Theorem 1.5
Let the conditions in Theorem 1.4 and (1.9) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\) hold. Assume that system (1.1) admits a unique global classical solution (u, v, w, z) with the property (1.11). Suppose that the conditions (1.12) and (1.13) are satisfied, then there exist \(C>0\) and \(\kappa _{1}>0\) such that
for all \(t>t_{4}\), where \(t_{4}>0\) is some fixed time and \((u_{*},v_{*},w_{*},z_{*})\) is given in (1.10).
Theorem 1.6
Let the conditions in Theorem 1.4 and (1.15) with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}<0(j=3,4)\) hold. Assume that system (1.1) admits a unique global classical solution (u, v, w, z) with the property (1.11). Suppose that the conditions (1.17) and (1.18) are satisfied. Then the unique global solution (u, v, w, z) has the following properties:
-
(i)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}\), then for some fixed time \(t_{5}>0\) there exist \(C>0\) and \(\kappa _{2}>0\) satisfying
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le Ce^{-\kappa _{2}t} \end{aligned} \end{aligned}$$(1.24)for all \(t>t_{5}\), where \((u_{\star },z_{\star })\) is given by (1.16).
-
(ii)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}=\frac{a_{0}}{b_{0}}\), then for some fixed time \(t_{6}>0\) there exist \(C>0\) and \(\kappa _{3}>0\) such that
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C(t-t_{6})^{-\kappa _{3}} \end{aligned}\nonumber \\ \end{aligned}$$(1.25)for all \(t>t_{6}\), where \((u_{\star },z_{\star })\) is given by (1.16).
Remark 1.3
In this paper, we only consider the large time behavior of the global solutions to system (1.1) under the \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{0}}{b_{0}}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}\) and \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}\le \frac{a_{0}}{b_{0}}\), respectively. For the strong competition case, it is not difficult to obtain the large time behavior by the same method used in the proof of Theorem 1.2 (or Theorem 1.6). However, for the fully strong competition case \(\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}<\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}\), there is still an open problem about stabilization of global bounded solutions.
This paper is organized as follows. In Sect. 2, we give the local existence of solution to system (1.1) and some preliminary lemmas. In Sect. 3, we study the global existence and boundedness of solutions to system (1.1) with \(\tau =0\), and prove Theorem 1.1. In Sect. 4, we study the asymptotic behavior of global solutions to system (1.1) with \(\tau =0\), and prove Theorem 1.2 and Theorem 1.3. In Sect. 5, we discuss the global existence and boundedness of solutions to system (1.1) with \(\tau =1\), and prove Theorem 1.4. In Sect. 6, we consider the asymptotic behavior of global solutions to system (1.1) when \(\tau =1\), and prove Theorem 1.5 and Theorem 1.6. In addition, we let \(u(\cdot ,t):=u(x,t)\) and omit signs dx during integrating for brevity thoughout this paper.
2 Preliminaries
In this section, we shall give several preliminary lemmas. Firstly, we state the local existence of solutions for (1.1).
Lemma 2.1
Let \(\tau \in \{0,1\},\xi _{i},\chi _{i}>0(i=1,2)\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that \(f_{1},f_{2}\) satisfy (1.2), and the initial data (\(u_{0},\tau v_{0},w_{0},\tau z_{0}\)) satisfies (1.3). Then there exist \(T_{\max }\in (0,\infty ]\) and uniquely determined nonnegative functions
such that (u, v, w, z) solves system (1.1) classically in \(\Omega \times (0,T_{\max })\), where \(q>n\). Moreover, if \(T_{max}<\infty \), then
Proof
This proof is similar to [42, Theorem 1.1] or [34, Lemma 2.1]. For more details, please refer to [14, Lemma 2.1]. The proof of Lemma 2.1 is complete. \(\square \)
Secondly, we shall give a uniform \(L^{1}\)-bound of (u, v, w, z) for (1.1).
Lemma 2.2
Let (u, v, w, z) be a solution to system (1.1) and the nonnegative initial data \((u_{0},\tau v_{0},w_{0},\tau z_{0})\) satisfies (1.3) with \(\tau \in \{0,1\}\). Suppose that the condition (1.6) holds. Then
and
for all \(t\in (0,T_{\max })\), where
Proof
Integrating the first equation in (1.1), we deduce from Young’s and Hölder’s inequalities that
for all \(t\in (0,T_{\max })\), where we have used that \(a_{2}>0\). By the similar method to \(w-\)equation and \(b_{1}>0\), we get
Then, combining (2.5) and (2.6), we have
Then the following proof of this lemma is similar to [14, Lemma 2.2]. Here we omit the details. The proof of Lemma 2.2 is complete. \(\square \)
Lemma 2.3
(See [37, ChapterIII, Lemma 5.1]) Let \(\Phi (t)\ge 0\) satisfy
for \(\Phi _{0}\ge 0\) with some constants \(k_{1},k_{2}>0\) and \(\theta >0\). Then
Lemma 2.4
(See [8]) Let \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a bounded domain with smooth boundary. Assume that \(p,k>0,m\in [0,k)\) and \(q,r\in [1,\infty ]\) hold. Then for any \(\Psi \in W^{k,q}(\Omega )\cap L^{r}(\Omega )\), there exists \(C_{GN}=C(k,q,r,\Omega )>0\) such that
where \(\alpha \) satisfies
and \(D^{k}\psi \) is expressed as Fr\(\acute{e}\)chet derivative of order k.
Lemma 2.5
(See [24, Lemma 4.2]) Let \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a bounded domain with smooth boundary and \(\Phi \in C^{2}(\Omega )\) satisfy \(\frac{\partial \Phi }{\partial \nu }=0\). Then
where \(C_{\Omega }\) is a positive constant depending only on the curvatures of \(\partial \Omega \).
Lemma 2.6
(See [15]) Let \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a bounded domain with smooth boundary. Then for any \(\varepsilon >0\), there exists \(C_{\varepsilon }>0\) such that
Lemma 2.7
(See [40, Lemma 2.3]) Let \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain and \(0\le t_{0}<T_{\max }\le \infty \). Assume that \(u_{0}\in W^{2,p}(\Omega )(p>n)\) with \(\partial _{\nu }u_{0}=0\) on \(\partial \Omega \). Then for each \(g\in L^{p}\big ((0,T_{\max });L^{p}(\Omega )\big )\), the problem
possesses a unique solution \(u\in W^{1,p}\big ((0,T_{\max });L^{p}(\Omega )\big )\cap L^{p}\big ((0,T_{\max }); W^{2,p}(\Omega )\big ).\) Moreover, if \(u(\cdot ,t_{0})\in W^{2,p}(\Omega )\) satisfies \(\frac{\partial u(\cdot ,t_{0})}{\partial \nu }=0\) on \(\partial \Omega ,\) then there exists \(C_{S}=C(p)>0\) such that
for all \(t\in (t_{0},T_{\max })\).
3 Boundedness for \(\tau =0\)
The aim of this section is to show the global boundedness of solution for (1.1) with \(\tau =0\) and prove Theorem 1.1.
Lemma 3.1
Let \(\tau =0\) and \(\Omega \subseteq {\mathbb {R}}\) be a smoothly bounded domain. Suppose that \(a_{2},b_{1}>0\) and (1.6) hold. Then there exists \(C_{1}>0\) such that
for all \(t\in (0,T_{\max })\).
Proof
By a straightforward computation and Lemma 2.2, we deduce from \(a_{2}>0\) and Young’s inequality that
for all \(t\in (0,T_{\max })\), where \(a_{5}:=\max \big \{(a_{3})_{+},(a_{4})_{+}\big \}\) and \(C_{2}=\frac{(2a_{0}+1+4a_{5}M_{0})^{3}}{27a_{1}^{2}}|\Omega |\). Similarly, we get
with \(C_{3}=\frac{(2b_{0}+1+4b_{5}M_{0})^{3}}{27b_{2}^{2}}|\Omega |\), where we have used that \(b_{1}>0\).
Combining (3.2) with (3.3), we get
for all \(t\in (0,T_{\max })\).
Then the following proof of this lemma is similar to [14, Lemma 3.1]. Here we omit the details. The proof of Lemma 3.1 is complete. \(\square \)
Lemma 3.2
Let \(\tau =0\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 2)\) be a smoothly bounded domain. Suppose that \(a_{2},b_{1}>0\), (1.6) and (1.7) or (1.8) hold. Then for some \(p>p_{0}\), there exists \(C_{4}>0\) such that
for all \(t\in (0,T_{\max })\), where
Proof
Multiplying the first equation in (1.1) by \(pu^{p-1}(p>p_{0}\ge 1)\) and integrating by parts over \(\Omega \), it follows from Young’s inequality and Lemma 2.2 that
for all \(t\in (0,T_{\max })\), where \(a_{5}:=\max \big \{(a_{3})_{+},(a_{4})_{+}\big \}\). Using Lemma 2.2, Lemma 2.4 and Young’s inequality, there exists a positive constant \(C_{5}\) such that
with \(C_{6}=(1-\alpha _{1})C_{5}^{\frac{1}{1-\alpha _{1}}}\bigg (\dfrac{p\alpha _{1}}{4(p-1)}\bigg )^{\frac{\alpha _{1}}{1-\alpha _{1}}}+C_{5}M_{0}^{p}>0\), where \(\alpha _{1}=\dfrac{\frac{np}{2}-\frac{n}{2}}{1-\frac{n}{2}+\frac{np}{2}}\in (0,1)\) because of \(p>1\). Inserting (3.7) into (3.6), one has
for all \(t\in (0,T_{\max })\).
Similarly, there is a positive constant \(C_{7}\) such that
for all \(t\in (0,T_{\max })\).
for all \(t\in (0,T_{\max })\).
Case 1: \(\xi _{1}+\xi _{2}<\min \big \{\chi _{1}+a_{1},\chi _{2}+b_{2}\big \}\).
It follows from (1.7) that
which implies that the constants \(K_{1}:=a_{1}p+\chi _{1}(p-1)-\big (\xi _{1}+\xi _{2}\big )(p-1)\) and \(K_{2}:=b_{2}p+\chi _{2}(p-1)-\big (\xi _{1}+\xi _{2}\big )(p-1)\) are positive. Then we make use of the method in [14, Lemma 3.2] to obtain (3.5). Here we omit the details.
Case 2: \(\xi _{1}<\frac{na_{2}}{n-2},\xi _{2}<\frac{nb_{1}}{n-2}\).
It follows from the condition (1.8) with \(n\ge 2\) that the definitions
are well-defined. Then we fix \(p\in I_{1}\cap I_{2}\), which implies \(\xi _{1}(p-1)-a_{2}p<0\) and \(\xi _{2}(p-1)-b_{1}p<0\). By (3.10) and Young’s inequality, we have
With aids of the Agmon-Douglis-Nirenberg \(L^{p}\) estimates (see [1, 2]) and (3.20) of [14], then for some \(p>\frac{n}{2}\), there exist some positive constants \(C_{9}\) and \(C_{10}\) such that
and
for all \(t\in (0,T_{\max })\).
Combining (3.11)–(3.13) yields
with \(C_{11}=C_{8}+C_{9}+C_{10}\). Then according to Lemma 2.3, one obtains
for all \(t\in (0,T_{\max })\) and some \(p>\frac{n}{2}\), which implies (3.5). The proof of Lemma 3.2 is complete. \(\square \)
Lemma 3.3
Let \(\tau =0\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\), (1.6) and (1.7) or (1.8) hold. Then for all nonnegative initial data \((u_{0},w_{0})\) satisfying (1.3), there exists \(C_{12}>0\) such that
for all \(t\in (0,T_{\max })\).
Proof
When \(n=1\), relying on Lemma 3.1 and the method in [51, Lemma 3.6], we can obtain that (3.15) holds. When \(n\ge 2\), it follows from Lemma 3.2 and the Moser-Alikakos iteration in [3] (or [35, Lemma A.1]) that (3.15) holds. The proof of Lemma 3.3 is complete. \(\square \)
Proof of Theorem 1.1
By Lemma 3.3, we obtain the boundedness of \(\Vert u\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}\) for all \(t\in (0,T_{\max })\). By the well-known elliptic maximum principle and global boundedness of (u, w), we derive that (v, z) is bounded in \(W^{1,q}(\Omega )(q>n)\). Hence it follows from Lemma 2.1 that \(T_{\max }=\infty \). The proof of Theorem 1.1 is complete. \(\square \)
4 Asymptotic Behavior for \(\tau =0\)
In this section, we prove Theorems 1.2 and 1.3 by constructing some suitable energy functionals. To do this, we need the following key lemmas.
Lemma 4.1
(See [4, Lemma 3.1]) Let \(f(t):(1,\infty )\rightarrow {\mathbb {R}}\) be a nonnegative and uniformly continuous function that satisfies \(\int _{1}^{\infty }f(t)dt<\infty .\) Then \(f(t)\rightarrow 0\) as \(t\rightarrow \infty .\)
4.1 Proof of Theorem 1.2
Lemma 4.2
Suppose that the conditions of Theorem 1.2 hold. Then there exists \(\delta _{1}>0\) such that
for all \(t>0\), where
and \((u_{*},v_{*},w_{*},z_{*})\) is given by (1.10). Moreover, we have
Proof
Letting
then \({\mathcal {F}}_{1}(t)\) can be rewritten as
for all \(t>0\).
Firstly, by (5.4) of [14], we get the nonnegativity of \({\mathcal {F}}_{1}(t)\) for all \(t\ge 0\).
Now, we will prove (4.1). By a simple calculation with (1.1) that
It follows from Hölder’s and Young’s inequalities that
due to \(a_{3},a_{4}<0\) and \(a_{0}=(a_{1}-a_{3}|\Omega |)u_{*}+(a_{2}-a_{4}|\Omega |)w_{*}\). Therefore, we have
Making use of the similar method for \({\mathcal {B}}_{1}(t)\), we obtain
because of \(b_{0}=(b_{1}-b_{3}|\Omega |)u_{*}+(b_{2}-b_{4}|\Omega |)w_{*}\) and \(b_{3},b_{4}<0\).
Collecting (4.3), (4.5) and (4.6) yields
By Young’s inequality, we derive
and
as well as
Then,
In view of \(0=\Delta v-v+w\) and \(v_{*}=w_{*}\), we obtain
Similarly, we have
Together with (4.8)–(4.10), we get
Since the conditions (1.12) and (1.13) hold, then the constants
are positive, and there exist \(\theta _{1},\theta _{2}\in (0,1)\) such that
and
By (4.12)–(4.14) and \(u_{*},w_{*}>0\), one can find some \(\delta _{1}>0\) to satisfy
Therefore, we get
where
For each the discriminant of \(y_{i}(i=1,2,3)\), we dedure from (4.15) that
which concludes
By (4.16) and (4.18), we directly obtain (4.1). Finally, integrating (4.1) over \((0,\infty )\), we get
The proof of Lemma 4.2 is complete. \(\square \)
Lemma 4.3
Suppose that the conditions of Theorem 1.2 hold. Then the solution (u, v, w, z) to (1.1) satisfies
as \(t\rightarrow \infty \).
Proof
A combination of Lemmas 4.1 and 4.2 implies this lemma. The proof of Lemma 4.3 is complete. \(\square \)
Lemma 4.4
Suppose that the conditions of Theorem 1.2 hold. Then there exist \(C_{1}>0\) and \(\lambda _{1}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
for all \(t>t_{1}\), where \(t_{1}>0\) is some fixed time.
Proof
This idea of proof is similar to [4, Lemma 3.7].
Set \(g_{1}(s)=s-u_{*}\ln s\) and \(g_{2}(s)=s-w_{*}\ln s\) for \(s>0\). By L’Hôpital’s rule, we have
and
By Lemma 4.3, one can see that there exist some \(t_{1}>0\) and \(C_{2},C_{3}>0\) such that
and
for all \(t>t_{1}\).
By means of the definitions of \({\mathcal {F}}_{1}(t)\), it follows from the second inequalities in (4.23) and (4.24) that there exists \(C_{4}>0\) such that
for all \(t>t_{1}\). With the aid of Lemma 4.2, we get
which implies
for all \(t>t_{1}\). Then combining (4.27) with the first inequalities in (4.23) and (4.24), there exists \(C_{5}>0\) such that
for all \(t>t_{1}\).
By applying the Gagliardo-Nirenberg inequality and (1.15), there exist some positive constants \(C_{6},C_{7}\) and \(C_{8}\) such that
for all \(t>t_{1}\).
By the application of the elliptic maximum principle and (4.29), we get
where we have used that \(\tau =0\) and \(v_{*}=w_{*},z_{*}=u_{*}\). The proof of Lemma 4.4 is complete. \(\square \)
Proof of Theorem 1.2
Lemma 4.4 directly shows the results of Theorem 1.2. \(\square \)
4.2 Proof of Theorem 1.3
Now, we introduce the following functionals
where \((u_{\star },z_{\star })\) is given by (1.16).
Lemma 4.5
Let (u, v, w, z) be a global bounded classical solution to (1.1). Suppose that the conditions of Theorem 1.3 hold. Then there exists \(\delta _{2}>0\) such that
for all \(t>0\). Moreover, we have
Proof
This proof is similar to Lemma 4.2. For reader’s convenience, we give the sketch of the proof. Firstly, \({\mathcal {F}}_{2}(t)\) can be rewritten as
where \({\mathcal {A}}_{2}(t):=\int _{\Omega } \bigg (u-u_{\star }-u_{\star }\ln \dfrac{u}{u_{\star }}\bigg )\) and \({\mathcal {B}}_{2}(t):=\int _{\Omega }w\). Similarly, we obtain the nonnegativity of \({\mathcal {F}}_{2}(t)\) by (5.4) of [14].
Next, we will prove (4.31). It follows from Young’s and Hölder’s inequalities that
where we have used that \(a_{0}=a_{1}u_{\star }-a_{3}u_{\star }|\Omega |\) and \(a_{3},a_{4}<0\).
Based on the third equation of (1.1), we deduce from \(b_{3},b_{4}<0\), Hölder’s and Young’s inequalities that
Combining (4.33) with (4.34) obtains
Then making once again use of (4.9) and (4.10), we get
It follows from (1.17) and (1.18) that \({\mathcal {K}}_{3}:=a_{1}+\frac{a_{4}}{2}|\Omega |+\frac{b_{3}}{2}|\Omega |,{\mathcal {K}}_{4}:=b_{2}+\frac{a_{4}}{2}|\Omega |+\frac{b_{3}}{2}|\Omega |\) are positive, and there exist \(\theta _{3},\theta _{4}\in (0,1)\) such that
and
as well as
According to (4.36)–(4.38) and \(u_{\star },w_{\star }>0\), one can find some \(\delta _{2}>0\) to satisfy
Therefore, we get
where
For each the discriminant of \(h_{i}(i=1,2,3)\) and by (4.39), we have
which concludes
By (4.40) and (4.42), we directly obtain (4.31). Moreover, we get
by integrating (4.31) over \((0,\infty )\). The proof of Lemma 4.5 is complete. \(\square \)
Lemma 4.6
Suppose that the conditions of Theorem 1.3 hold. Then the solution (u, v, w, z) to (1.1) satisfies
as \(t\rightarrow \infty \).
Proof
A combination of Lemma 4.1 and Lemma 4.5 implies this lemma. The proof of Lemma 4.6 is complete. \(\square \)
Lemma 4.7
Let the conditions of Theorem 1.3 hold.
-
(i)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}\), then there exist \(C_{9}>0\) and \(\lambda _{2}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C_{9}e^{-\lambda _{2}t} \end{aligned} \end{aligned}$$(4.44)for all \(t>t_{2}\), where \(t_{2}>0\) is some fixed time.
-
(ii)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}=\frac{a_{0}}{b_{0}}\), then there exist \(C_{10}>0\) and \(\lambda _{3}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C_{10}(t-t_{3})^{-\lambda _{3}} \end{aligned}\nonumber \\ \end{aligned}$$(4.45)for all \(t>t_{3}\), where \(t_{3}>0\) is some fixed time.
Proof
-
(i)
When \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}\), by using Lemma 4.6 and a similar argument as in the proof of [25, Lemma 4.3], there exist some \(t_{2}>0\) and \(C_{11},C_{12}>0\) such that
$$\begin{aligned}&C_{11}\int _{\Omega }\bigg ((u-u_{\star })^{2}+v^{2}+w^{2}+(z-z_{\star })^{2}+w\bigg )\le {\mathcal {F}}_{2}(t)\nonumber \\&\quad \le C_{12}\int _{\Omega }\bigg ((u-u_{\star })^{2}+v^{2}+w^{2}+(z-z_{\star })^{2}+w\bigg ) \end{aligned}$$(4.46)for all \(t>t_{2}\). Then we infer from Lemma 4.4 that there exists \(C_{13}>0\) such that
$$\begin{aligned} \frac{d}{dt}{\mathcal {F}}_{2}(t)\le -C_{13}\delta _{1} {\mathcal {F}}_{2}(t), \end{aligned}$$(4.47)which implies
$$\begin{aligned} {\mathcal {F}}_{2}(t)\le {\mathcal {F}}_{2}(t_{2})e^{-C_{13}\delta _{1}(t-t_{2})} \end{aligned}$$(4.48)for all \(t>t_{2}\). Finally, by the same argument as in the proof of Lemma 4.4, we obtain (4.44).
-
(ii)
When \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}=\frac{a_{0}}{b_{0}}\), it follows from the definition of \({\mathcal {F}}_{2}(t)\), (4.23) and Hölder’s inequality that there exist \(t_{3}>0\) and \(C_{14}>0\) such that
$$\begin{aligned} \begin{aligned} {\mathcal {F}}_{2}(t)\le&\int _{\Omega }\bigg (C_{3}(u-u_{\star })^{2}+v^{2}+w+(z-z_{\star })^{2}\bigg )\\&\le C_{3}\bigg (\int _{\Omega }(u-u_{\star })^{2}\bigg )^{\frac{1}{2}}\bigg (\int _{\Omega }(u-u_{\star })^{2}\bigg )^{\frac{1}{2}}+\bigg (\int _{\Omega }v^{2}\bigg )^{\frac{1}{2}}\bigg (\int _{\Omega }v^{2}\bigg )^{\frac{1}{2}}\\&\quad +|\Omega |^{\frac{1}{2}}\bigg (\int _{\Omega }w^{2}\bigg )^{\frac{1}{2}}+\bigg (\int _{\Omega }(z-z_{\star })^{2}\bigg )^{\frac{1}{2}}\bigg (\int _{\Omega }(z-z_{\star })^{2}\bigg )^{\frac{1}{2}}\\&\le C_{14}\bigg (\int _{\Omega }(u-u_{\star })^{2}+\int _{\Omega }v^{2}+\int _{\Omega }w^{2}+\int _{\Omega }(z-z_{\star })^{2}\bigg ) \end{aligned} \end{aligned}$$(4.49)for all \(t>t_{3}\), where we have used the boundedness of solution (u, v, w, z). Thus we duduce from Lemma 4.5 that there exists \(C_{15}>0\) such that
$$\begin{aligned} \frac{d}{dt}{\mathcal {F}}_{2}(t)\le -C_{15}\delta _{1}{\mathcal {F}}_{2}^{2}(t), \end{aligned}$$(4.50)which implies
$$\begin{aligned} {\mathcal {F}}_{2}(t)\le \dfrac{C_{16}}{t-t_{3}} \end{aligned}$$(4.51)for all \(t>t_{3}\) with some \(C_{16}>0\).
With the aids of (4.23) and (4.51), we find \(C_{17}>0\) such that
for all \(t>t_{3}\).
By the similar method as (4.29) and (4.30), we obtain (4.45). \(\square \)
Proof of Theorem 1.3
Lemma 4.7 directly gives the statement of Theorem 1.3. \(\square \)
5 Boundedness for \(\tau =1\)
The aim of this section is to show the global boundedness of solution to (1.1) with \(\tau =1\) and prove Theorem 1.4. Firstly, we give the extensibility and regularity of solution to system (1.1), which will be used to confirm the global existence and boundedness of solutions.
Lemma 5.1
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.6) hold. Suppose that there exists \(p_{0}\ge 1\) such that \(p_{0}>\frac{n}{2}\) and
Then \(T_{\max }=\infty \) and
Moreover, there exist \(\vartheta \in (0,1)\) and \(C_{1}>0\) independent of t such that
for all \(t>1\).
Proof
For the last term \(f_{1}(u,w)\) in the \(u-\)equation of (1.1), we deduce
with \(a_{5}:=\max \big \{(a_{3})_{+},(a_{4})_{+}\big \}\), where we have used \(a_{2}>0\) and Lemma 2.2. We use the same method for \(w-\)equation. Then the following proof of this lemma is similar to [4, Lemma 2.6]. The proof of Lemma 5.1 is complete. \(\square \)
Lemma 5.2
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.6) hold. Then there exists \(C_{2}>0\) such that
for all \(t\in (0,T_{\max }-\tau _{0})\), where \(\tau _{0}:=\min \{1,\frac{T_{\max }}{2}\}\).
Proof
This proof is similar to [14, Lemma 4.2]. Here we omit the details for brevity. \(\square \)
Lemma 5.3
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.6) hold. Then there exists \(C_{3}>0\) such that
and
for all \(t\in (0,T_{\max })\), as well as
and
for all \(t\in (0,T_{\max }-\tau _{0})\), where \(\tau _{0}:=\min \{1,\frac{T_{\max }}{2}\}\).
Proof
This proof is similar to [14, Lemma 4.3]. The proof of Lemma 5.3 is complete. \(\square \)
When \(n=2\), we shall establish \(L^{2}-\)bound of u and w, which is essential to obtain \(L^{\infty }-\)bound of u and w.
Lemma 5.4
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{2}\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.6) hold. Then there exists \(C_{4}>0\) such that
and
for all \(t\in (0,T_{\max })\).
Proof
Multiplying the first equation in (1.1) by 2u and integrating by parts over \(\Omega \), we deduce from Lemma 2.2 and Young’s inequality that
for all \(t\in (0,T_{\max })\), where we have used that \(a_{2}>0\) and \(a_{5}:=\max \big \{(a_{3})_{+},(a_{4})_{+}\big \}\).
Similarly, we get
by \(b_{1}>0\) for all \(t\in (0,T_{\max })\), where \(b_{5}:=\max \big \{(b_{3})_{+},(b_{4})_{+}\big \}\). Then combining (5.12) with (5.13) and using Young’s inequality, we have
for all \(t\in (0,T_{\max })\), where \(C_{5}>0\).
Then the following proof of this lemma is similar to [14, Lemma 4.4]. Here we omit the details. The proof of Lemma 5.4 is complete. \(\square \)
When \(n=3\), in order to obtain the \(L^{\infty }-\)bound of u and w, we still need to establish \(L^{2}-\)bound of u and w.
Lemma 5.5
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{3}\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.21) holds. Then there exists \(C_{6}>0\) such that
for all \(t\in (0,T_{\max })\).
Proof
We shall prove this lemma from the following steps:
Step 1. Based one (5.12), we apply Young’s inequality and Lemma 2.2 to obtain
for all \(t\in (0,T_{\max })\), where \(a_{5}:=\max \big \{(a_{3})_{+},(a_{4})_{+}\big \}\) and we have used the condition that \(a_{2}>0\). By the similar way on \(w-\)equation, we obtain
for all \(t\in (0,T_{\max })\), where \(b_{5}:=\max \big \{(b_{3})_{+},(b_{4})_{+}\big \}\) and we have used the condition that \(b_{1}>0\).
Step 2. Notice that
for all \(t\in (0,T_{\max })\). Relying on the first and third equations in (1.1), it follows from Lemma 2.2 that
for all \(t\in (0,T_{\max })\). Applying Young’s inequality once again, we obtain
and
as well as
for all \(t\in (0,T_{\max })\).
It follows from the second and fourth equation in (1.1) that
for all \(t\in (0,T_{\max })\), where we have used that \(2\nabla v\cdot \nabla \Delta v=\Delta |\nabla v|^{2}-2\left| D^{2} v\right| ^{2}\) and \(2\nabla z\cdot \nabla \Delta z=\Delta |\nabla z|^{2}-2\left| D^{2} z\right| ^{2}\). Likewise, applying Young’s inequality for the last term in (5.23), there is
for all \(t\in (0,T_{\max })\).
Thus, in view of (5.18)–(5.24), we derive
for all \(t\in (0,T_{\max })\).
Step 3. It follows from the second equation in (1.1) and \(2\nabla v\cdot \nabla \Delta v=\Delta |\nabla v|^{2}-2\left| D^{2} v\right| ^{2}\) that
for all \(t\in (0,T_{\max })\). By Young’s inequality and the pointwise inequality \(|\Delta v|^{2}\le n|D^{2}v|^{2}\) with \(n=3\),
and
Thus, combining (5.26)–(5.28) implies
for all \(t\in (0,T_{\max })\).
Similarly,
for all \(t\in (0,T_{\max })\).
Step 4. As a consequence of (5.16), (5.17), (5.25), (5.29) and (5.30), we get
for all \(t\in (0,T_{\max })\).
It follows from (1.21) that these constants \(7\xi _{1}^{2}+2b_{1}^{2}-a_{1}b_{1},7\chi _{1}^{2}+2b_{1}^{2}+28(a_{2}^{2}+b_{1}^{2})-a_{1}b_{1},7\chi _{2}^{2}+2a_{2}^{2}-a_{2}b_{2},7\xi _{2}^{2}+2a_{2}^{2}+28(a_{2}^{2}+b_{1}^{2})-a_{2}b_{2}\) are negative. Then, from (5.31) we obtain
for all \(t\in (0,T_{\max })\).
Step 5. Let
and
for all \(t\in (0,T_{\max })\). Hence, we deduce from (5.32) that
for all \(t\in (0,T_{\max })\).
It follows from Young’s inequality, Lemmas 2.4 and 4.3 that
with some constants \(C_{7}>0\). Applying Young’s inequality, Lemma 2.2, Lemma 2.5 and Lemma 2.6 as well as Lemma 4.3, we obtain
for all \(t\in (0,T_{\max })\), where \(C_{8},C_{9}\) and \(C_{10}\) are positive constants.
Inserting (5.34) and (5.35) into (5.33), we apply Young’s inequality to obtain
with some \(C_{11}>0\). One can get
in view of Lemma 2.3. The proof of Lemma 5.5 is complete. \(\square \)
When \(n\ge 4\), we shall show \(L^{p}-\)bound so as to obtain \(L^{\infty }-\)bound of u and w.
Lemma 5.6
Let \(\tau =1\) and \(\Omega \subseteq {\mathbb {R}}^{n}(n\ge 4)\) be a smoothly bounded domain. Assume that \(a_{2},b_{1}>0\) and (1.22) holds. Then for all \(p>n\), there exists \(C_{12}>0\) such that
for all \(t\in (0,T_{\max })\).
Proof
This proof is similar to [51, Lemma 4.1]. For reader’s convenience, we give the sketch of the proof.
Multiplying the first equation in (1.1) by \(pu^{p-1}(p>n)\) and integrating by parts over \(\Omega \), we infer from Young’s inequality and Lemma 2.2 that
for all \(t\in (0,T_{\max })\), where we have used the condition that \(a_{2}>0\). Applying Lemmas 2.2, 2.4 and Young’s inequality, there exists a positive constant \(C_{13}\) such that
with \(C_{14}=(1-\alpha _{1})C_{13}^{\frac{1}{1-\alpha _{1}}}\bigg (\dfrac{p\alpha _{1}}{4(p-1)}\bigg )^{\frac{\alpha _{1}}{1-\alpha _{1}}}+C_{13}M_{0}^{p}>0\), where \(\alpha _{1}=\dfrac{\frac{np}{2}-\frac{n}{2}}{1-\frac{n}{2}+\frac{np}{2}}\). Here it follows from \(p>n\) that \(\alpha _{1}=\dfrac{\frac{np}{2}-\frac{n}{2}}{1-\frac{n}{2}+\frac{np}{2}}\in (0,1)\). By (5.39) and (5.40),
for all \(t\in (0,T_{\max })\).
Let \(t_{0}\in (0,T_{\max })\) such that \(t_{0}\le 1\). Applying the variation of constants formula to (5.41), there exists a positive constant \(C_{15}\) such that
for all \(t\in (t_{0},T_{\max })\).
Similarly,
for all \(t\in (t_{0},T_{\max })\), where \(C_{16}=\int _{\Omega }w^{p}(t_{0})+\frac{C_{11}}{p+1}\).
Combining (5.42) with (5.43) yields
for all \(t\in (t_{0},T_{\max })\), where \(C_{17}=C_{15}+C_{16}\).
Then the following proof of this lemma is similar to [14, Lemma 4.5]. Here we omit the details. The proof of Lemma 5.6 is complete. \(\square \)
Proof of Theorem 1.4
For the case \(n=1\), it follows from Lemmas 2.2 and 5.1 that the solution of (1.1) is globally bounded. For the case \(n=2\), it follows from Lemma 5.1 and Lemma 5.4 that the solution of (1.1) is globally bounded. When \(n=3\), the global boundedness of solutions for (1.1) is obtained by Lemmas 5.1 and 5.5. When \(n\ge 4\), it follows from Lemma 5.1 and Lemma 5.6 that the solution of (1.1) is globally bounded. The proof of Theorem 1.4 is complete. \(\square \)
6 Asymptotic Behavior for \(\tau =1\)
In this section, we discuss the asymptotic behavior for the fully parabolic system (1.1), and prove Theorem 1.5 and Theorem 1.6.
6.1 Proof of Theorem 1.5
Similar to the way used in Lemma 4.2, we introduce the following functionals
where \((u_{*},v_{*},w_{*},z_{*})\) is given by (1.10) and \(\rho _{1},\rho _{2}>0\) shall be determined.
Lemma 6.1
Let (u, v, w, z) be a global bounded classical solution to (1.1). Suppose that the conditions of Theorem 1.5 hold. Then there exists \(\beta _{1}>0\) such that
for all \(t>0\). Moreover,
Proof
This proof is similar to [14, Lemma 5.5] (or [4, Lemma 3.2]). The proof of Lemma 6.1 is complete. \(\square \)
Lemma 6.2
Suppose that the conditions of Theorem 1.5 hold. Then the solution (u, v, w, z) to (1.1) satisfies
as \(t\rightarrow \infty \).
Proof
A combination of Lemmas 4.1 and 6.1 implies this lemma. The proof of Lemma 6.2 is complete. \(\square \)
Lemma 6.3
Suppose that the conditions of Theorem 1.5 hold. Then there exist \(C_{1}>0\) and \(\kappa _{1}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
for all \(t>t_{4}\), where \(t_{4}>0\) is some fixed time.
Proof
This proof is similar to [14, Lemma 5.7] (or [4, Lemma 3.6]). Here we omit the details. The proof of Lemma 6.3 is complete. \(\square \)
Proof of Theorem 1.5
Lemma 6.3 directly shows the statement of Theorem 1.5. \(\square \)
6.2 Proof of Theorem 1.6
Lemma 6.4
Let (u, v, w, z) be a global bounded classical solution to (1.1). Suppose that the conditions of Theorem 1.6 hold. Then there exists \(\beta _{2}>0\) such that
for all \(t>0\), where
Here \(\rho _{1},\rho _{2}>0\) shall be determined and \((u_{\star },z_{\star })\) is given by (1.16). Moreover,
Proof
This proof is similar to [14, Lemma 5.5] (or [4, Lemma 3.2]). Here we omit the details. The proof of Lemma 6.4 is complete. \(\square \)
Lemma 6.5
Suppose that the conditions of Theorem 1.6 hold. Then the solution (u, v, w, z) to (1.1) satisfies
as \(t\rightarrow \infty \).
Proof
A combination of Lemmas 5.1 and 6.4 implies this lemma. The proof of Lemma 6.5 is complete. \(\square \)
Lemma 6.6
Suppose that the conditions of Theorem 1.6 hold.
-
(i)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}<\frac{a_{0}}{b_{0}}\), then there exist \(C_{2}>0\) and \(\kappa _{2}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C_{2}e^{-\kappa _{2}t} \end{aligned} \end{aligned}$$(6.6)for all \(t>t_{5}\), where \(t_{5}>0\) is some fixed time.
-
(ii)
If \(\frac{a_{2}-a_{4}|\Omega |}{b_{2}-b_{4}|\Omega |}<\frac{a_{1}-a_{3}|\Omega |}{b_{1}-b_{3}|\Omega |}=\frac{a_{0}}{b_{0}}\), then there exist \(C_{3}>0\) and \(\kappa _{3}>0\) such that the solution (u, v, w, z) to (1.1) satisfies
$$\begin{aligned} \begin{aligned} \Vert u-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v\Vert _{L^{\infty }(\Omega )}+\Vert w\Vert _{L^{\infty }(\Omega )}+\Vert z-z_{\star }\Vert _{L^{\infty }(\Omega )}\le C_{3}(t-t_{6})^{-\kappa _{3}} \end{aligned} \end{aligned}$$(6.7)for all \(t>t_{6}\), where \(t_{6}>0\) is some fixed time.
Proof
This proof is similar to Lemma 4.7 (or [4, Lemma 3.6]). Here we omit the details. The proof of Lemma 6.6 is complete. \(\square \)
Proof of Theorem 1.6
Lemma 6.6 directly gives the results of Theorem 1.6. \(\square \)
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Acknowledgements
The authors would like to deeply thank the editor and anonymous reviewers for their insightful and constructive comments. Pan Zheng is also deeply grateful to Professor Renjun Duan for his help and support at The Chinese University of Hong Kong.
Funding
The work is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program (Grant Nos: XJ2021042, 2021-005), Young Hundred Talents Program of CQUPT in 2022–2024, Chongqing Municipal Education Commission Science and Technology Research Project (Grant No. KJZD-K202200602) and Chongqing Postgraduate Research and Innovation Project in 2022 (Grant Nos: CYS22451).
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Hu, R., Zheng, P. Global Stability in a Two-species Attraction–Repulsion System with Competitive and Nonlocal Kinetics. J Dyn Diff Equat 36, 2555–2592 (2024). https://doi.org/10.1007/s10884-022-10215-5
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DOI: https://doi.org/10.1007/s10884-022-10215-5