Citation: Masaaki Mizukami. Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system[J]. AIMS Mathematics, 2016, 1(3): 156-164. doi: 10.3934/Math.2016.3.156
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Nowadays, mathematics is useful in many things, for example, physics, chemistry, biology, computer, medical, architecture, and so on (see e.g., [3, 7, 14]). Here we focus on biology. One of the important models in biology is the logistic equation ut=u(1-u). Some of biological models have the logistic term, e.g., the Fisher-KPP equation
ut=Δu+u(1−u). |
On the other hand, many mathematicians study a chemotaxis system lately, which describes a part of the life cycle of cellular slime molds with chemotaxis. After the pioneering work of Keller-Segel [8], a number of variations of the chemotaxis system are proposed and investigated (see e.g., [2, 4, 5]). Also, multi-species chemotaxis systems have been studied by e.g., [6, 15]. In this paper we focus on a twospecies chemotaxis system with logistic term which describes a situation in which multi populations react on a single chemoattractant.
We consider the two-species chemotaxis system
{ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u),x∈Ω,t>0,vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v),x∈Ω,t>0,wt=dΔw+h(u,v,w),x∈Ω,t>0,∂u∂n=∂v∂n=∂w∂n=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω, | (1.1) |
where Ω is a bounded domain in RN (N∈N) with smooth boundary ∂Ω and n denotes the unit outer normal vector of ∂Ω. The initial data u0, v0 and w0 are assumed to be nonnegative functions. The unknown functions u(x, t) and v(x, t) represent the population densities of two species and w(x, t) shows the concentration of the substance at place x and time t.
In mathematical view, global existence and behavior of solutions are fundamental theme. Recently, Negreanu-Tello [12, 13] built a technical way to prove global existence and asymptotic behavior of solutions to (1.1). In [13] they dealt with (1.1) when d=0, µi > 0 under the condition
∃ˉw≥w0;h(ˉu,ˉv,ˉw)≤0, |
where ˉu, ˉv satisfy some representations determined by ˉw. In [12] they studied (1.1) when 0 < d < 1, µi=0 under similar conditions as in [13] and
χi′+11−dχ2i≤0(i=1,2). | (1.2) |
They supposed in [12, 13] that the functions h, χi for i=1, 2 generalize of the prototypical case χi(w)=Ki(1+w)σi (Ki>0,σi≥1), h(u,v,w)=u+v−w. These days, the restriction of 0≤d < 1 for global existence is completely removed and asymptotic stability of solutions to (1.1) is established for the first time under a smallness condition for the function χi generalizing of χi(w)=Ki(1+w)σi (Ki>0,σi>1) ([11]).
The purpose of this paper is to improve a way in [11] for obtaining asymptotic stability of solutions to (1.1) under a more general and sharp smallness condition for the sensitivity function χi(w). We shall suppose throughout this paper that h, χi (i=1, 2) satisfy the following conditions:
χi∈C1+ω([0,∞))∩L1(0,∞)(0<∃ω<1),χi>0(i=1,2), | (1.3) |
h∈C1([0,∞)×[0,∞)×[0,∞)),h(0,0,0)≥0, | (1.4) |
∃γ>0; ∂h∂u(u,v,w)≥0,∂h∂v(u,v,w)≥0,∂h∂w(u,v,w)≤−γ, | (1.5) |
∃δ>0,∃M>0;|h(u,v,w)+δw|≤M(u+v+1), | (1.6) |
∃ki>0;−χi(w)h(0,0,w)≤ki(i=1,2). | (1.7) |
We also assume that
∃p>N; 2dχi′(w)+((d−1)p+√(d−1)2p2+4dp)[χi(w)]2≤0(i=1,2). | (1.8) |
The above conditions cover the prototypical example χi(w)=Ki(1+w)σi (Ki>0, σi>1), h(u,v,w)=u+v−w. We assume that the initial data u0, v0, w0 satisfy
0≤u0∈C(ˉΩ)∖{0}, 0≤v0∈C(ˉΩ)∖{0}, 0≤w0∈W1,q(Ω)(∃q>N). | (1.9) |
The following result which is concerned with global existence and boundedness in (1.1) was established in [11].
Theorem 1.1 ([11, Theorem 1.1]). Let d≥0, µi > 0 (i=1, 2). Assume that h, χi satisfy (1.3)-(1.8). Then for any u0, v0, w0 satisfying (1.9) for some q > N, there exists an exactly one pair (u, v, w) of nonnegative functions
u,v,w∈C(ˉΩ×[0,∞))∩C2,1(ˉΩ×(0,∞))when d>0,u,v,w∈C([0,∞);W1,q(Ω))∩C1((0,∞);W1,q(Ω))when d=0, |
which satisfy (1.1). Moreover, the solution (u; v; w) is uniformly bounded, i.e., there exists a constant C1 > 0 such that
‖u(t)‖L∞(Ω)+‖v(t)‖L∞(Ω)+‖w(t)‖L∞(Ω)≤C1forall t≥0. |
Since Theorem 1.1 guarantees that u, v and w exist globally and are bounded and nonnegative, it is possible to define nonnegative numbers α, β by
α:=max(u,v,w)∈Ihu(u,v,w),β:=max(u,v,w)∈Ihv(u,v,w), | (1.10) |
where I=(0, C1)3 and C1 is defined in Theorem 1.1.
Now the main result reads as follows. The main theorem is concerned with asymptotic stability in (1.1).
Theorem 1.2. Let d > 0, µi > 0 (i=1, 2). Under the conditions (1.3)-(1.9) and
α>0,β>0,χ1(0)2<16μ1dγα2+β2+2αβ,χ2(0)2<16μ2dγα2+β2+2αβ, | (1.11) |
the unique global solution (u, v, w) of (1.1) satisfies that there exist C > 0 and λ>0 such that
‖u(t)−1‖L∞(Ω)+‖v(t)−1‖L∞(Ω)+‖w(t)−˜w‖L∞(Ω)≤Ce−λt(t>0), |
where ˜w≥0 such that h(1, 1, ˜w)=0.
Remark 1.1. This result improves the previous result [11, Theorem 1.2]. Indeed, the condition (1.11) is sharper than“χi(0) are suitably small”assumed in [11]. Moreover, this result attains to show the convergence rate which cannot be given in [11].
Remark 1.2. From (1.4)-(1.6) there exists ˜w≥0 such that h(1,1,˜w)=0. Indeed, if we choose ˉw≥3M/δ, then (1.6) yields that h(1,1,ˉw)≤3M−δˉw≤0. On the other hand, (1.4) and (1.5) imply that h(1, 1, 0)≥h(0, 0, 0)≥0. Hence, by the intermediate value theorem there exists ˜w≥0 such that h(1, 1, ˜w)=0.
The strategy for the proof of Theorem 1.2 is to modify an argument in [10]. The key for this strategy is to construct the following energy estimate which was not given in [11]:
ddtE(t)≤−ε(∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2) |
with some function E(t)≥0 and some ε>0. This strategy enables us to improve the conditions assumed in [11].
In this section we will establish asymptotic stability of solutions to (1.1). For the proof of Theorem 1.2, we shall prepare some elementary results.
Lemma 2.1 ([1, Lemma 3.1]). Suppose that f : (1, ∞)→R is a uniformly continuous nonnegative function satisfying ∫∞1f(t)dt<∞. Then f(t)→0 as t→∞.
Lemma 2.2. Let a1,a2,a3,a4,a5∈R. Suppose that
a1>0,a3>0,a5−a224a1−a244a3>0. | (2.1) |
Then
a1x2+a2xz+a3y2+a4yz+a5z2≥0 | (2.2) |
holds for all x,y,z∈R.
Proof. From straightforward calculations we obtain
a1x2+a2xz+a3y2+a4yz+a5z2=a1(x+a2z2a1)2+a3(y+a4z2a3)2+(a5−a224a1−a244a3)z2. |
In view of the above equation, (2.1) leads to (2.2).
Now we will prove the key estimate for the proof of Theorem 1.2.
Lemma 2.3. Let (u, v, w) be a solution to (1.1). Under the conditions (1.3)-(1.9) and (1.11), there exist δ1,δ2>0andε>0 such that the nonnegative functions E1 and F1 defined by
E1(t):=∫Ω(u−1−logu)+δ1μ1μ2∫Ω(v−1−logv)+δ22∫Ω(w−˜w)2 |
and
F1(t):=∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2%+∫Ω|∇w|2 |
satisfy
ddtE1(t)≤−εF1(t)(t>0). | (2.3) |
Proof. Thanks to (1.11), we can choose δ1=βα>0andδ2>0 satisfying
max{χ1(0)2(1+δ1)4d,μ1χ2(0)2(1+δ1)4μ2d}<δ2<4μ1γδ1α2δ1+β2. | (2.4) |
We denote by A1(t), B1(t), C1(t) the functions defined as
A1(t):=∫Ω(u−1−logu),B1(t)=∫Ω(v−1−logv),C1(t):=12∫Ω(w−˜w)2, |
and we write as
E1(t)=A1(t)+δ1μ1μ2B1(t)+δ2C1(t). |
The Taylor formula applied to H(s)=s-log s (s≥0) yields A1(t)=∫Ω(H(u)−H(1)) is a nonnegative function for t > 0 (more detail, see [1, Lemma 3.2]). Similarly, we have that B1(t) is a positive function. By straightforward calculations we infer
ddtA1(t)=−μ1∫Ω(u−1)2−∫Ω|∇u|2u2+∫Ωχ1(w)u∇u⋅∇w,ddtB1(t)=−μ2∫Ω(v−1)2−∫Ω|∇v|2v2+∫Ωχ2(w)v∇v⋅∇w,ddtC1(t)=∫Ωhu(u−1)(w−˜w)+∫Ωhv(v−1)(w−˜w)+∫Ωhw(w−˜w)2−d∫Ω|∇w|2 |
with some derivatives hu, hv and hw. Hence we have
ddtE1(t)=I1(t)+I2(t), | (2.5) |
where
I1(t):=−μ1∫Ω(u−1)2−δ1μ1∫Ω(v−1)2+δ2∫Ωhu(u−1)(w−˜w)+δ2∫Ωhv(v−1)(w−˜w)+δ2∫Ωhw(w−˜w)2 |
and
I2(t):=−∫Ω|∇u|2u2+∫Ωχ1(w)u∇u⋅∇w−δ1μ1μ2∫Ω|∇v|2v2+δ1μ1μ2∫Ωχ2(w)v∇v⋅∇w−dδ2∫Ω|∇w|2. | (2.6) |
At first, we shall show from Lemma 2.2 that there exists "1 > 0 such that
I1(t)≤−ε1(∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2). | (2.7) |
To see this, we put
g1(ε):=μ1−ε,g2(ε):=δ1μ1−ε,g3(ε):=(−δ2hw−ε)−h2u4(μ1−ε)δ22−h2v4(δ1μ1−ε)δ22. |
Since µ1 > 0 and δ1=βα>0, we have g1(0)=µ1 > 0 and g2(0)=δ1μ1>0. In light of (1.5) and the definitions of δ2,α,β>0 (see (1.10) and (2.4)) we obtain
g3(0)=δ2(−hw−(h2u4μ1+h2v4δ1μ1)δ2)≥δ2(γ−(α2δ1+β4δ1μ1)δ2)>0. |
Combination of the above inequalities and the continuity of gi for i=1, 2, 3 yield that there exists ε1>0 such that gi(ε1) > 0 hold for i=1, 2, 3. Thanks to Lemma 2.2 with
a1=μ1−ε1,a2=−δ2hu,a3=δ1μ1−ε1,a4=−δ2hv,a5=−δ2hw−ε1,x=u(t)−1,y=v(t)−1,z=w(t)−˜w, |
we obtain (2.7) with ε1>0. Lastly we will prove
I2(t)≤0. | (2.8) |
Noting that χ′i<0 (from (1.8)) and then using the Young inequality, we have
∫Ωχ1(w)u∇u⋅∇w≤χ1(0)∫Ω|∇u⋅∇w|u≤χ1(0)2(1+δ1)4dδ2∫Ω|∇u|2u2+dδ21+δ1∫Ω|∇w|2 |
and
δ1μ1μ2∫Ωχ2(w)v∇v⋅∇w≤χ2(0)δ1μ1μ2∫Ω|∇v⋅∇w|v≤χ2(0)2δ1(1+δ1)4dδ2(μ1μ2)2∫Ω|∇v|2v2+dδ1δ21+δ1∫Ω|∇w|2. |
Plugging these into (2.6) we infer
I2(t)≤−(1−χ1(0)2(1+δ1)4dδ2)∫Ω|∇u|2u2−δ1μ1μ2(1−μ1χ2(0)2(1+δ1)4dμ2δ2)∫Ω|∇v|2v2. |
We note from the definition of δ2>0 that
1−χ1(0)2(1+δ1)4dδ2>0,1−μ1χ2(0)2(1+δ1)4dμ2δ2>0. |
Thus we have (2.8). Combination of (2.5), (2.7) and (2.8) implies the end of the proof.
Lemma 2.4. Let (u, v, w) be a solution to (1.1). Under the conditions (1.3)-(1.9) and (1.11), (u, v, w) has the following asymptotic behavior:
‖u(t)−1‖L∞(Ω)→0,‖v(t)−1‖L∞(Ω)→0,‖w(t)−˜w‖L∞(Ω)→0(t→∞). |
Proof. Firstly the boundedness of u, v, ∇w and a standard parabolic regularity theory ([9]) yield that there exist θ∈(0,1) and C > 0 such that
‖u‖C2+θ,1+θ2(ˉΩ×[1,t])+‖v‖C2+θ,1+θ2(ˉΩ×[1,t])+‖w‖C2+θ,1+θ2(ˉΩ×[1,t])≤C for all t≥1. |
Therefore in view of the Gagliardo-Nirenberg inequality
‖φ‖L∞(Ω)≤c‖φ‖NN+2W1,∞(Ω)‖φ‖2N+2L∞(Ω)(φ∈W1,∞(Ω)), | (2.9) |
it is sufficient to show that
‖u(t)−1‖L2(Ω)→0,‖v(t)−1‖L2(Ω)→0,‖w(t)−˜w‖L2(Ω)→0(t→∞). |
We let
f1(t):=∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2. |
We have that f1(t) is a nonnegative function, and thanks to the regularity of u, v, w we can see that f1(t) is uniformly continuous. Moreover, integrating (2.3) over (1, ∞), we infer from the positivity of E1(t) that
∫∞1f1(t)dt≤1εE1(1)<∞. |
Therefore we conclude from Lemma 2.1 that f1(t)→0 (t→∞), which means
∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2→0(t→∞). |
This implies the end of the proof.
Lemma 2.5. Let (u, v, w) be a solution to (1.1). Under the conditions (1.3)-(1.9) and (1.11), there exist C > 0 and λ>0 such that
‖u(t)−1‖L∞(Ω)+‖v(t)−1‖L∞(Ω)+‖w(t)−˜w‖L∞(Ω)≤Ce−λt(t>0). |
Proof. From the L’Hôpital theorem applied to H(s) :=s-log s we can see
lims→1H(s)−H(1)(s−1)2=lims→1H″(s)2=12. | (2.10) |
In view of the combination of (2.10) and ‖u−1‖L∞(Ω)→0 from Lemma 2.4 we obtain that there exists t0 > 0 such that
14∫Ω(u−1)2≤A1(t)=∫Ω(H(u)−H(1))≤∫Ω(u−1)2(t>t0). | (2.11) |
A similar argument, for the function v, yields that there exists t1 > t0 such that
14∫Ω(v−1)2≤B1(t)≤∫Ω(v−1)2(t>t1). | (2.12) |
We infer from (2.11) and the definitions of E1(t), F1(t) that
E1(t)≤c6F1(t) |
for all t > t1 with some c6 > 0. Plugging this into (2.3), we have
ddtE1(t)≤−εF1(t)≤−εc6E1(t)(t>t1), |
which implies that there exist c7 > 0 and ℓ>0 such that
E1(t)≤c7e−ℓt(t>t1). |
Thus we obtain from (2.11) and (2.12) that
∫Ω(u−1)2+∫Ω(v−1)2+∫Ω(w−˜w)2≤c8E1(t)≤c7c8e−ℓt |
for all t > t1 with some c8 > 0. From the Gagliardo-Nirenberg inequality (2.9) with the regularity of u, v, w, we achieve that there exist C > 0 and λ>0 such that
‖u(t)−1‖L∞(Ω)+‖v(t)−1‖L∞(Ω)+‖w(t)−˜w‖L∞(Ω)≤Ce−λt(t>0). |
This completes the proof of Lemma 2.5.
Proof of Theorem 1.2. Theorem 1.2 follows directly from Lemma 2.5.
The authors would like to thank the referee for valuable comments improving the paper.
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