Existence of steady-state solutions for a class of competing systems with cross-diffusion

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Existence of steady-state solutions for a class of competing systems with cross-diffusion

and self-diffusion

Ningning Zhu

^B

School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, P.R. China Received 10 November 2020, appeared 10 January 2021

Communicated by Eduardo Liz

Abstract. We focus on a system of two competing species with cross-diffusion and self- diffusion. By constructing an appropriate auxiliary function, we derive the sufficient conditions such that there are no coexisting steady-state solutions to the model. It is worth noting that the auxiliary function constructed above is applicable to Dirichlet, Neumann and Robin boundary conditions.

Keywords: cross-diffusion, self-diffusion, steady-state solution, maximum principle.

2020 Mathematics Subject Classification: 35J57, 35B50.

1 Introduction

In this work, we study the steady-state solutions of the following competing systems with cross-diffusion and self-diffusion:











∂u

∂t =_∆[(d₁+a₁₁u+a₁₂v)u] +u(a₁−b₁u−c₁v), x∈_Ω,t>0,

∂v

∂t =_∆[(d₂+a₂₁u+a₂₂v)v] +v(a₂−b₂u−c₂v)_, x∈_Ω,t>_0, α₁u+β₁∂u

∂ν =α₂v+β₂∂v

∂ν =0, x∈_∂Ω,t >0, u(x, 0) =u₀(x)≥0, v(x, 0) =v₀(x)≥0, x∈_Ω,

(1.1)

whereΩ⊂_Rⁿ(n≥1)is a bounded domain with smooth boundary,uandvare the densities of two competing species, α_i,β_i and a_ij (i,j = 1, 2) are nonnegative constants, a_i,b_i,c_i and d_i (i= _{1, 2})are all positive constants, a₁₁ anda₂₂stand for the self-diffusion pressures, while a₁₂ and a₂₁ are the cross-diffusion pressures, a₁,a₂ represent the intrinsic growth rates of the two species, b₁,c₂ describe the intra-specific competitions, whileb₂,c₁ describe the inter- specific competitions, andd₁,d₂are their diffusion rates.

BEmail: zhunn5@mail2.sysu.edu.cn

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System (1.1) was first proposed by Shigesada, Kawasaki and Teramoto [10] in 1979 to investigate the spatial segregation of interacting species. In the last several decades, a great deal of mathematical effort has been devoted to the study of the model. For the smooth solutions of the system (1.1) with homogeneous Neumann boundary conditions, [4] and [11]

obtained the global existence and boundedness in a bounded convex domain. We refer to [2,3,5–7] for the study of the positive steady-state solutions. For instance, Lou and Ni [2]

established the sufficient conditions for the existence and nonexistence of nonconstant steady- state solutions in the strong and weak competition case, respectively. Whena₂₁= a22 =0, Lou et al. [5] provided the parameters ranges such that the system has no nonconstant positive solutions fora₁₁=0 anda₁₁6=0, respectively.

For literatures about the system (1.1) under homogeneous Dirichlet boundary conditions, see [1,8,12,14] and references therein. In [9], by the decomposing operators and the theory of fixed point, Ryn and Ahn discussed the existence of the positive coexisting steady-state of system (1.1) for two competing species or predator-prey species.

Motivated by [5], we introduce the effect of self-diffusion and consider the model under three boundary conditions. Our purpose is to establish the sufficient conditions such that the following system has no coexisting solutions:











∆[(d₁+αv)u] +u(a₁−b₁u−c₁v) =0, x ∈_Ω,

∆[(d₂+βv)v] +v(a₂−b₂u−c₂v) =0, x ∈_Ω, α₁u+β₁∂u

∂ν

=α₂v+β₂∂v

∂ν

=0, x ∈∂Ω,

(1.2)

where α_i ≥ 0,β_i ≥ 0 and α_i+β_i > 0 for i = 1, 2. In what follows, we always assume that α ≥ 0,β ≥ 0,a_i > 0,b_i > 0,c_i > 0 and d_i > 0 for i = 1, 2. To achieve that, the main tools we use are the strong maximum principle, Hopf’s boundary lemma and the divergence theorem. Since u and v represent species densities, we are interested in the nonnegative classical solution(u,v) of (1.2), which means that (u,v) ∈ (C¹(_Ω)∩C²(_Ω))², u,v ≥ 0 in Ω, and satisfies (1.2) in the pointwise sense.

The remainder of this work is organized as follows. In Section 2, we show that the nonnegative classical solutions are strictly positive if they are not identically equal to zero, which plays a key role in the proof of main theorems. Section 3 constructs an auxiliary function, which can be used to produce contradictions, and thus parameter ranges for nonexistence of coexisting steady-state solutions will be obtained under three boundary conditions.

2 Preliminaries

Let us first give the following proposition by applying the strong maximum principle, which indicates that nonnegative classical solutions are strictly positive if they are nontrivial.

Proposition 2.1. Suppose that(u,v)is a nonnegative classical solution of (1.2). Then if u 6≡0, we have u>0inΩ, and if v6≡0, we have v>0inΩ.

Proof. We only proveu>_{0 in} _Ω_wheneveru6≡0, since the positivity ofvinΩcan be proved in a similar way. Letw= (d₁+αv)u. Due tod₁>0,α≥0 andv≥0 inΩ, it suffices to prove w>0 inΩ. Otherwise, there isx₀∈ _Ωsuch thatw(x₀) =min_x_∈_Ωw(x) =0.

It follows from the first equation of (1.2) that

∆w+u(a₁−b₁u−c₁v) =_∆w+ ^a¹−b₁u−c₁v

d₁+αv ·w=0.

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Let

Lw =−_∆w+cw with c= ^b¹^u+c₁v d₁+αv . Then

c≥0 and Lw = ^a¹^w

d₁+αv ≥0 inΩ.

So, an application of the strong maximum principle shows that w is constant in Ω, and thusw=0, a contradiction tou6≡0. This completes the proof.

Remark 2.2. When α_i = 0 and β_i > 0 for i = 1, 2, that is, in the case of Neumann boundary conditions, we can get further thatu,v>0 inΩby Hopf’s boundary lemma.

Next, we list two lemmas about the existence of positive solutions for single equation under Dirichlet or Robin boundary conditions, which can reveal the existence of semi steady-state solution of system (1.2). The following lemma comes from Theorem 2.1 in [8].

Lemma 2.3. Consider the following problem:

(−_∆[(d+γw)w] =w(a−bw), x∈_Ω,

w=0, x∈_∂Ω, ^(2.1)

where a,b,d are positive constants andγ is nonnegative constant. Letλ^d₁ > 0denote the first eigenvalue of−_∆with the homogeneous Dirichlet boundary condition on∂Ω. If

λ^d₁< ^a d, then problem(2.1)has a unique positive solution.

From now on, ifλ^d₁ < _d^a¹

1 andλ^d₁ < ^a_d²

2, we denoteu^∗andv^∗as the unique positive solution of systems

(−d₁∆u+ (b₁u−a₁)u=0, x∈ _Ω,

u=0, x∈ _∂Ω,

and

(−_∆[(d₂+βv)v] + (c₂v−a₂)v =0, x∈_Ω,

v=0, x∈_∂Ω,

respectively.

For Robin boundary conditions, the corresponding result can be found in Theorem 2.10 of [9].

Lemma 2.4. Consider the following system:







−_∆[(d(x) +γw)w] =w(a(x)−bw), x ∈_Ω, δw+η∂w

∂ν =0, x ∈_∂Ω, ^(2.2)

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whereγis a nonnegative constant, b,δ,ηare positive constants and d(x),a(x)∈C²(_Ω)with d(x)>

0for all x ∈_{Ω. If} _∂ν^∂(d(x))≤0on∂Ωandλ₁(d(x),a(x),δ,η)>0, then(2.2)has a unique positive solution, where

λ₁(d(x),a(x),δ,η) = R

Ω

− |∇[d(x)φ₁]|²+d(x)a(x)φ²₁ −R

∂Ωd(x)^h^δ

ηd(x)−^∂d⁽^x⁾

∂ν

i φ²₁ k^pd(x)φ₁k²

L²(_Ω)

denotes the principal eigenvalue with eigenfunctionφ₁ of the following eigenvalue problem:







∆[(d(x)φ] +a(x)φ= λφ, x∈_Ω, δφ+η∂φ

∂ν =0, x∈_∂Ω.

Similarly, if λ₁(d₁,a₁,α₁,β₁) > 0 and λ₁(d₂,a₂,α₂,β₂) > 0, we write u^∗∗ and v^∗∗ as the unique positive solution of systems







−d₁∆u+ (b₁u−a₁)u=0, x∈ _Ω, α₁u+β₁∂u

∂ν =0, x∈ ∂Ω, and







−_∆[(d2+βv)v] + (c2v−a2)v =0, x∈_Ω, α₂v+β₂∂v

∂ν =0, x∈_∂Ω,

respectively.

3 Steady-state solutions

Now we give the main theorems of this work. When the intra-competition and inter-competition parameters of one species are greater than inter-competition and intra-competition of the other, respectively, whereas the intrinsic growth rate is less than that of the other, we explore two different sufficient criteria for nonexistence of coexisting solutions of system (1.2).

3.1 Dirichlet boundary conditions

Theorem 3.1. Letα_i >0,β_i =0for i=1, 2,α≥0andβ≥0. Assume that(u,v)is a nonnegative classical solution of (1.2). If

(i) b₁ >b₂, c₁>c₂, a₁ <a₂, d₁ ≥d₂ and α≥β or

(ii) b₁<b2, c₁< c2, a₁ >a2, d₁≤d2 and α≤ β, then we have either

(u,v)≡(0, 0), or

(u,v) = (u^∗, 0) if λ^d₁ < ^a¹ d₁, or

(u,v) = (0,v^∗) if λ^d₁ < ^a² d₂.

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Proof. (i) By way of contradiction, suppose thatu 6≡ 0 and v 6≡0. Thus, u andv are positive in Ωby Proposition2.1. So, it is apparent from system (1.2) that:











∆[(d₁+αv)u]

u =−a₁+b₁u+c₁v, x∈_Ω,

∆[(d₂+βv)v]

v =−a₂+b₂u+c₂v, x∈_Ω,

u=v=_0, x∈∂Ω.

(3.1)

Let

w= (d₁+αv)u. (3.2)

Then, by (3.1) and conditionsb₁>b₂, c₁ >c₂anda₁<a₂, we have

∆w

w > ^∆[(d2+βv)v]

(d₁+αv)v inΩ. (3.3)

We now define a function p(s) =s

d2−d1

d1 (d₁+αs)

2β−α−^d² d1α

α fors >0. (3.4)

It is easy to verify that p(s)>0 for any s>0. Moreover, a direct calculation implies that p⁰(s) = p(s) ^d²−d₁

d₁s + ^2β−α− ^d_d²

1α d₁+αs

! . This, together with (3.3), yields that

div[(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]]

= (d₁+αv)vp(v)_∆w+∇[(d₁+αv)vp(v)]· ∇w

−wp(v)_∆[(d₂+βv)v]− ∇[wp(v)]· ∇[(d₂+βv)v]

>∇[(d₁+αv)vp(v)]· ∇w− ∇[wp(v)]· ∇[(d2+βv)v] inΩ.

Furthermore, we can see that

∇[(d₁+αv)vp(v)]· ∇w− ∇[wp(v)]· ∇[(d₂+βv)v]

=^hαvp(v)∇v+ (d₁+αv)p(v)∇v+ (d₁+αv)vp⁰(v)∇vi

·^hαu∇v+ (d₁+αv)∇ui

−^hαup(v)∇v+ (d₁+αv)p(v)∇u+ (d₁+αv)up⁰(v)∇vi

·^hβv∇v+ (d₂+βv)∇vi

=|∇v|²^h2α²uvp(v) +d₁αup(v) +d₁αuvp⁰(v) +α²uv²p⁰(v)−2αβuvp(v)

−2d₁βuvp⁰(v)−2αβuv²p⁰(v)−d₂αup(v)−d₁d₂up⁰(v)−d₂αuvp⁰(v)ⁱ +∇u· ∇vh

3d₁αvp(v) +d²₁p(v) +d²₁vp⁰(v) +2d₁αv²p⁰(v) +_2α²v²p(v) +α²v³p⁰(v)−2d₁βvp(v)−2αβv²p(v)−d₁d₂p(v)−d₂αvp(v)ⁱ

,|∇v|²M(u,v) +∇u· ∇vN(u,v).

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Since

N(u,v) =p(v)^h3d₁αv+d²₁+2α²v²−2d₁βv−2αβv²−d₁d₂−d₂αvi

+p⁰(v)v(d₁+αv)²

= p(v)^h(d²₁−d₁d₂) + (3d₁α−2d₁β−d₂α)v+ (_2α²−_2αβ)v² +^d²−d₁

d₁ (d₁+αv)²+v(d₁+αv)(2β−α− ^d² d₁α)ⁱ

=0, and

M(u,v)

= αup(v)(d₁−d2+2αv−2βv) +up⁰(v)−d₁d2+ (d₁α−2d₁β−d2α)v+ (α²−2αβ)v²

= αup(v)(d₁−d₂+2αv−2βv) +up⁰(v)(d₁+αv)(−d₂+αv−2βv)

= αup(v)(d₁−d2+2αv−2βv) +up(v)(d2−d₁−2αv+2βv)

v (−d2+αv−2βv)

=up(v)(d₁−d₂+2αv−2βv) d2

v +2β

≥0,

we conclude that divh

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱ>0 inΩ. (3.5) Now, let

Ωε ={x∈_Ω|dist(_x,∂Ω)>ε} for any smallε>_0.

Since(u,v)∈ (C¹(_Ω)∩C²(_Ω))², we know thath

(d₁+αv)vp(v)∇w−wp(v)∇[(d2+βv)v]ⁱ∈ C¹(_Ω_ε). Then it follows from divergence theorem that

Z

Ωdivh

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱdx (3.6)

= _lim

ε→0

Z

Ωε

div h

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱdx

= lim

ε→0

Z

∂Ωε

h

(d₁+αv)vp(v)∇w−wp(v)∇[(d2+βv)v]ⁱ·νdS

= lim

ε→0

Z

∂Ωε

(d₁+αv)vp(v)^∂w

∂ν −wp(v)^∂[(d₂+βv)v]

∂ν

dS

= lim

ε→0

Z

∂Ωε

"

∂w

∂νv

d2

d1(d₁+αv)

2β−^d² d1α

α −βu∂v

∂νv

d2

d1(d₁+αv)

2β−^d² d1α α

− ^u

v(d2+βv)^∂v

∂νv

d2

d1(d₁+αv)

2β−^d_d² 1α α

# dS , ^lim

ε→0(I₁(ε) +I₂(ε) +I₃(ε)).

Obviously, I₁(ε)and I₂(ε)both tend to zero asε→0. To deal with the term I₃(ε), we take V=

ϕ(x)∈C¹(_Ω)

ϕ|_Ω>0, ϕ|_∂_Ω =0, ∂ϕ

∂ν ∂Ω <0

.

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Then Hopf’s boundary lemma tells us that ^∂u_∂ν⁽^x⁰⁾ <0 and ^∂v_∂ν⁽^x⁰⁾ <0 for anyx₀ ∈∂Ω, and thus u∈Vandv∈V. Define

g(x):=









 u(x)

v(x)^, ^x∈_Ω,

∂u(x)

∂ν

∂v(x)

∂ν , x∈_∂Ω.

Then by applying Lemma 2.4 in [13], we get g(x) ∈ C Ω,(0,+_∞). Therefore I₃(ε) also approaches to zero asε→0.

As a result,R

Ωdivh

(d₁+αv)vp(v)∇w−wp(v)∇[(d2+βv)v]ⁱdx =0 because of Lebesgue dominated convergence theorem and the boundary conditions in (1.2), which contradicts (3.5).

So either (u,v) = (0, 0), or only one of them is equal to zero. Whenv≡0, we have







−_∆u+ ¹

d₁(b₁u−a₁)u=0, x∈_Ω,

u=0, x∈_∂Ω.

By Lemma 2.3, we can see (u,v) = (u^∗, 0) if λ^d₁ < _d^a¹

1. Similarly, if u ≡ 0 and λ^d₁ < _d^a²

2, then (u,v) = (0,v^∗). This finishes the proof of the first part.

(ii) Now, we also assume that u 6≡ 0 and v 6≡ 0. Then an application of Proposition2.1 provides thatuandvare positive inΩ. Hence,

∆w

w < ^∆[(d2+βv)v]

(d₁+αv)v inΩ,

according to the hypotheses b₁ <b2,c₁< c2anda₁ >a2, wherewis defined by (3.2).

Givend₁ ≤d₂ andα≤ β, we know that div

h

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱ<₀ _in_Ω, where p(v)is defined as in (i).

Furthermore, again by divergence theorem, we can prove that Z

Ωdivh

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱdx =0,

a contradiction. By repeating the argument in (i), we complete the proof of Theorem3.1.

3.2 Neumann boundary conditions

In (3.6), if we consider Neumann boundary conditions, we can directly check that (3.6) equals to zero. Consequently, the following theorem is stated without proof.

Theorem 3.2. Suppose thatα_i =0,β_i > 0for i =1, 2and(u,v)is a nonnegative classical solution of (1.2). If

(_i) b₁> b₂, c₁ >c₂, a₁< a₂, d₁≥ d₂ and α≥ β or

(ii) b₁ <b2, c₁<c2, a₁ >a2, d₁ ≤d2 and α≤β, then either(u,v)≡ (_{0, 0}), or(u,v) = ^a_b¹

1, 0

, or(u,v) = _0,^a_c²

2

.

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3.3 Robin boundary conditions

In this subsection, we consider the case in whichα_i andβ_i (i=1, 2) are both positive.

Theorem 3.3. Letα_i >0,β_i >0for i=1, 2and(u,v)be a nonnegative classical solution of (1.2). If (i) b₁ >b₂, c₁ >c₂, a₁< a₂, d₁ ≥d₂,α≥β and α₁

β₁ ≥ ^α² β₂

or

(ii) b₁ <b2, c₁<c2, a₁ >a2, d₁ ≤d2,α≤β and α₁ β₁ ≤ ^α²

β₂, then we have either

(u,v)≡(0, 0), or

(u,v) = (u^∗∗, 0) if λ₁(d₁,a₁,α₁,β₁)>0, or

(u,v) = (0,v^∗∗) if λ₁(d₂,a₂,α₂,β₂)>0.

Proof. We only prove (i), as (ii) can be proved in a same manner. According to the arguments of the proof of Theorem3.1, we can obtain from the hypothesisb₁ >b₂,c₁ > c₂,a₁ < a₂,d₁ ≥ d₂ andα≥βthat

divh

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱ>0, where the function p(v)is introduced in (3.4).

We mention that ^∂u_∂ν =−^α¹

β1u,^∂v_∂ν =−^α²

β2von ∂Ω. The boundary integral becomes that Z

Ωdivh

(d₁+αv)vp(v)∇w−wp(v)∇[(d₂+βv)v]ⁱdx

=

Z

∂Ωv

d2

d1(d₁+αv)

2β−^d_d² 1α α

αu∂v

∂ν+ (d₁+αv)^∂u

∂ν −2βu∂v

∂ν−d2

u v

∂v

∂ν

dS

=

Z

∂Ωv

d2

d1(d₁+αv)

2β−^d² d1α α

uv

2βα₂

β₂ −αα₁ β₁ −αα₂

β₂

+u

d₂α₂

β₂ −d₁α₁ β₁

dS

≤0,

due to d₁ ≥ d2,α ≥ β and ^α_β¹

1 ≥ ^α_β²

2, a contradiction. Thus, ifv = 0, we have(u,v) = (u^∗∗, 0) whenλ₁(d₁,a₁,α₁,β₁)>0. Similarly,(u,v) = (0,v^∗∗)ifu = 0 andλ₁(d₂,a₂,α₂,β₂) >0. This completes the proof.

Acknowledgements

The author would like to deeply appreciate Prof. Taishan Yi for valuable guidance and comments. And the author thanks the anonymous referee for her/his comments and suggestions.

Research was supported partially by the National Natural Science Foundation of China (Grant No. 11971494).

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