Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions

doi:10.1016/S1468-1218(02)00014-7

Nonlinear Analysis: Real World Applications

Volume 4, Issue 1, March 2003, Pages 65-86

https://doi.org/10.1016/S1468-1218(02)00014-7 Get rights and content

Abstract

Uniform boundedness and convergence of global solutions are proved for quasilinear parabolic systems with cross-diffusions dominated by self-diffusions in population dynamics. Gagliardo–Nirenberg-type inequalities are used in the estimates of solutions in order to establish W₂¹-bounds uniform in time. In each step of estimates the contribution of the diffusion coefficients is emphasized, and it is concluded that the uniform bound remains independent of the growth of the diffusion coefficient in the system. Hence, convergence of solutions is established for systems with large diffusion coefficients.

Introduction

In this paper we study the following quasilinear parabolic system in population dynamics: $u_{t} =Δ[(d_{1} +α_{11} u+α_{12} v)u]+(a_{1} −b_{1} u−c_{1} v)u in Ω×(0,∞), v_{t} =Δ[(d_{2} +α_{21} u+α_{22} v)v]+(a_{2} −b_{2} u−c_{2} v)v in Ω×(0,∞), ∂u ∂ν = ∂v ∂ν =0 on ∂Ω×(0,∞), u(x,0)=u_{0} (x)⩾0, v(x,0)=v_{0} (x)⩾0 in Ω ̄,$ where $Ω⊂ R^{n}$ is a bounded smooth domain and $α_{ij} ⩾0, d_{i}, a_{i}, b_{i}, c_{i}$ are positive constants for i,j=1,2. System (1.1) was proposed by Shigesada et al. [13] in 1979 in an attempt to model spatial segregation phenomena between two competing species. In system (1.1) u and v are nonnegative functions which represent the population densities of two competing species. d₁ and d₂ are the diffusion rates of the two species, respectively. a₁ and a₂ denote the intrinsic growth rates, b₁ and c₂ account for intra-specific competitions, b₂ and c₁ are the coefficients for inter-specific competitions. When α₁₁=α₁₂=α₂₁=α₂₂=0, (1.1) reduces to the well-known Lotka–Voltera competition–diffusion system. α₁₁ and α₂₂ are usually referred as self-diffusion, and α₁₂,α₂₁ are cross-diffusion pressures. By adopting the coefficients $α_{ij} (i,j=1,2)$ system (1.1) takes into account the pressures created by mutually competing species. For more details on the backgrounds of this model, we refer the reader to [11], [13].

To describe results on system (1.1) we use the following notation throughout this paper.

Notation. Let $Ω$ be a region in $R^{n}$ . The norm in $L_{p} (Ω)$ is denoted by $|·|_{L_{p}(Ω)}$ , 1⩽p⩽∞. The usual Sobolev spaces of real valued functions in $Ω$ with exponent k⩾0 are denoted by $W_{p}^{k} (Ω), 1⩽p<∞$ . And $||·||_{W_{p}^{k}(Ω)}$ represents the norm in the Sobolev space $W_{p}^{k} (Ω)$ . For $Ω=[0,1]⊂ R^{1}$ we shall use the simplified notation ||·||_k,p for $||·||_{W_{p}^{k}(Ω)}$ and |·|_p for $|·|_{L_{p}(Ω)}$ .

The local existence of solutions to (1.1) was established by Amann [2], [3], [4]. According to his results system (1.1) has a unique nonnegative solution u(·,t),v(·,t) in $C([0,T),W_{p}^{1} (Ω))∩C^{∞} ((0,T),C^{∞} (Ω))$ , where T∈(0,∞] is the maximal existence time for the solution $u, v$ . The following result is also due to Amann [3].

Theorem 1.1

Let u₀ and v₀ be in $W_{p}^{1} (Ω)$ . System (1.1) possesses a unique nonnegative maximal smooth solution $u(x,t),v(x,t)∈C([0,T),W_{p}^{1} (Ω))∩C^{∞} (Ω ̄ ×(0,T))$ for 0⩽t<T, where p>n and 0<T⩽∞. If the solution satisfies the estimates $sup_{0<t<T} ||u(·,t)||_{W_{p}^{1}(Ω)} <∞, sup_{0<t<T} ||v(·,t)||_{W_{p}^{1}(Ω)} <∞$ , then T=+∞. If, in addition, u₀ and v₀ are in $W_{p}^{2} (Ω)$ , then $u(x,t),v(x,t)∈C([0,∞),W_{p}^{2} (Ω))$ , and $sup_{0⩽t<∞} ||u(·,t)||_{W_{p}^{2}(Ω)} <∞$ , $sup_{0⩽t<∞} ||v(·,t)||_{W_{p}^{2}(Ω)} <∞$ .

System (1.1) is a special case of the concrete example (7), (8) in the Introduction of [3], and the results stated in Theorem 1.1 are from the theorem in the Introduction of [3].

So far the existence of nonnegative global solutions for system (1.1) has been proved under very restrictive hypotheses only. Kim [7] proved the global existence of smooth nonnegative solutions for $n=1, d_{1} =d_{2}$ and α₁₁=α₂₂=0. Deuring [5] showed the global existence of classical positive solutions to system (1.1) with α₁₁=α₂₂=0 and small coefficients depending on the initial values for n⩾1. In the case n=2, Lou et al. [9] proved that system (1.1) with α₂₁=0 has a unique smooth global solution. Yagi [14] established the global existence of smooth nonnegative solutions to system (1.1) for $Ω⊂ R^{2}$ under the condition either $0<α_{21} <8α_{11}, 0<α_{12} <8α_{22}$ or $α_{21} =α_{22} =0, α_{11} >0$ . Yagi [14] obtained estimates of Gronwall's type depending on T to prove the global existence. His proof holds for $Ω⊂ R^{1}$ with a minor modification. And yet the qualitative properties of those global solutions have not been proved for system (1.1) in the special cases mentioned above.

We study in this paper system (1.1) for the spatial domain $[0,1]⊂ R^{1}$ under the condition $0<α_{21} <8α_{11}, 0<α_{12} <8α_{22}$ as Yagi [14]. System (1.1) is rewritten as follows: $u_{t} =(d_{1} u+α_{11} u^{2} +α_{12} uv)_{xx} +u(a_{1} −b_{1} u−c_{1} v) in [0,1]×(0,∞), v_{t} =(d_{2} v+α_{21} uv+α_{22} v^{2})_{xx} +v(a_{2} −b_{2} u−c_{2} v) in [0,1]×(0,∞), u_{x} (x,t)=v_{x} (x,t)=0 at x=0,1, u(x,0)=u_{0} (x)⩾0, v(x,0)=v_{0} (x)⩾0 in [0,1],$ where α_ij,d_i,a_i,b_i,c_i are positive constants for i,j=1,2. Throughout this paper we assume that the initial functions u₀(x),v₀(x) are not identically zero and in the function space W₂¹([0,1]).

We first prove the uniform boundedness of the global solutions of system (1.2) under condition (α). Applying Gagliardo–Nirenberg-type inequalities we establish the uniform W₂¹-bound independent of T, the maximal existence time, for the solutions obtained in Theorem 1.1. Thus, we have the global existence and the uniform L_∞-bound of the solutions from Theorem 1.1 and the Sobolev embedding theorems. Especially in each step of estimates of the solution we look for the contribution of the diffusion coefficients $d_{1}, d_{2}$ and conclude that the uniform bound of the solution is independent of d₁,d₂ if d₁,d₂⩾1. Hence for large d₁,d₂ we obtain convergence results on the solution in the weak competition case. In the strong competition case we construct invariant sets and prove the convergence of the global solutions in those invariant sets for a special case of system (1.2) by using the independence of the uniform bound of the solution of d₁,d₂⩾1.

Remark

Throughout this paper we refer to the condition c₁/c₂<a₁/a₂<b₁/b₂ as the weak competition condition, and the condition b₁/b₂<a₁/a₂<c₁/c₂ as the strong competition condition.

Here we state the main theorems of this paper.

Theorem 1.2

Suppose condition (α) for system (1.2). For the maximal solution (u(x,t),v(x,t)) of system (1.2) as in Theorem 1.1 there exist positive constants $t_{0}, M′=M′(d_{i},α_{ij},a_{i},b_{i},c_{i},i,j=1,2)$ , and M=M(d_i,α_ij,a_i,b_i,c_i,i,j=1,2) such that $max {||u(·,t)||_{1,2},||v(·,t)||_{1,2} : t∈(t_{0},T)}⩽M′, max {u(x,t), v(x,t) : (x,t)∈[0,1]×(t_{0},T)}⩽M$ and T=+∞. In the case d₁,d₂⩾1, and $d ̄ ⩽d_{2} /d_{1} ⩽ d ̄$ , where $d ̄, d ̄$ are positive constants, the constants $M′, M$ are independent of d₁,d₂⩾1, that is, M′ and M depend only on $d ̄, d ̄, α_{ij}, a_{i}, b_{i}, c_{i}, i,j=1,2$ .

Theorem 1.3

Suppose condition (α), the weak competition condition, and that $u_{0}, v_{0}$ are in W₂²([0,1]) for system (1.2). If d₁,d₂⩾1 satisfy that $(b_{2}^{2} α_{12}^{2} u ̄^{2} +c_{1}^{2} α_{21}^{2} v ̄^{2})M^{2} <4b_{2} c_{1} u ̄ v ̄ d_{1} d_{2},$ where M is the positive constant in Theorem 1.2, $(u ̄, v ̄)= a_{1} c_{2} −a_{2} c_{1} b_{1} c_{2} −b_{2} c_{1}, b_{1} a_{2} −b_{2} a_{1} b_{1} c_{2} −b_{2} c_{1},$ then the solution (u(t),v(t)) converges to $(u ̄, v ̄)$ uniformly in [0,1] as t→∞, and $(u ̄, v ̄)$ is globally asymptotically stable.

Remark

Lou and Ni [8] have drawn the same convergence result as in Theorem 1.3 from different assumptions that α₁₁⩾(γ/8)α₂₁ and α₂₂⩾(1/8γ)α₁₂, where $γ=c_{1} v ̄ /b_{2} u ̄$ .

Theorem 1.4

Suppose condition (α) and the strong competition condition for system (1.2) and also that $d_{1} =d_{2} =d, α_{11} =α_{21} =α, α_{12} =α_{22} =β$ . We assume that b₁/b₂<1⩽a₁/a₂, and $4a_{2} −4 a_{1} b_{2} b_{1} + a_{1} c_{1}^{2} b_{1}^{2} <0.$ Let d⩾1 satisfies that $d> M 2 α^{2} +β^{2},$ where M is the positive constant in Theorem 1.2 independent of d⩾1. For $k⩽(b_{2} −b_{1})/(c_{1} −c_{2}), k<k_{0}$ , where $k_{0} =(a_{1} /a_{2}^{2} b_{1})(2a_{1} c_{2} −a_{2} c_{1})+(2 a_{1} /a_{2}^{2} b_{1}) a_{2}^{2} b_{1} (a_{1} b_{2} −a_{2} b_{1})+a_{1}^{2} c_{2} (a_{1} c_{2} −a_{2} c_{1})$ , the region Σ_k≔{(u,v)|u⩾0,v⩾0,v⩽ku} is a domain of attraction of (a₁/b₁,0) for system (1.2) in the sense that if {(u₀(x),v₀(x))|x∈[0,1]}⊂Σ_k and u₀,v₀∈W₂²([0,1]), then the solution (u(x,t),v(x,t)) of (1.2) converges to (a₁/b₁,0) uniformly in [0,1] as t→∞.

Remark

The following are some example sets of constants which satisfy all the conditions in Theorem 1.4. $α=β=1, d⪢1, α=β=1, d⪢1, a_{1} =1, b_{1} =1, c_{1} =2, a_{1} =1, b_{1} =1, c_{1} =3, a_{2} =1, b_{2} =3, c_{2} =1, a_{2} = 12, b_{2} =3, c_{2} =1, k_{0} =2, b_{2} −b_{1} c_{1} −c_{2} =2. k_{0} =4(12 + 12)≈4.82843, b_{2} −b_{1} c_{1} −c_{2} =1.$

The proofs of Theorem 1.3, Theorem 1.4 are given in 2 Convergence in the weak competition case, 3 Convergence in the strong competition case, respectively. In Section 4 we collect calculus inequalities which are necessary for the proof of Theorem 1.2. Theorem 1.2 is proved in Section 5.

Section snippets

Convergence in the weak competition case

Proof of Theorem 1.3

In this proof we consider the weak competition case of the competition coefficients, that is, c₁/c₂<a₁/a₂<b₁/b₂. By the strong maximum principle and the Hopf boundary lemma for parabolic equations ([12]), u(x,t)>0 and v(t,x)>0 in [0,1]×(0,∞). Using the functional H(u,v) defined in the following we observe the convergence of global solutions of system (1.2): $H(u,v)= ∫ 01 b_{2} u− u ̄ − u ̄ log u u ̄ +c_{1} v− v ̄ − v ̄ log v v ̄ d x,$ where $(u ̄, v ̄)= a_{1} c_{2} −a_{2} c_{1} b_{1} c_{2} −b_{2} c_{1}, b_{1} a_{2} −b_{2} a_{1} b_{1} c_{2} −b_{2} c_{1}$ is the stable constant steady state of system

Convergence in the strong competition case

We study in this section convergence of solutions to system (1.2) in the strong competition case, that is, b₁/b₂<a₁/a₂<c₁/c₂. In order to construct an invariant set for the solution of system (1.2) we assume in this section that $d_{1} =d_{2} =d, α_{11} =α_{21} =α, α_{12} =α_{22} =β$ and rewrite system (1.2) into the following system: $u_{t} =[(d+αu+βv)u]_{xx} +uf(u,v) in [0,1]×(0,∞), v_{t} =[(d+αu+βv)v]_{xx} +vg(u,v) in [0,1]×(0,∞), u_{x} (x,t)=v_{x} (x,t)=0 at x=0,1, u(x,0)=u_{0} (x)⩾0, v(x,0)=v_{0} (x)⩾0 in [0,1],$ where $f(u,v)=a_{1} −b_{1} u−c_{1} v, g(u,v)=a_{2} −b_{2} u−c_{2} v$ .

In Lemma

Calculus inequalities

Theorem 4.1

Let $Ω∈ R^{n}$ be a bounded domain with $∂Ω$ in C^m. For every function u in $W^{m,r} (Ω), 1⩽q,r⩽∞$ , the derivative $D^{j} u, 0⩽j<m$ , satisfies the inequality $|D^{j} u|_{p} ⩽C(|D^{m} u|_{r}^{a} |u|_{q}^{1−a} +|u|_{q}),$ where 1/p=j/n+a(1/r−m/n)+(1−a)1/q, for all a in the interval j/m⩽a<1, provided one of the following three conditions is satisfied:

(i)
r⩽q,
(ii)
0<n(r−q)/mrq<1, or
(iii)
n(r−q)/mrq=1 and m−n/q is not a nonnegative integer.

(The positive constant C depends only on

n, m, j, q, r, a.

)

Proof

We refer the reader to Friedman [6] or Nirenberg [10] for the proof of this

Uniform boundedness

Proof of Theorem 1.2

Step 1: Taking integration of the first equation of (1.2) over the domain [0,1], we have $d d t ∫ 01 u(t) d x= ∫ 01 (a_{1} u−b_{1} u^{2} −c_{1} uv) d x⩽a_{1} ∫ 01 u(t) d x−b_{1} ∫ 01 u^{2} (t) d x⩽a_{1} ∫ 01 u(t) d x−b_{1} ∫ 01 u(t) d x^{2} .$ In the case that $∫ 01 u_{0} d x<a_{1} /b_{1}$ we have that $∫ 01 u(t) d x⩽a_{1} /b_{1}$ for all t>0. In the case that $∫ 01 u_{0} d x⩾a_{1} /b_{1}$ there exist positive constants δ and τ₀′ such that $∫ 01 u(t) d x<δ+a_{1} /b_{1}$ for all t∈(τ₀′,∞).

We estimate $∫ 01 v(t) d x$ similarly and conclude that there exist positive constants τ₀ and M₀=M₀(a_i,b_i,c_i,i,j=1,2) such that $∫ 01 u(t) d x<M_{0}, ∫ 01$

Acknowledgements

The author thanks Professor Wei-Ming Ni for his helpful discussions. This work is partially supported by NSF.

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Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions

Abstract

Introduction

Section snippets

Convergence in the weak competition case

Convergence in the strong competition case

Calculus inequalities

Uniform boundedness

Acknowledgements

J. Differential Equations

J. Theoret. Biol.

Nonlinear Anal., Theory, Methods & Applications

Sobolev Spaces

Dynamic theory of quasilinear parabolic equations, III. Global Existence

Math. Z.

Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems

Differential Integral Equations