This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
{ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)−χ3∇⋅(w∇v)+μ2w(1−w−a2u),x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0,
under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with n≥1, where the parameters a1,a2, χ1,χ2,χ3, μ1,μ2 are positive constants. We first showed some conditions between χ1μ1, χ2μ2, χ3μ2 and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.
Citation: Rong Zhang, Liangchen Wang. Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop[J]. Electronic Research Archive, 2021, 29(6): 4297-4314. doi: 10.3934/era.2021086
[1] | Rong Zhang, Liangchen Wang . Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop. Electronic Research Archive, 2021, 29(6): 4297-4314. doi: 10.3934/era.2021086 |
[2] | Chang-Jian Wang, Yuan-Hao Zang . Boundedness of solutions in a two-species chemotaxis system. Electronic Research Archive, 2025, 33(5): 2862-2880. doi: 10.3934/era.2025126 |
[3] | Chun Huang . Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, 2021, 29(5): 3261-3279. doi: 10.3934/era.2021037 |
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[7] | Bin Wang . Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions. Electronic Research Archive, 2023, 31(6): 3097-3122. doi: 10.3934/era.2023157 |
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[9] | Yunfei Yuan, Changchun Liu . Optimal control for the coupled chemotaxis-fluid models in two space dimensions. Electronic Research Archive, 2021, 29(6): 4269-4296. doi: 10.3934/era.2021085 |
[10] | Zhonghua Qiao, Xuguang Yang . A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28(3): 1207-1225. doi: 10.3934/era.2020066 |
This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
{ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)−χ3∇⋅(w∇v)+μ2w(1−w−a2u),x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0,
under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with n≥1, where the parameters a1,a2, χ1,χ2,χ3, μ1,μ2 are positive constants. We first showed some conditions between χ1μ1, χ2μ2, χ3μ2 and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.
In this paper, we consider the two-species chemotaxis- competition system with two chemicals
{ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)−χ3∇⋅(w∇v)+μ2w(1−w−a2u),x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=∂z∂ν=0,x∈∂Ω,t>0,(u,w)(x,0)=(u0(x),w0(x)),x∈Ω, | (1.1) |
where
Chemotaxis describes oriented movement of cells along the concentration gradient of a chemical signal produced by the cells, which is important in a large variety of fields within the life cycle of most multicellular organisms. Based on a well-known chemotaxis model for the chemotactic movement of one specie [11], a generalization of chemotaxis model for multi-species or multi-chemical-signal chemotaxis system was proposed [9,24,42]. System (1.1) describes the communication between breast tumor cells and macrophages in close proximity via a short-ranged chemical signaling loop according to the classical Lotka-Volterra dynamics [22], which was proposed by Knútsdóttir et al. [12]. The key physical variables in (1.1) are assumed to be the density of the macrophages (denoted by
From a mathematical point of view, the system (1.1) contains mainly two subsystems. When
{wt=Δw−χ3∇⋅(w∇v)+μ2w(1−w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0, | (1.2) |
which describe the aggregation of the paradigm species Dictyostelium discoideum. Model (1.2) has been extensively studied during past four decades. Boundedness of global solutions and unbounded solutions for (1.2) have been extensively investigated (see the surveys [2,6,8] and the references therein). When
{ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,0=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)+μ2w(1−w−a2u),x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0, | (1.3) |
which describes the spatio-temporal evolution of two populations which on the one hand proliferate and mutually compete with Lotka-Volterra kinetics and on the other hand the individuals shall move according to random diffusion and migrate toward a chemical signal produced by the opposing species. Without kinetic terms, that is, when
Compared with the chemotaxis-only system and the two-species chemotaxis system with two chemicals, the coupled a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop the system (1.1) is much less understood. For the simplified version (1.1) in the unit disk
0=Δv−m2|Ω|+wand0=Δz−m1|Ω|+u, |
respectively, where
In order to better understand (1.1), we should mention two biological species which compete for the resources and migrate towards a higher concentration of a chemical produced by themselves was proposed by Tello and Winkler [33]
{ut=Δu−χ1∇⋅(u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=Δv−χ2∇⋅(v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,0=Δw−w+u+v,x∈Ω,t>0, | (1.4) |
this model has been extensively studied. When
The focus of this paper is to establish the global existence, large time behavior, and the rates of convergence of solution to (1.1). The first of our result asserts global existence of a bounded solution based on the comparison methods in [38,37,4,30].
Theorem 1.1. Suppose that
(i)
(ii)
(iii)
Then for the nonnegative initial data
‖u‖C2+θ,1+θ2(¯Ω×[t,t+1])+‖v‖C2+θ,1+θ2(¯Ω×[t,t+1])+‖w‖C2+θ,1+θ2(¯Ω×[t,t+1])+‖z‖C2+θ,1+θ2(¯Ω×[t,t+1])≤Mfor allt≥1. | (1.5) |
The large time behavior and convergence rates of solutions to (1.1) is mathematically and biologically interesting. We first give the result of competitive coexistence case
Theorem 1.2. Let
μ1μ2a1(1−a1a2)−a2μ2χ21u∗16−w∗μ1a1χ238>0 | (1.6) |
and
μ1μ22a1a2(1−a1a2)−a22μ22χ21u∗16−μ1μ2a1a2χ23w∗8−μ1μ2a21χ22w∗8+μ2a1a2χ21χ22u∗w∗128+μ1a21χ22χ23w∗64>0, | (1.7) |
then one can find
‖u−u∗‖L∞(Ω)+‖v−v∗‖L∞(Ω)+‖w−w∗‖L∞(Ω)+‖z−z∗‖L∞(Ω)≤Ce−μt |
for all
u∗=z∗=1−a11−a1a2,v∗=w∗=1−a21−a1a2. | (1.8) |
Remark 1. Since
limμ1→+∞,μ2→+∞f(μ1,μ2)μ1μ22=a1a2(1−a1a2)>0. |
Hence, there exist some constants
The following result is on the competitive exclusion case
Theorem 1.3. Let
(i) If
μ22a2(1−a′1a2)−a2μ2χ238+a′1χ22χ2364−μ2a′1χ228>0andμ2>χ238(1−a′1a2), | (1.9) |
there exist
‖u‖L∞(Ω)+‖v−1‖L∞(Ω)+‖w−1‖L∞(Ω)+‖z‖L∞(Ω)≤Ce−λtfor allt>0. |
(ii) If
‖u‖L∞(Ω)+‖v−1‖L∞(Ω)+‖w−1‖L∞(Ω)+‖z‖L∞(Ω)≤C(1+t)−κfor allt>0. |
This paper is organized as follows. In Section 2, we show local existence of a solution to (1.1) and use comparison methods to prove global existence and boundedness of (1.1) (Theorems 1.1). Section 3 is devoted to the proof of asymptotic stability to (1.1) (Theorems 1.2 and 1.3).
The local existence of solutions to (1.1) which can be achieved similarly by using well-established methods in [30,31,41].
Lemma 2.1. Let
u,v,w,z∈C(¯Ω×[0,Tmax))∩C2,1(¯Ω×(0,Tmax)), | (2.1) |
and which is such that either
‖u(⋅,t)‖L∞(Ω)+‖w(⋅,t)‖L∞(Ω)→∞ast↗Tmax. |
Moreover, if the nonnegative initial data
According to comparison methods in [38,37,4,30], we will prove boundedness of solution to model (1.1). First we let the parabolic operator
L1u:=Δu−χ1∇u⋅∇v,(x,t)∈Ω×(0,Tmax) | (2.2) |
and
L2w:=Δw−χ2∇w⋅∇z−χ3∇w⋅∇v,(x,t)∈Ω×(0,Tmax). | (2.3) |
Thus from the model (1.1) shows that
ut−L1u=u{−χ1Δv+μ1(1−u−a1w)}=u{μ1−μ1u−χ1v+(χ1−a1μ1)w} | (2.4) |
and
wt−L2w=w{−χ2Δz−χ3Δv+μ2(1−w−a2u)}=w{μ2−(μ2−χ3)w+(χ2−a2μ2)u−χ2z−χ3v} | (2.5) |
for all
We first consider the conditions
Lemma 2.2. Let
Proof. Since
p1u:=ut−L1u−u{μ1−μ1u}≤0,(x,t)∈Ω×(0,Tmax) | (2.6) |
and
p2w:=wt−L2w−w{μ2−(μ2−χ3)w+(χ2−a2μ2)u}≤0,(x,t)∈Ω×(0,Tmax), | (2.7) |
where
Case 1.
ξ≥max{‖u0‖L∞(Ω)A,1A,‖w0‖L∞(Ω),μ2μ2−χ3}. | (2.8) |
Hence, we can define constant functions
¯u=¯u(x,t):=Aξand¯w=¯w(x,t):=ξ,(x,t)∈¯Ω×[0,Tmax), | (2.9) |
which satisfy
¯u(x,0)=Aξ≥u0(x)and¯w(x,0)=ξ≥w0(x)for allx∈Ω. | (2.10) |
Due to the definition of
p1¯u=−Aξ{μ1−μ1Aξ}≥0,(x,t)∈Ω×(0,Tmax). | (2.11) |
According to (2.6), (2.10) and (2.11), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that
u(x,t)≤¯u(x,t)=Aξfor all(x,t)∈Ω×(0,Tmax). | (2.12) |
Using the definition of
p2¯w=−ξ{μ2−(μ2−χ3)ξ+(χ2−a2μ2)u}≥−ξ{μ2−(μ2−χ3)ξ}≥0,(x,t)∈Ω×(0,Tmax). | (2.13) |
According to (2.7), (2.10) and (2.13), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that
w(x,t)≤¯w(x,t)=ξfor all(x,t)∈Ω×(0,Tmax). | (2.14) |
Case 2.
0<A<μ2−χ3χ2−a2μ2. | (2.15) |
Let
ξ≥max{‖u0‖L∞(Ω)A,1A,‖w0‖L∞(Ω),μ2μ2−χ3−(χ2−a2μ2)A}. | (2.16) |
We again let
¯u(x,0)=Aξ≥u0(x)and¯w(x,0)=ξ≥w0(x)for allx∈Ω. | (2.17) |
Similar to (2.11) and (2.12), we infer
u(x,t)≤¯u(x,t)=Aξfor all(x,t)∈Ω×(0,Tmax). | (2.18) |
Hence, using
p2¯w=−ξ{μ2−(μ2−χ3)ξ+(χ2−a2μ2)u}≥−ξ{μ2−(μ2−χ3)ξ+(χ2−a2μ2)Aξ}=−ξ{μ2−[μ2−χ3−(χ2−a2μ2)A]ξ}>0,(x,t)∈Ω×(0,Tmax), | (2.19) |
thus by (2.7) and (2.17) we have
w(x,t)≤¯w(x,t)=ξfor all(x,t)∈Ω×(0,Tmax). | (2.20) |
Using (2.12), (2.14), (2.18) and (2.20) along with extensibility criterion in Lemma 2.1, this entails that
Next, we consider the conditions
Lemma 2.3. Let
Proof. Since
p3u:=ut−L1u−u{μ1−μ1u+(χ1−a1μ1)w}≤0,(x,t)∈Ω×(0,Tmax) | (2.21) |
and
p4w:=wt−L2w−w{μ2−(μ2−χ3)w}≤0,(x,t)∈Ω×(0,Tmax), | (2.22) |
where
A>χ1−a1μ1+μ1ξμ1 | (2.23) |
and
ξ≥max{‖u0‖L∞(Ω)A,‖w0‖L∞(Ω),μ2μ2−χ3}. | (2.24) |
We again let
¯u(x,0)=Aξ≥u0(x)and¯w(x,0)=ξ≥w0(x)for allx∈Ω. | (2.25) |
Using (2.22) and (2.24) we conclude
p4¯w=−ξ{μ2−(μ2−χ3)ξ}>0,(x,t)∈Ω×(0,Tmax), | (2.26) |
so according to (2.22) and (2.25) imply
w(x,t)≤¯w(x,t)=ξfor all(x,t)∈Ω×(0,Tmax). | (2.27) |
Hence, using
p3¯u=−Aξ{μ1−μ1Aξ+(χ1−a1μ1)w}≥−Aξ{μ1−μ1Aξ+(χ1−a1μ1)ξ}≥0,(x,t)∈Ω×(0,Tmax). | (2.28) |
From (2.21), (2.25) and (2.28) and the comparison principle, we infer
u(x,t)≤¯u(x,t)=Aξfor all(x,t)∈Ω×(0,Tmax). | (2.29) |
Then using (2.27) and (2.29) along with extensibility criterion in Lemma 2.1, this entails that
Finally, we consider the conditions
Lemma 2.4. Let
Proof. The conditions
χ1−a1μ1μ1<μ2−χ3χ2−a2μ2. | (2.30) |
Hence, we can choose
ξ≥max{μ1χ1−a1μ1‖u0‖L∞(Ω),‖w0‖L∞(Ω)} | (2.31) |
and
χ1−a1μ1+μ1ξμ1<μ2−χ3−μ2ξχ2−a2μ2, |
which enables us to choose
χ1−a1μ1+μ1ξμ1<A<μ2−χ3−μ2ξχ2−a2μ2. | (2.32) |
Hence, (2.32) implies
{−Aξ{μ1−μ1Aξ+(χ1−a1μ1)ξ}>0,−ξ{μ2−(μ2−χ3)ξ+(χ2−a2μ2)Aξ}>0. | (2.33) |
We again let
¯u(x,0)=Aξ>χ1−a1μ1μ1ξ≥u0(x)and¯w(x,0)=ξ≥w0(x)for allx∈Ω. | (2.34) |
Due to
{ut−L1u−u{μ1−μ1u+(χ1−a1μ1)w}≤¯ut−L1¯u−¯u{μ1−μ1¯u+(χ1−a1μ1)¯w},wt−L2w−w{μ2−(μ2−χ3)w+(χ2−a2μ2)u}≤¯wt−L2¯w−¯w{μ2−(μ2−χ3)¯w+(χ2−a2μ2)¯u} |
in
u(x,t)≤¯u(x,t)=Aξandw(x,t)≤¯w(x,t)=ξfor all(x,t)∈Ω×(0,Tmax). |
Therefore, using the extensibility criterion in Lemma 2.1, we have
Proof of Theorem 1.1. We only need to use Lemmas 2.2-2.4 to obtain the global bounded solutions of (1.1). Finally, the Hölder continuity of the solution
In this section, we will derive the asymptotic behavior of the solutions to the model (1.1), the ideas mainly come from [1,21,38,39]. To achieve our goals, we first recall the following lemma which is important for the proof of Theorems 1.2 and 1.3 (see [7,Lemma 4.6] or [20,Lemma 3.1]).
Lemma 3.1. Suppose that
‖φ(x,t)‖Cσ,σ2(¯Ω×[t,t+1])≤Cfor allt≥1. |
Moreover, assume that there exists some constant
∫∞0∫Ω(φ(x,t)−M)2dxdt<∞. |
Then
φ(⋅,t)→MinC0(¯Ω)ast→∞. |
This following lemma is a straightforward result from [40,Lemma 5.1].
Lemma 3.2. Let
a11>0,a22>0,a11a22a33−a22a2134−a11a2234>0 |
and
a11a22a33a44−a22a44a2134−a11a44a2234−a22a33a2144+a214a22316>0. |
Then there exists
a11x21+a22x22+a33x23+a44x24+a13x1x3+a14x1x4+a23x2x3≥ε(x21+x22+x23+x24) |
for all
In order to prove stabilization of solutions to (1.1), we will divide the proof into two cases.
In this subsection, we will study the asymptotic behavior of the solution of (1.1) with
E1(t):=a2μ2∫Ω(u−u∗−u∗lnuu∗)+a1μ1∫Ω(w−w∗−w∗lnww∗), | (3.1) |
where
Lemma 3.3. Under the assumptions of Theorem 1.2, then there exists
E1(t)≥0andddtE1(t)≤−εF1(t)for allt>0, | (3.2) |
where the function
F1(t):=∫Ω(u−u∗)2+∫Ω(v−v∗)2+∫Ω(w−w∗)2+∫Ω(z−z∗)2. | (3.3) |
Proof. From Taylor's formula we see that
ddtE1(t)=−u∗μ2a2∫Ω|∇u|2u2+u∗μ2a2χ1∫Ω∇u⋅∇vu−w∗μ1a1∫Ω|∇w|2w2+w∗μ1a1χ2∫Ω∇w⋅∇zw+w∗μ1a1χ3∫Ω∇w⋅∇vw+μ1μ2a1∫Ω(w−w∗)(1−w−a2u)+μ1μ2a2∫Ω(u−u∗)(1−u−a1w)=−μ1μ2a2∫Ω(u−u∗)2−μ1μ2a1∫Ω(w−w∗)2−u∗μ2a2∫Ω|∇u|2u2−2μ1μ2a1a2∫Ω(u−u∗)(w−w∗)+u∗μ2a2χ1∫Ω∇u⋅∇vu−w∗μ1a1∫Ω|∇w|2w2+w∗μ1a1χ2∫Ω∇w⋅∇zw+w∗μ1a1χ3∫Ω∇w⋅∇vw. | (3.4) |
In order to obtain (3.2), we first deal with the last five parts on the right of (3.4), by a simple computation we conclude
−u∗μ2a2∫Ω|∇u|2u2+u∗μ2a2χ1∫Ω∇u⋅∇vu=−u∗μ2a2∫Ω(∇uu−χ12∇v)2+u∗μ2a2χ214∫Ω|∇v|2≤u∗μ2a2χ214∫Ω|∇v|2 | (3.5) |
and
−w∗μ1a1∫Ω|∇w|2w2+w∗μ1a1χ2∫Ω∇w⋅∇zw+w∗μ1a1χ3∫Ω∇w⋅∇vw=−w∗μ1a1∫Ω(∇ww−χ22∇z−χ32∇v)2+w∗μ1a1χ224∫Ω|∇z|2+w∗μ1a1χ234∫Ω|∇v|2+w∗μ1a1χ2χ32∫Ω∇v⋅∇z≤w∗μ1a1χ224∫Ω|∇z|2+w∗μ1a1χ234∫Ω|∇v|2+w∗μ1a1χ2χ32∫Ω∇v⋅∇z≤w∗μ1a1χ222∫Ω|∇z|2+w∗μ1a1χ232∫Ω|∇v|2. | (3.6) |
Since
∫Ω|∇v|2=−∫Ω(v−v∗)2+∫Ω(w−w∗)(v−v∗). | (3.7) |
Similar to the fourth equation in (1.1) yields
∫Ω|∇z|2=−∫Ω(z−z∗)2+∫Ω(u−u∗)(z−z∗). | (3.8) |
Inserting (3.5)-(3.8) into (3.4) we conclude
ddtE1(t)≤−a2μ1μ2∫Ω(u−u∗)2−a1μ1μ2∫Ω(w−w∗)2−(a2μ2χ21u∗4+w∗μ1a1χ232)∫Ω(v−v∗)2−a1μ1χ22w∗2∫Ω(z−z∗)2+a1μ1χ22w∗2∫Ω(u−u∗)(z−z∗)−2a1a2μ1μ2∫Ω(u−u∗)(w−w∗)+(a2μ2χ21u∗4+w∗μ1a1χ232)∫Ω(w−w∗)(v−v∗). | (3.9) |
To see (3.2), we will show that there exists
−a2μ1μ2∫Ω(u−u∗)2−a1μ1μ2∫Ω(w−w∗)2−2a1a2μ1μ2∫Ω(u−u∗)(w−w∗)−(a2μ2χ21u∗4+w∗μ1a1χ232)∫Ω(v−v∗)2−a1μ1χ22w∗2∫Ω(z−z∗)2+a1μ1χ22w∗2∫Ω(u−u∗)(z−z∗)+(a2μ2χ21u∗4+w∗μ1a1χ232)∫Ω(w−w∗)(v−v∗)≤−ε(∫Ω(u−u∗)2+∫Ω(v−v∗)2+∫Ω(w−w∗)2+∫Ω(z−z∗)2). | (3.10) |
To confirm that the assumptions of Lemma 3.2 are satisfied, let
a11:=μ1μ2a2,a22:=a2μ2χ21u∗4+w∗μ1a1χ232,a33:=μ1μ2a1,a44:=a1μ1χ22w∗2, |
a13:=2μ1μ2a1a2,a14:=−a1μ1χ22w∗2,a23:=−a2μ2χ21u∗4−w∗μ1a1χ232 |
and
x1:=u−u∗,x2:=v−v∗,x3:=w−w∗,x4:=z−z∗. |
Since
a11=μ1μ2a2>0,a22=a2μ2χ21u∗4+w∗μ1a1χ232>0. | (3.11) |
Thanks to (1.6) we have
a11a22a33−a22a2134−a11a2234=(μ21μ22a1a2−μ21μ22a21a22)(a2μ2χ21u∗4+w∗μ1a1χ232)−μ1μ2a24(a2μ2χ21u∗4+w∗μ1a1χ232)2=μ1μ2a2(a2μ2χ21u∗4+w∗μ1a1χ232)(μ1μ2a1(1−a1a2)−a2μ2χ21u∗16−w∗μ1a1χ238)>0. | (3.12) |
Using (1.7), we obtain
a11a22a33a44−a22a44a2134−a11a44a2234−a22a33a2144+a214a22316=μ21μ2a1a2χ22w∗2(a2μ2χ21u∗4+w∗μ1a1χ232)(μ1μ2a1(1−a1a2)−a2μ2χ21u∗16−w∗μ1a1χ238)+μ21a21χ42w2∗16(a2μ2χ21u∗4+w∗μ1a1χ232)(a2μ2χ21u∗16+w∗μ1a1χ238−μ1μ2a1)=μ21a1χ22w∗2(a2μ2χ21u∗4+w∗μ1a1χ232){μ1μ22a1a2(1−a1a2)−a22μ22χ21u∗16−μ1μ2a1a2χ23w∗8+μ2a1a2χ21χ22u∗w∗128+μ1a21χ22χ23w∗64−μ1μ2a21χ22w∗8}>0. | (3.13) |
Hence, combining (3.11)-(3.13) and Lemma 3.2 we have (3.10), which concludes the proof of Lemma 3.3.
Next, we will establish convergence rates for the solution to the model (1.1).
Lemma 3.4. Under the assumptions of Theorem 1.2, then there exist
‖u−u∗‖L∞(Ω)+‖v−v∗‖L∞(Ω)+‖w−w∗‖L∞(Ω)+‖z−z∗‖L∞(Ω)≤Ce−μt |
for all
Proof. The proof is similar to the corresponding proofs of [20,Lemmas 3.5 and 3.6] or [38,Lemmas 3.4 and 3.5], for the convenience of the readers, we give a sketch the proof. We divide the proof into two steps.
Step 1. We derive the large time behavior of solution to (1.1).
Integrating the second part of (3.2) and using (3.3) to see that
∞∫1∫Ω(u−u∗)2+∞∫1∫Ω(v−v∗)2+∞∫1∫Ω(w−w∗)2+∞∫1∫Ω(z−z∗)2≤E1(1)ε. |
According to Lemma 3.1 and (1.5), we obtain
‖u−u∗‖L∞(Ω)+‖v−v∗‖L∞(Ω)+‖w−w∗‖L∞(Ω)+‖z−z∗‖L∞(Ω)→0ast→∞. | (3.14) |
Step 2. We derive the convergence rates of solution to (1.1).
Let
lims→u∗H(s)−H(u∗)(s−u∗)2=lims→u∗H′(s)2(s−u∗)=12u∗. | (3.15) |
Hence, using (3.14) and (3.15) and the definitions
min{a2μ24u∗,a1μ14w∗}(∫Ω(u−u∗)2+∫Ω(w−w∗)2)≤E1(t)≤max{a2μ2u∗,a1μ1w∗}(∫Ω(u−u∗)2+∫Ω(w−w∗)2) | (3.16) |
for all
which implies
(3.17) |
with some
(3.18) |
we can find some
(3.19) |
Then by the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) enables us to obtain
(3.20) |
with some
When
(3.21) |
where
Lemma 3.5. Under the assumptions of Theorem 1.3, then there exists
(3.22) |
where
Proof. Using Taylor's formula to see that
(3.23) |
Using the third equation in (1.1), we obtain
(3.24) |
Combining (3.23) and (3.24), we have
(3.25) |
Using Young's inequality we infer that
(3.26) |
Multiplying the second equation in (1.1) with
(3.27) |
Testing the fourth equation in (1.1) with
(3.28) |
Substituting (3.26)-(3.28) into (3.25), we have
(3.29) |
In order to prove (3.22), we will show that there exists
(3.30) |
Then using the same argument as in the proof of Lemma 3.4, we put
and
Since
(3.31) |
Thanks to the second part of (1.9) we have
(3.32) |
By the first part of (1.9), we infer that
(3.33) |
Then collecting (3.31)-(3.33) we have (3.30), which implies the end of the proof.
According ideas come from [20,Lemmas 3.8,3.9 and 3.10] or [38,Lemmas 3.7 and 3.8], we shall give the convergence rates for the case
Lemma 3.6. Under the assumptions of Theorem 1.3.
(i) Let
(ii) Suppose that
Proof. (i) This part can be proved by a similar proof in Lemma 3.4.
(ii) Since
(3.34) |
Similar to (3.14), integrating (3.34) and using Lemma 3.1 enable us to obtain
(3.35) |
Hence, by the definition of
(3.36) |
Combining (3.34) and (3.36), using the definition of
which implies there exists
(3.37) |
By L'Hôpital's theorem and (3.37), we can find
By (1.5) and the Gagliardo-Nirenberg inequality, we can find
Using the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) implies
with some
Proof of Theorem 1.2. The proof of Theorem 1.2 follows from Lemma 3.4.
Proof of Theorem 1.3. We only need to use Lemma 3.6 to obtain the proof of Theorem 1.3.
The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work.
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1. | Xu Pan, Chunlai Mu, Weirun Tao, Global dynamics and spatiotemporal patterns of a two‐species chemotaxis system with chemical signaling loop and Lotka–Volterra competition, 2024, 0022-2526, 10.1111/sapm.12746 |