Improvement of conditions for asymptotic stability in a two-species   chemotaxis-competition model with signal-dependent sensitivity

Masaaki Mizukami

arXiv:1706.04774·math.AP·June 16, 2017

Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

Masaaki Mizukami

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TL;DR

This paper improves the mathematical conditions required for ensuring the long-term stability of a two-species chemotaxis-competition system with signal-dependent sensitivity, advancing understanding of such biological models.

Contribution

It refines the existing conditions for asymptotic stability in a chemotaxis-competition model, narrowing the gap between previous theoretical results.

Findings

01

Enhanced conditions for asymptotic stability are established.

02

The results extend previous stability criteria.

03

The work provides a more comprehensive understanding of the model's behavior.

Abstract

This paper deals with the two-species chemotaxis-competition system. About the problem, Bai--Winkler first obtained asymptotic stability under some conditions. Recently, the conditions assumed in the previous work were improved; however, there is a gap between the conditions assumed in the previous works. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior.

Equations144

⎩ ⎨ ⎧ u_{t} = d_{1} Δ u - \nabla \cdot (u χ_{1} (w) \nabla w) + μ_{1} u (1 - u - a_{1} v) v_{t} = d_{2} Δ v - \nabla \cdot (v χ_{2} (w) \nabla w) + μ_{2} v (1 - a_{2} u - v) w_{t} = d_{3} Δ w + α u + β v - γ w in Ω \times (0, \infty), in Ω \times (0, \infty), in Ω \times (0, \infty),

⎩ ⎨ ⎧ u_{t} = d_{1} Δ u - \nabla \cdot (u χ_{1} (w) \nabla w) + μ_{1} u (1 - u - a_{1} v) v_{t} = d_{2} Δ v - \nabla \cdot (v χ_{2} (w) \nabla w) + μ_{2} v (1 - a_{2} u - v) w_{t} = d_{3} Δ w + α u + β v - γ w in Ω \times (0, \infty), in Ω \times (0, \infty), in Ω \times (0, \infty),

⎩ ⎨ ⎧ u_{t} = d_{1} Δ u - \nabla \cdot (u χ_{1} (w) \nabla w) + μ_{1} u (1 - u - a_{1} v), v_{t} = d_{2} Δ v - \nabla \cdot (v χ_{2} (w) \nabla w) + μ_{2} v (1 - a_{2} u - v), w_{t} = d_{3} Δ w + α u + β v - γ w, \nabla u \cdot ν = \nabla v \cdot ν = \nabla w \cdot ν = 0, u (x, 0) = u_{0} (x), v (x, 0) = v_{0} (x), w (x, 0) = w_{0} (x) x \in Ω, t > 0, x \in Ω, t > 0, x \in Ω, t > 0, x \in \partial Ω, t > 0, x \in Ω,

⎩ ⎨ ⎧ u_{t} = d_{1} Δ u - \nabla \cdot (u χ_{1} (w) \nabla w) + μ_{1} u (1 - u - a_{1} v), v_{t} = d_{2} Δ v - \nabla \cdot (v χ_{2} (w) \nabla w) + μ_{2} v (1 - a_{2} u - v), w_{t} = d_{3} Δ w + α u + β v - γ w, \nabla u \cdot ν = \nabla v \cdot ν = \nabla w \cdot ν = 0, u (x, 0) = u_{0} (x), v (x, 0) = v_{0} (x), w (x, 0) = w_{0} (x) x \in Ω, t > 0, x \in Ω, t > 0, x \in Ω, t > 0, x \in \partial Ω, t > 0, x \in Ω,

u_{t} = u (1 - u - a_{1} v), v_{t} = v (1 - a_{2} u - v) .

u_{t} = u (1 - u - a_{1} v), v_{t} = v (1 - a_{2} u - v) .

Δ w = α u + β v - γ w .

Δ w = α u + β v - γ w .

μ_{1} > \frac{d _{2} χ _{1}^{2} u ^{*}}{\frac{4 a _{1} γ ( 1 - a _{1} a _{2} ) d _{1} d _{2} d _{3}}{( a _{1} α ^{2} + a _{2} β ^{2} - 2 a _{1} a _{2} α β )} - \frac{d _{1} a _{1} χ _{2}^{2} v ^{*}}{4 μ _{2} a _{2}}}, μ_{2} > \frac{χ _{2}^{2} v ^{*} ( a _{1} α ^{2} + a _{2} β ^{2} - 2 a _{1} a _{2} α β )}{16 d _{2} d _{3} a _{2} γ ( 1 - a _{1} a _{2} )} .

μ_{1} > \frac{d _{2} χ _{1}^{2} u ^{*}}{\frac{4 a _{1} γ ( 1 - a _{1} a _{2} ) d _{1} d _{2} d _{3}}{( a _{1} α ^{2} + a _{2} β ^{2} - 2 a _{1} a _{2} α β )} - \frac{d _{1} a _{1} χ _{2}^{2} v ^{*}}{4 μ _{2} a _{2}}}, μ_{2} > \frac{χ _{2}^{2} v ^{*} ( a _{1} α ^{2} + a _{2} β ^{2} - 2 a _{1} a _{2} α β )}{16 d _{2} d _{3} a _{2} γ ( 1 - a _{1} a _{2} )} .

μ_{1} > \frac{M _{1}^{2} u ^{*} ( 1 + δ _{1} ) ( α _{2}^{2} a _{1} δ _{1} + β _{2}^{2} a _{2} - α _{1} β _{1} a _{1} a _{2} ( 1 + δ _{1} ))}{4 a _{1} d _{1} d _{3} γ ( 4 δ _{1} - a _{1} a _{2} ( 1 + δ _{1} ) ^{2} )},

μ_{1} > \frac{M _{1}^{2} u ^{*} ( 1 + δ _{1} ) ( α _{2}^{2} a _{1} δ _{1} + β _{2}^{2} a _{2} - α _{1} β _{1} a _{1} a _{2} ( 1 + δ _{1} ))}{4 a _{1} d _{1} d _{3} γ ( 4 δ _{1} - a _{1} a _{2} ( 1 + δ _{1} ) ^{2} )},

μ_{2} > \frac{M _{2}^{2} v ^{*} ( 1 + δ _{1} ) ( α _{2}^{2} a _{1} δ _{1} + β _{2}^{2} a _{2} - α _{1} β _{1} a _{1} a _{2} ( 1 + δ _{1} ))}{4 a _{2} d _{2} d _{3} γ ( 4 δ _{1} - a _{1} a _{2} ( 1 + δ _{1} ) ^{2} )},

(\frac{u ^{*} χ _{1}^{2}}{4 d _{1} d _{3} a _{1} μ _{1}}, \frac{v ^{*} χ _{2}^{2}}{4 d _{2} d _{3} a _{2} μ _{2}})

(\frac{u ^{*} χ _{1}^{2}}{4 d _{1} d _{3} a _{1} μ _{1}}, \frac{v ^{*} χ _{2}^{2}}{4 d _{2} d _{3} a _{2} μ _{2}})

(\frac{u ^{*} M _{1}^{2}}{4 d _{1} d _{3} a _{1} μ _{1}}, \frac{v ^{*} M _{2}^{2}}{4 d _{2} d _{3} a _{2} μ _{2}})

(\frac{u ^{*} M _{1}^{2}}{4 d _{1} d _{3} a _{1} μ _{1}}, \frac{v ^{*} M _{2}^{2}}{4 d _{2} d _{3} a _{2} μ _{2}})

\displaystyle=\left\{(s,t)\in\mathbb{R}^{2}\,\Big{|}\,0\leq s<\frac{f(q_{0})}{1+q_{0}},\ 0\leq t<\frac{f(q_{0})}{1+q_{0}}\right\},

I := {q > 0 ∣ 4 q - (1 + q)^{2} a_{1} a_{2} > 0}, f (q) := \frac{γ ( 4 q - ( 1 + q ) ^{2} a _{1} a _{2} )}{a _{1} α ^{2} q + a _{2} β ^{2} - a _{1} a _{2} α β ( 1 + q )}

I := {q > 0 ∣ 4 q - (1 + q)^{2} a_{1} a_{2} > 0}, f (q) := \frac{γ ( 4 q - ( 1 + q ) ^{2} a _{1} a _{2} )}{a _{1} α ^{2} q + a _{2} β ^{2} - a _{1} a _{2} α β ( 1 + q )}

χ_{1} (s) \leq M_{1}, χ_{2} (s) \leq M_{2} \mbox f or a l l s > 0,

χ_{1} (s) \leq M_{1}, χ_{2} (s) \leq M_{2} \mbox f or a l l s > 0,

(\frac{u ^{*} M _{1}^{2}}{4 d _{1} d _{3} a _{1} μ _{1}}, \frac{v ^{*} M _{2}^{2}}{4 d _{2} d _{3} a _{2} μ _{2}}) \in q \in I ⋃ {(s, t) \in R^{2} ∣ s, t \geq 0, s + q t < f (q)},

0 \leq u_{0} \in C (\overline{Ω}) ∖ {0}, 0 \leq v_{0} \in C (\overline{Ω}) ∖ {0}, 0 \leq w_{0} \in W^{1, q} (Ω) (\exists q > n) .

0 \leq u_{0} \in C (\overline{Ω}) ∖ {0}, 0 \leq v_{0} \in C (\overline{Ω}) ∖ {0}, 0 \leq w_{0} \in W^{1, q} (Ω) (\exists q > n) .

∥ u ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ v ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ w ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} \leq M \mbox f or a l l t \geq 1

∥ u ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ v ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ w ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} \leq M \mbox f or a l l t \geq 1

∥ u (\cdot, t) - u^{*} ∥_{L^{\infty} (Ω)} + ∥ v (\cdot, t) - v^{*} ∥_{L^{\infty} (Ω)} + ∥ w (\cdot, t) - w^{*} ∥_{L^{\infty} (Ω)} \leq C e^{- ℓ t} \mbox f or a l l t > 0,

∥ u (\cdot, t) - u^{*} ∥_{L^{\infty} (Ω)} + ∥ v (\cdot, t) - v^{*} ∥_{L^{\infty} (Ω)} + ∥ w (\cdot, t) - w^{*} ∥_{L^{\infty} (Ω)} \leq C e^{- ℓ t} \mbox f or a l l t > 0,

u^{*} := \frac{1 - a _{1}}{1 - a _{1} a _{2}}, v^{*} := \frac{1 - a _{2}}{1 - a _{1} a _{2}}, w^{*} := \frac{α u ^{*} + β v ^{*}}{γ} .

u^{*} := \frac{1 - a _{1}}{1 - a _{1} a _{2}}, v^{*} := \frac{1 - a _{2}}{1 - a _{1} a _{2}}, w^{*} := \frac{α u ^{*} + β v ^{*}}{γ} .

q \in I ⋃ {(s, t) \in R^{2} ∣ s, t \geq 0, s + q t < f (q)}

q \in I ⋃ {(s, t) \in R^{2} ∣ s, t \geq 0, s + q t < f (q)}

\displaystyle=\left\{(s,t)\in\mathbb{R}^{2}\,\Big{|}\,s\geq 0,\ 0\leq t<\frac{a_{2}\gamma}{\alpha(a_{2}\beta-\alpha)_{+}},\ h_{1}(s,t)>0,\ h_{2}^{+}(s,t)>0,\ h_{2}^{-}(s,t)<0\right\},

h_{1} (s, t) :=

h_{1} (s, t) :=

- 4 γ (2 a_{1} α^{2} - 2 a_{1} a_{2} α β + a_{1} a_{2}^{2} β^{2} - a_{1}^{2} a_{2} α^{2}) s

- 4 γ (2 a_{2} β^{2} γ - 2 a_{1} a_{2} α β + a_{1}^{2} a_{2} α^{2} - a_{1} a_{2}^{2} β^{2}) t + 16 γ^{2} (1 - a_{1} a_{2})

h_{2}^{\pm} (s, t) :=

h_{2}^{\pm} (s, t) :=

\pm 4 γ 1 - a_{1} a_{2} .

a_{1} α^{2} q + a_{2} β^{2} - a_{1} a_{2} α β (1 + q) > 0

a_{1} α^{2} q + a_{2} β^{2} - a_{1} a_{2} α β (1 + q) > 0

D_{α} := - a_{1} a_{2} β^{2} (4 q - (1 + q)^{2} a_{1} a_{2}) < 0.

D_{α} := - a_{1} a_{2} β^{2} (4 q - (1 + q)^{2} a_{1} a_{2}) < 0.

χ_{i} \in C^{1 + η} ([0, \infty)) \cap L^{1} (0, \infty),

χ_{i} \in C^{1 + η} ([0, \infty)) \cap L^{1} (0, \infty),

2 d_{i} d_{3} χ_{i}^{'} (w) + ((d_{3} - d_{i}) p + (d_{3} - d_{i})^{2} p^{2} + 4 d_{i} d_{3} p) [χ_{i} (w)]^{2} \leq 0,

w χ_{i} (w) \leq C_{χ} \mbox f or a l l w \geq 0 \mbox an d i = 1, 2, 3

\frac{d}{d t} E (t) \leq - ε \int_{Ω} [(u (\cdot, t) - u^{*})^{2} + (v (\cdot, t) - v^{*})^{2} + (w (\cdot, t) - w^{*})^{2}] \mbox f or a l l t > 0

\frac{d}{d t} E (t) \leq - ε \int_{Ω} [(u (\cdot, t) - u^{*})^{2} + (v (\cdot, t) - v^{*})^{2} + (w (\cdot, t) - w^{*})^{2}] \mbox f or a l l t > 0

\int_{Ω} \frac{χ _{1} ( w )}{u} \nabla u \cdot \nabla w \mbox an d \int_{Ω} \frac{χ _{2} ( w )}{v} \nabla v \cdot \nabla w .

\int_{Ω} \frac{χ _{1} ( w )}{u} \nabla u \cdot \nabla w \mbox an d \int_{Ω} \frac{χ _{2} ( w )}{v} \nabla v \cdot \nabla w .

∥ u ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ v ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ w ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} \leq M \mbox f or a l l t \geq 1

∥ u ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ v ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} + ∥ w ∥_{C^{2 + θ, 1 + \frac{θ}{2}} (\overline{Ω} \times [t, t + 1])} \leq M \mbox f or a l l t \geq 1

a > 0, a d - \frac{b ^{2}}{4} > 0, a df + \frac{b ce}{4} - \frac{c d ^{2}}{4} - \frac{b ^{2} f}{4} - \frac{a e ^{2}}{4} > 0.

a > 0, a d - \frac{b ^{2}}{4} > 0, a df + \frac{b ce}{4} - \frac{c d ^{2}}{4} - \frac{b ^{2} f}{4} - \frac{a e ^{2}}{4} > 0.

a y_{1}^{2} + b y_{1} y_{2} + c y_{1} y_{3} + d y_{2}^{2} + e y_{2} y_{3} + f y_{3}^{2} \geq ε (y_{1}^{2} + y_{2}^{2} + y_{3}^{2})

a y_{1}^{2} + b y_{1} y_{2} + c y_{1} y_{3} + d y_{2}^{2} + e y_{2} y_{3} + f y_{3}^{2} \geq ε (y_{1}^{2} + y_{2}^{2} + y_{3}^{2})

^{t} X A_{ε} X = (a - ε) y_{1}^{2} + b y_{1} y_{2} + c y_{1} y_{3} + (d - ε) y_{2}^{2} + e y_{2} y_{3} + (f - ε) y_{3}^{2} \geq 0

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0002010Mathematics Subject Classification. Primary: 35K51; Secondary: 92C17, 35B40. 000*Key words and phrases: chemotaxis; Lotka–Volterra; global existence; stabilization. *

**Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model

with signal-dependent sensitivity **

Masaaki Mizukami 000 E-mail: [email protected]

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Abstract. This paper deals with the two-species chemotaxis-competition system

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$ , $n\geq 2$ ; $\chi_{i}$ are functions satisfying some conditions. About this problem, Bai–Winkler [1] first obtained asymptotic stability in (1.1) under some conditions in the case that $a_{1},a_{2}\in(0,1)$ . Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that $a_{1},a_{2}\in(0,1)$ .

1 Introduction

This paper presents improvement of [1, 6]. In this paper we consider the two-species chemotaxis system with competitive kinetics

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ ( $n\geq 2$ ) with smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector to $\partial\Omega$ ; $d_{1},d_{2},d_{3},\mu_{1},\mu_{2},a_{1},a_{2}$ and $\alpha,\beta,\gamma$ are positive constants; $\chi_{1},\chi_{2},u_{0},v_{0},w_{0}$ 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 chemical substance at place $x$ and time $t$ .

The problem (1.1), which is proposed by Tello–Winkler [12], is a problem on account of the influence of chemotaxis, diffusion, and the Lotka–Volterra competitive kinetics, i.e., with coupling coefficients $a_{1},a_{2}>0$ in

[TABLE]

The mathematical difficulties of the problem (1.1) are to deal with the chemotaxis term $\nabla\cdot(u\nabla w)$ and the competition term $u(1-u-a_{1}v)$ . To overcome these difficulties, firstly, the parabolic-parabolic-elliptic problem (i.e., $w_{t}$ is replaced with [math] in (1.1)) was studied and some conditions for global existence and stabilization in (1.1) were established ([2, 11, 12]). In the parabolic-parabolic-elliptic case global existence of classical solutions to (1.1) and their asymptotic behavior were obtained in the case that $a_{1},a_{2}\in(0,1)$ ([2, 12]) and the case that $a_{1}>1>a_{2}$ ([11]). Recently, these results which give global existence and stabilization in (1.1) were improved in some cases ([7]).

On the other hand, in general, the fully parabolic problem (1.1) is a more difficult problem than the parabolic-parabolic-elliptic case; because we cannot use the relation

[TABLE]

About this problem, global existence and boundedness were shown in the 2-dimensional case ([1]) and the $n$ -dimensional case ([4]). Moreover, in the case that $a_{1},a_{2}\in(0,1)$ , Bai–Winkler [1] obtained asymptotic stability in (1.1) under the conditions

[TABLE]

Recently, in [6], the conditions (1.3) were improved; asymptotic behavior of solutions holds when there exists $\delta_{1}>0$ satisfying $4\delta_{1}-a_{1}a_{2}(1+\delta_{1})^{2}>0$ ,

[TABLE]

where $M_{1},M_{2}>0$ are some constants satisfying $\chi_{1}(s)\leq M_{1}$ , $\chi_{2}(s)\leq M_{2}$ for all $s\geq 0$ . Here we note that the conditions (1.3) and (1.4)–(1.5) can be rewritten as

[TABLE]

and

[TABLE]

respectively, where

[TABLE]

and $q_{0}\in I$ is a maximizer of $\frac{f(q)}{1+q}$ . The regions derived from (1.6) and (1.7) are described in Figure 1.

These results [1, 6] were also concerned with asymptotic stability in (1.1) in the case that $a_{1}\geq 1>a_{2}$ . More related works can be found in [5, 6, 8, 9, 10, 13]; global existence and boundedness in (1.1) with general sensitivity functions can be found in [6, 13]; related works which treated the non-competition case are in [5, 8, 9, 10].

In summary the conditions for asymptotic stability in (1.1) are known; however, there is a gap between the conditions (1.3) and (1.4)–(1.5) (for more details, see Figure 1). The purpose of this work is to improve the conditions assumed in [1] and [6] for asymptotic behavior in the case that $a_{1},a_{2}\in(0,1)$ . In order to attain this purpose we shall assume throughout this paper that there exists $M_{1},M_{2}>0$ satisfying

[TABLE]

where the interval $I$ and the function $f$ are defined as (1.8). The region derived from the condition (1.10) is described in Figure 2, and include the regions derived from (1.6) and (1.7).

Now the main results read as follows. We suppose that the initial data $u_{0},v_{0},w_{0}$ satisfy

[TABLE]

The first theorem is concerned with asymptotic behavior in (1.1) in the case $a_{1},a_{2}\in(0,1)$ .

Theorem 1.1.

Let $d_{1},d_{2},d_{3},\mu_{1},\mu_{2},\alpha,\beta,\gamma>0$ , $a_{1},a_{2}\in(0,1)$ be constants, let $\chi_{1},\chi_{2}$ be nonnegative functions and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that there exists a unique global classical solution $(u,v,w)$ of (1.1) satisfying

[TABLE]

with some $\theta\in(0,1)$ and $M>0$ . Then under the conditions (1.9)–*(1.11), the solution $(u,v,w)$ satisfies that there exist $C>0$ and $\ell>0$ such that *

[TABLE]

where

[TABLE]

Remark 1.1.

The condition (1.10) improves the conditions assumed in [1] and [6] (for more details, see Section 3). Moreover, from the careful calculations we have that

[TABLE]

where

[TABLE]

and

[TABLE]

Remark 1.2.

We note from $q\in I$ that

[TABLE]

holds. Indeed, the discriminant of $a_{1}q\alpha^{2}-a_{1}a_{2}\beta(1+q)\alpha+a_{2}\beta^{2}$ is negative:

[TABLE]

Then a combination of results concerned with global existence and boundedness in (1.1) ([1, 4, 6]) and Theorem 1.1 implies the following results.

Theorem 1.2.

Let $d_{1},d_{2},d_{3}>0$ , $\mu_{1},\mu_{2}>0$ , $a_{1},a_{2}\in(0,1)$ , $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that $\chi_{1},\chi_{2}>0$ are constants and one of the following two properties is satisfied:**

(i)

$n=2$ ,

(ii)

$\Omega$ * is a convex domain and $\mu_{1}>\frac{n\chi_{1}}{4}$ , $\mu_{2}>\frac{n\chi_{2}}{4}$ , $\mu_{1}+\frac{a_{1}\mu_{1}}{2}+\frac{a_{2}\mu_{2}\chi_{1}}{2\chi_{2}}>\frac{n\chi_{1}}{2}$ and $\mu_{2}+\frac{a_{2}\mu_{2}}{2}+\frac{a_{1}\mu_{1}\chi_{2}}{\chi_{1}}>\frac{n\chi_{2}}{2}$ hold.*

Then under the conditions (1.9)–(1.11), the same conclusion as in Theorem 1.1 holds.

Theorem 1.3.

Let $d_{1},d_{2},d_{3}>0$ , $\mu_{1},\mu_{2}>0$ , $a_{1},a_{2}\in(0,1)$ , $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that the functions $\chi_{1},\chi_{2}$ satisfy the following conditions:**

[TABLE]

with some $\eta\in(0,1)$ , $p>n$ and $C_{\chi}>0$ . Then under the conditions (1.9)–(1.11), the same conclusion as in Theorem 1.1 holds.

Remark 1.3.

We note from the global-in-time lower estimate for $w$ ([3]) that the same arguments as in the proof of Theorem 1.1 can also be applied to the case that $\chi_{i}(w)=\frac{K_{i}}{w}$ .

The strategy of the proof of Theorem 1.1 is to modify the methods in [1, 6]. One of the keys for the proof of Theorem 1.1 is to derive the following energy estimate:

[TABLE]

with some positive function $E$ and some constant $\varepsilon>0$ . Thanks to (1.12), we can obtain Theorem 1.1. The key for the improvement is to provide the best estimate for the terms

[TABLE]

This paper is organized as follows. In Section 2 we prove asymptotic stability in the case that $a_{1},a_{2}\in(0,1)$ (Theorems 1.1, 1.2 and 1.3). Section 3 is devoted to discussions; we confirm that the condition (1.10) improves the conditions assumed in [1] and [6].

2 Proof of Theorem 1.1

In this section we prove stabilization in (1.1) in the case that $a_{1},a_{2}\in(0,1)$ . Here we assume that there exists a unique global classical solution $(u,v,w)$ of (1.1) satisfying

[TABLE]

with some $\theta\in(0,1)$ and $M>0$ . We first provide the following lemma which will be used later.

Lemma 2.1.

Let $a,b,c,d,e,f\in\mathbb{R}$ . Suppose that

[TABLE]

Then there exists $\varepsilon>0$ such that

[TABLE]

holds for all $y_{1},y_{2},y_{3}\in\mathbb{R}$ .

Proof.

In order to prove this lemma we shall see that there is $\varepsilon>0$ such that

[TABLE]

holds for all $y_{1},y_{2},y_{3}\in\mathbb{R}$ , where

[TABLE]

To confirm that there is $\varepsilon>0$ such that (2.2) holds for all $y_{1},y_{2},y_{3}\in\mathbb{R}$ we put

[TABLE]

and shall show the existence of $\varepsilon_{1}>0$ satisfying $g_{i}(\varepsilon_{1})>0$ for $i=1,2,3$ . Now thanks to (2.1), we can see that

[TABLE]

and

[TABLE]

Thus a combination of the above inequalities and the continuity argument yields that there is $\varepsilon_{1}>0$ such that $g_{i}(\varepsilon_{1})>0$ hold for $i=1,2,3$ . Therefore aided by the Sylvester criterion, we have (2.2) for all $y_{1},y_{2},y_{3}\in\mathbb{R}$ , which means the end of the proof. ∎

Then we will prove the following energy estimate which leads to asymptotic behavior of solutions to (1.1). The proof is mainly based on the methods in [1, 6].

Lemma 2.2.

Let $a_{1},a_{2}\in(0,1)$ and let $(u,v,w)$ be a solution to (1.1). Then under the conditions (1.9)–(1.11), there exist a nonnegative function $E:(0,\infty)\to\mathbb{R}$ and a constant $\varepsilon>0$ such that

[TABLE]

holds for all $t>0$ .

Proof.

In light of (1.10) we can take $q>0$ such that

[TABLE]

and

[TABLE]

hold and $\delta>0$ satisfying

[TABLE]

For all $t>0$ we denote by $A(t)$ , $B(t)$ , $C(t)$ the functions defined as

[TABLE]

and shall confirm that the function $E(t)$ defined as

[TABLE]

satisfies (2.3) with some $\varepsilon>0$ . Firstly the Taylor formula enables us to see that $E(t)$ is a nonnegative function for $t>0$ (for more details, see [1, Lemma 3.2]). From the straightforward calculations we infer

[TABLE]

Hence we have

[TABLE]

where

[TABLE]

and

[TABLE]

In order to confirm that there is $\varepsilon_{1}>0$ such that

[TABLE]

for all $t>0$ we will see that the assumption of Lemma 2.1 is satisfied with

[TABLE]

and

[TABLE]

From the definitions of $q,\delta>0$ we can see that

[TABLE]

and

[TABLE]

Hence thanks to Lemma 2.1, there exists $\varepsilon_{1}>0$ such that (2.4) holds. We next verify that there is $\varepsilon_{2}>0$ such that

[TABLE]

By virtue of the Young inequality, we infer from (1.9) that

[TABLE]

and

[TABLE]

which implies that

[TABLE]

Therefore plugging the definition of $\delta>0$ into (2.6) leads to the existence of $\varepsilon_{2}>0$ such that (2.5) holds, which concludes the proof of this lemma. ∎

Then we have the following desired estimate.

Lemma 2.3.

Let $a_{1},a_{2}\in(0,1)$ and assume that (1.9)–(1.11) are satisfied. Then there exist $C>0$ and $\ell>0$ such that

[TABLE]

Proof.

The same arguments as in the proofs of [1, Theorems 3.3, 3.6 and 3.7] enable us to obtain this lemma. ∎

Proof of Theorem 1.1.

Lemma 2.3 immediately leads to Theorem 1.1. ∎

Proof of Theorems 1.2 and 1.3.

A combination of results concerned with global existence and boundedness in (1.1) ([1, 4, 6]) and the standard parabolic regularity argument, along with Theorem 1.1 implies Theorems 1.2 and 1.3. ∎

3 Discussions

In this section we shall confirm that the condition (1.10) improves the conditions assumed in the previous works aided by the view of (1.6) and (1.7). Here since

[TABLE]

holds for every $q\in I$ , we can see that the condition (1.10) improves the conditions assumed in [6]. In order to accomplish the purpose of this section, noting that

[TABLE]

we will confirm that

[TABLE]

To see (3.1) we shall see that $q=1$ is not the best choice, i.e., $q=1$ is not a maximizer of the functions $f(q)$ and

[TABLE]

From the straightforward calculations we infer that

[TABLE]

Now we will divide into two cases and show that (3.1) holds for each cases.

Case 1: $a_{1}\alpha\neq\beta$ and $a_{2}\beta\neq\alpha$ .** In this case, by virtue of (3.2), we can see that**

[TABLE]

which means that $q=1$ is not a maximizer of $f$ and $g$ . Thus one of the following two properties holds:

[TABLE]

Therefore we obtain that (3.1) holds in this case.

Case 2: $a_{1}\alpha=\beta$ or $a_{2}\beta=\alpha$ .** We first deal with the case that $a_{1}\alpha=\beta$ . In this case, in view of the fact that**

[TABLE]

$q=1$ ** is a maximizer of $f$ . On the other hand, thanks to (3.2) together with the fact that $\alpha-a_{2}\beta=\alpha(1-a_{1}a_{2})>0$ , we can see that $q=1$ is not a maximizer of $g$ ; there is $q_{2}\in I$ such that**

[TABLE]

Similarly, in the case that $a_{2}\beta=\alpha$ , we infer that $q=1$ is a maximizer of $g$ and is not a maximizer of $f$ ; there exists $q_{3}\in I$ such that

[TABLE]

Hence we derive that (3.1) holds also in this case.

According to the above two cases, we can obtain that (3.1) holds, which means the condition (1.10) improves the conditions assumed in [1]. Therefore we attain the purpose of this paper.

Bibliography13

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] X. Bai and M. Winkler. Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. , 65:553–583, 2016.
2[2] T. Black, J. Lankeit, and M. Mizukami. On the weakly competitive case in a two-species chemotaxis model. IMA J. Appl. Math. , 81:860–876, 2016.
3[3] K. Fujie. Boundedness in a fully parabolic chemotaxis system with singular sensitivity. J. Math. Anal. Appl. , 424:675–684, 2015.
4[4] K. Lin, C. Mu, and L. Wang. Boundedness in a two-species chemotaxis system. Math. Methods Appl. Sci. , 38:5085–5096, 2015.
5[5] M. Mizukami. Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system. AIMS Mathematics , 1:156–164, 2016.
6[6] M. Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete Contin. Dyn. Syst. Ser. B , 22:2301–2319, 2017.
7[7] M. Mizukami. Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type. submitted. ar Xiv:1703.08389 [math.AP].
8[8] M. Mizukami and T. Yokota. Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion. J. Differential Equations , 261:2650–2669, 2016.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

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1 Introduction

Theorem 1.1**.**

Remark 1.1**.**

Remark 1.2**.**

Theorem 1.2**.**

Theorem 1.3**.**

Remark 1.3**.**

2 Proof of Theorem 1.1

Lemma 2.1**.**

Proof.

Lemma 2.2**.**

Proof.

Lemma 2.3**.**

Proof.

Proof of Theorem 1.1.

Proof of Theorems 1.2 and 1.3.

3 Discussions

Theorem 1.1.

Remark 1.1.

Remark 1.2.

Theorem 1.2.

Theorem 1.3.

Remark 1.3.

Lemma 2.1.

Lemma 2.2.

Lemma 2.3.