Skip to content

Advertisement

  • Research
  • Open Access

Global behavior for a strongly coupled model of plankton allelopathy

Advances in Difference Equations20122012:89

https://doi.org/10.1186/1687-1847-2012-89

  • Received: 3 May 2012
  • Accepted: 27 June 2012
  • Published:

Abstract

In this article, we consider a strongly coupled model of plankton allelopathy. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

AMS Subject Classification (2000): 35K57; 35B35; 92D25.

Keywords

  • plankton
  • allelopathy
  • cross-diffusion
  • global solution
  • uniform boundedness
  • stability

1 Introduction

The effects of toxic substances on ecological communities is an important problem from an environmental point of view. In 1996, Chattopadhyay [1] modified two species Lotka-Volterra competitive system by considering that each species produces a substance toxic to the other but only when the other is present and the modified model takes the following form
d u d t = u ( a 1 - b 11 u - b 12 v - e 1 u v ) , d v d t = v ( a 2 - b 21 u - b 22 v - e 2 u v ) ,
(1.1)

where a i , b ij , and e i (i, j = 1, 2) are positive constants, u(t), v(t) denote the population density of two competing species; a1, a2 are the intrinsic growth rates of two competing species; b11, b22 are the rates of intra-specific competition of the first and the second species, respectively; b12, b21 are the rates of inter-specific competition of the first and the second species, respectively; e1 and e2 are, respectively, the rates of toxic inhibition of the first species by the second and vice versa. For more details on the backgrounds about this system, see [1].

The system (1.1) has a positive equilibrium E* = (u*, v*) if and only if
b 12 b 22 < a 1 a 2 < b 11 b 21 , b 12 b 22 < e 1 e 2 < b 11 b 21 ,
(1.2)
where
u * = ( q 12 q 12 2 4 p 12 r 12 ) / ( 2 p 12 ) , v * = ( q 21 q 21 2 4 p 21 r 21 ) / ( 2 p 21 ) ,
and
p i j = b i j e i - b i i e j , q i j = a i e j - a j e i - b i i b j j + b i j b j i , r i j = a i b j j - a j b i j , i , j = 1 , 2 .
Chattopadhyay [1] proved that the equilibrium (u*, v*) is globally asymptotically stable if
4 ( b 11 + e 1 v ) ( b 22 + e 2 u ) ( b 12 + b 21 + e 1 u * + e 2 v * ) 2 .
(1.3)
The corresponding weakly coupled reaction-diffusion system for (1.1) is as follows
u t = d 1 Δ u + u ( a 1 - b 11 u - b 12 v - e 1 u v ) , x Ω , t > 0 , v t = d 2 Δ v + v ( a 2 - b 21 u - b 22 v - e 2 u v ) , x Ω , t > 0 , n u = n v = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω ,
(1.4)

where Ω N is bounded smooth domain, n is the outward unit normal vector of the boundary Ω, n = ∂/∂n. The, constants d1 and d2, called diffusion coefficients, are positive, and u0(x) and v0(x) are non-negative functions which are not identically zero.

It is obvious that (u*, v*) is the unique positive equilibrium of the system (1.4) if (1.2) holds. Tian et al. [2] proved that the equilibrium (u*, v*) of the system (1.4) is locally asymptotically stable if (1.2) holds.

In recent years, the SKT-type cross-diffusion systems have attracted the attention of a great number of investigators and have been successfully developed on the theoretical backgrounds. The above work mainly concentrate on (1) The instability and stability induced by cross-diffusion, and the existence of non-constant positive steady-state solutions [35]; (2) the global existence of strong solutions [613]; (3) the global existence of weak solutions based on semi-discretization or finite element approximation [1417]; and (4) the dynamical behaviors [9, 10], etc.

Tian et al. [2] considered the following SKT-type cross-diffusion system
u t = d 1 Δ u + u ( a 1 - b 11 u - b 12 v - e 1 u v ) , x Ω , t > 0 , v t = d 2 Δ ( v + d 3 u v ) + v ( a 2 - b 21 u - b 22 v - e 2 u v ) , x Ω , t > 0 , n u = n v = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω ,
(1.5)
and they proved that:
  1. (1)
    If μ 2 < μ ̃ , then there exists a positive constant d 2 * such that the equilibrium (u*, v*) of the system (1.5) is unstable provided that d 2 d 2 * , the (1.2) and the following condition
    ( 1 + d 3 u * ) ( b 11 + e 1 v * ) < d 3 v * ( b 12 + e 1 u * )
    (1.6)
     
hold, where 0 = μ0 < μ1 < μ2 < · · · are the eigenvalues of the operator -Δ on Ω with the homogeneous Neumann boundary condition, μ ̃ = v * ( b 12 + e 1 u * ) / d 1 ;
  1. (2)

    the steady-state system of the system (1.4) has non-constant positive solution, if one of the following conditions is satisfied:

     
  2. (i)

    d1D1 for some positive D1(d2);

     
  3. (ii)

    d2D2 for some positive D2(d1); and

     
  4. (3)

    if μ ̃ ( μ n , μ n + 1 ) for some n ≥ 1, and the sum σ n = i = 2 n dim E ( μ i ) is odd, then there exists a positive number d 2 * such that the system (1.5) has at least one inhomogeneous positive steady-state solution if d 2 d 2 * , (1.2) and (1.6) hold, where d1 and d3 are fixed.

     
We are concerned with the following plankton allelopathy model with full cross-diffusion
u t = Δ ( d 1 u + α 11 u 2 + α 12 u v ) + u ( a 1 - b 11 u - b 12 v - e 1 u v ) , x Ω , t > 0 , v t = Δ ( d 2 v + α 21 u v + α 22 v 2 ) + v ( a 2 - b 21 u - b 22 v - e 2 u v ) , x Ω , t > 0 , n u = n v = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω ,
(1.7)

where Ω is a bounded domain N with smooth boundary ∂Ω, n is the outward unit normal vector of the boundary ∂Ω. d i , α ij , a i , b ij , e i (i, j = 1, 2) are positive constants and the initial data u0 and v0 are continuous non-negative functions which are not identically zero. The homogeneous Neumann boundary condition indicates that the system is self-contained with zero population flux across the boundary. The parameters d1, d2 are the diffusion rates, α ii (i = 1, 2) are referred as self-diffusion pressures, and α ij (i, j = 1, 2, ij) are cross-diffusion pressures. For more details on the backgrounds about self-diffusion and cross-diffusion, one can see [8].

The local existence of solutions for the system (1.7) is an immediate consequence of a series of important articles by Amann [1820]. Roughly speaking, if u0(x) and v0(x) in W p 1 ( Ω ) with p > n, then (1.7) has a unique non-negative solution u, v C ([0, T ), W P 1 ( Ω ) ) C ( ( 0 , T ) , C(Ω)), where T (0, ∞] is the maximal existence time for the solution. If the solution (u, v) satisfies the estimate
sup { | | u ( , t ) | | W p 1 ( Ω ) , | | v ( , t ) | | W p 1 ( Ω ) : 0 < t < T } < ,

then T = +∞. Moreover, if u0(x), v 0 ( x ) W P 2 ( Ω ) , then u, v C ([0, ∞), W P 2 ( Ω ) ) .

For the following SKT system
u t = d 1 Δ [ ( 1 + α v + γ u ) u ] + a u ( 1 - u - c v ) , x Ω , t > 0 , v t = d 2 Δ [ ( 1 + δ v ) v ] + b v ( 1 - d u - v ) , x Ω , t > 0 , n u = n v = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω
(P)
Yamada [13] proposed four open problems:
  1. (1)

    The global existence of solutions of (P) in the case δ > 0 and the space dimension N ≥ 6;

     
  2. (2)

    the global existence in the case γ = 0;

     
  3. (3)

    in order to study the asymptotic behavior of u, v as t → ∞, need to establish the uniform boundedness of global solutions; and

     
  4. (4)
    the global existence of solutions for the following full SKT system
    u t = d 1 Δ [ ( 1 + α v + γ u ) u ] + a u ( 1 - u - c v ) , x Ω , t > 0 , v t = d 2 Δ [ ( 1 + β u + δ v ) v ] + b v ( 1 - d u - v ) , x Ω , t > 0 , n u = n v = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω
     

with α, γ, β, δ > 0.

Very few global existence results for (1.7) are known. The main purpose of this article is to establish the uniform boundedness of global solutions for the system (1.7) in one space dimension. For convenience, we consider the following system
u t = ( d 1 u + α 11 u 2 + α 12 u v ) x x + u ( a 1 - b 11 u - b 12 v - e 1 u v ) , 0 < x < 1 , t > 0 , v t = ( d 2 v + α 21 u v + α 22 v 2 ) x x + v ( a 2 - b 21 u - b 22 v - e 2 u v ) , 0 < x < 1 , t > 0 , u x ( x , t ) = v x ( x , t ) = 0 , x = 0 , 1 , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , 0 < x < 1 .
(1.8)

We firstly investigate the global existence and the uniform boundedness of the solutions for (1.8), then prove the global asymptotic stability of the positive equilibrium (u*, v*) of (1.8) by an important lemma from [21]. The proof is complete and complement to the uniform convergence theorems in [2224].

It is obvious that (u*, v*) is the unique positive equilibrium of the system (1.8) if (1.2) holds.

For simplicity, we denote | | | | W p k ( 0 , 1 ) by | · | k,p and | | | | L p ( 0 , 1 ) by | · | p . Our main results are as follows.

Theorem 1.1. Let u 0 , v 0 W 2 2 ( 0 , 1 ) , (u, v) is the unique non-negative solution of system (1.2) in the maximal existence interval [0, T ). Assume that
8 α 11 α 21 > α 12 2 , 8 α 22 α 12 > α 21 2 .
(1.9)
Then there exist t0 > 0 and positive constants M, M' which depend on d i , α ij , a i , b ij , e i (i, j = 1, 2), such that
sup { | u ( , t ) | 1 , 2 , | v ( , t ) | 1 , 2 : t ( t 0 , T ) } M ,
(1.10)
max { u ( x , t ) , v ( x , t ) : ( x , t ) [ 0 , 1 ] × ( t 0 , T ) } M ,
(1.11)

and T = +∞. Moreover, in the case that d1, d2 ≥ 1, d 2 / d 1 [ d - , d ̄ ] , where d ¯ and d ̄ are positive constants, M', M depend on d - , d ̄ , but do not depend on d1, d2.

Remark 1.1. Since the continuous embedding H1(Ω) L(Ω) holds only in one space dimension, we can only establish the uniform maximum-norm estimates about time for the solution in one space dimension.

Theorem 1.2 Assume that all conditions in Theorem 1.1 are satisfied. Assume further that,
4 d 1 d 2 u * v * > M 2 ( α 12 u * + α 21 v * ) 2 ,
(1.12)
4 b 11 b 22 > ( b 12 + b 21 + e 1 u * + e 2 v * ) 2
(1.13)

and (1.2) hold, M is given by (1.11). Then the unique positive equilibrium (u*, v*) of (1.8) is globally asymptotically stable.

Remark 1.2. The system (1.8) has no non-constant positive steady-state solution if all conditions of Theorem 1.2 hold.

2 Global solutions

In order to establish the uniform W 2 1 -estimates of the solutions for system (1.2), the following Gagliardo-Nirenberg- type inequalities and the corresponding corollary play important roles (see [25]).

Theorem 2.1. Let Ω n be a bounded domain with ∂Ω C m . For every function u W r m ( 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 ) ,

provided one of the following three conditions is satisfied: (1) rq, (2) 0 < n(r-q)/(mrq) < 1, or (3) n(r-q)/(mrq) = 1 and m - n/q is not a non-negative integer, where 1/p = j/n + a(1/r - m/n) + (1 - a)/q for all a [j/m, 1), and the positive constant C depends on n, m, j, q, r, a.

Corollary 1. There exists a positive constant C such that
| u | 2 C ( | u x | 2 1 / 3 | u | 1 2 / 3 + | u | 1 ) , u W 2 1 ( 0 , 1 ) ,
(2.1)
| u | 4 C ( | u x | 2 1 / 2 | u | 1 1 / 2 + | u | 1 ) , u W 2 1 ( 0 , 1 ) ,
(2.2)
| u | 7 2 C ( | u x | 2 10 / 21 | u | 1 11 / 21 + | u | 1 ) , u W 2 1 ( 0 , 1 ) ,
(2.3)
| u x | 2 C ( | u x x | 2 3 / 5 | u | 1 2 / 5 + | u | 1 ) , u W 2 2 ( 0 , 1 ) .
(2.4)

Throughout this article, we always denote that C is Sobolev embedding constant or other kind of universal constant, A j , B j , C j are some positive constants which depend only on α ij , a i , b ij , e i (i, j = 1, 2), K j are positive constants depending on d i , α ij , a i , b ij , e i (i, j = 1, 2). When d1, d2 ≥ 1, d 2 / d 1 [ d - , d ̄ ] , K j depend on d - , d ̄ , but do not depend on d1 and d2.

Proof of Theorem 1.1. Step 1, estimate |u|1, |v|1.

Taking integration of the first, the first and the second equations in (1.8) over the domain (0,1), respectively, we have
d d t 0 1 u d x = 0 1 u ( a 1 - b 11 u - b 12 v - e 1 u v ) d x a 1 0 1 u d x - b 11 0 1 u d x 2 , d d t 0 1 v d x = 0 1 v ( a 2 - b 21 u - b 22 v - e 2 u v ) d x a 2 0 1 v d x - b 22 0 1 v d x 2 .
So, there exists a positive constant M0 which depends on a i , b ij (i, j = 1, 2), such that
0 1 u d x , 0 1 v d x M 0 , t τ 0 .
(2.5)
Moreover, there exists a positive constant M 0 which depends on a i , b ij (i, j = 1, 2) and L1-norm of u0, v0, such that
0 1 u d x , 0 1 v d x M 0 , t 0 .
(2.5a)

Step 2, estimate |u|2, |v|2.

Multiplying the first two equations in system (1.2) by u, v, respectively, and integrating over (0, 1) we have
1 2 d d t 0 1 u 2 d x - d 1 0 1 u x 2 d x - 0 1 [ ( 2 α 11 u + α 12 v ) u x 2 + α 12 v x u u x ] d x + a 1 0 1 u 2 d x , 1 2 d d t 0 1 v 2 d x - d 2 0 1 v x 2 d x - 0 1 [ ( α 21 u + 2 α 22 v ) v x 2 + α 21 u x v v x ] d x + a 2 0 1 v 2 d x ,
from which it follows that
1 2 d d t 0 1 ( u 2 + v 2 ) d x - d 0 1 ( u x 2 + v x 2 ) d x + a 1 0 1 u 2 d x + a 2 0 1 v 2 d x - 0 1 q ( u x , v x ) d x - d 0 1 ( u x 2 + v x 2 ) d x + ( a 1 + a 2 ) 0 1 ( u 2 + v 2 ) d x - 0 1 q ( u x , v x ) d x ,
where d = min{d1, d2}. Some tedious calculations yield that
q ( u x , v x ) = ( 2 α 11 u + α 12 v ) u x 2 + ( α 12 u + α 21 v ) u x v x + ( α 21 u + 2 α 22 v ) v x 2
is positive definite quadratic form of u x , v x if (1.9) holds. So (1.9) implies that
1 2 d d t 0 1 ( u 2 + v 2 ) d x - d 0 1 ( u x 2 + v x 2 ) d x + ( a 1 + a 2 ) 0 1 ( u 2 + v 2 ) d x .
(2.6)
Now, we proceed in the following two cases.
  1. (i)
    t ≥ τ0.The inequality (2.1) implies that | u | 2 6 C ( | u x | 2 2 | u | 1 4 + | u | 1 6 ) C M 0 4 ( | u x | 2 2 + M 0 2 ) . So we have 0 1 u x 2 d x 1 C M 0 4 0 1 u 2 d x 3 - M 0 2 , and
    - 0 1 ( u x 2 + v x 2 ) d x - 1 9 C M 0 4 0 1 ( u 2 + v 2 ) d x 3 + 2 M 0 2 .
    (2.7)
     
It follows from (2.5) and (2.6) that
1 2 d d t 0 1 ( u 2 + v 2 ) d x d - C 2 0 1 ( u 2 + v 2 ) d x 3 + 2 M 0 2 + 1 d ( a 1 + a 2 ) 0 1 ( u 2 + v 2 ) d x .
(2.8)
This means that there exist positive constants τ1 and M1 depending on d i , a i , b ij (i, j = 1, 2), such that
0 1 u 2 d x , 0 1 v 2 d x M 1 , t τ 1 .
(2.9)
When d ≥ 1, M1 is independent of d because the zero point of the right-hand side in (2.11) can be estimated by positive constants independent of d.
  1. (ii)
    t ≥ 0. Repeating estimates in (i) by (2.8)', we can obtain that there exists a positive constant M 1 depending on d i , a i , b ij (i, j = 1, 2) and the L1, L2-norm of u0, v0, such that
    0 1 u 2 d x , 0 1 v 2 d x M 1 , t 0 .
    (2.9a)
     

When d ≥ 1, M 1 is independent of d.

Step 3, estimate |u x |2, |v x |2.

Introducing the following scaling
ũ = u d 1 , = v d 1 , t ̃ = d 1 t .
(2.10)
Denoting ξ = d2/d1, and using u, v, t instead of ũ , , t ̃ , respectively, then system (1.2) reduces to
u t = P x x + f ( u , v ) , 0 < x < 1 , t > 0 , v t = Q x x + g ( u , v ) , 0 < x < 1 , t > 0 , u x ( x , t ) = v x ( x , t ) = 0 , x = 0 , 1 , t > 0 , u ( x , 0 ) = ũ 0 ( x ) , v ( x , 0 ) = 0 ( x ) , 0 < x < 1 ,
(2.11)

where P = u + α11u2 + α12uv, Q = ξv + α21uv + α22v2, f ( u , v ) = a 1 d 1 - 1 u - b 11 u 2 - b 12 u v - e 1 d 1 u 2 v , g ( u , v ) = a 2 d 1 - 1 v - b 21 u v - b 22 v 2 - e 2 d 1 u v 2 .

We still proceed in the following two cases.
  1. (i)
    t τ 1 * = d 1 τ 1 . It is clear that
    0 1 u d x , 0 1 v d x M 0 d 1 - 1 , 0 1 u 2 d x , 0 1 v 2 d x M 1 d 1 - 2 , | P | 1 , | Q | 1 A 1 K 1 d 1 - 1 ,
    (2.12)
     

where K 1 = ( 1 + ξ ) + M 1 d 1 - 1 and A1 = max{M0, α11 + α12, α21 + α22}.

Multiplying the first two equations in (2.11) by P t , Q t , integrating them over the domain (0, 1), respectively, and then adding up the two integration equalities, we have
1 2 ȳ ( t ) = - 0 1 u t 2 d x - ξ 0 1 v t 2 d x - 0 1 q ( u t , v t ) d x + 0 1 [ ( 1 + 2 α 11 u + α 12 v ) u t f + α 12 u v t f ] d x + 0 1 [ ( ξ + α 21 u + 2 α 22 v ) v t g + α 21 v u t g ] d x
where ȳ ( t ) = 0 1 ( P x 2 + Q x 2 ) d x . It is not hard verify by (1.9) that there exists a positive constant C3 depending only on α ij (i, j = 1, 2), such that
q ( u t , v t ) C 3 ( u + v ) ( u t 2 + v t 2 ) .
Thus,
1 2 ȳ ( t ) - 0 1 u t 2 d x - ξ 0 1 v t 2 d x - C 3 0 1 ( u + v ) ( u t 2 + v t 2 ) d x + 0 1 [ ( 1 + 2 α 11 u + α 12 v ) u t f + α 12 u v t f ] d x + 0 1 [ ( ξ + α 21 u + 2 α 22 v ) v t g + α 21 v u t g ] d x .
(2.13)
Using Young inequality, Hölder inequality, and (2.12), we can obtain the following estimates
0 1 u 3 d x 0 1 u 7 d x 1 / 5 0 1 u 2 d x 4 / 5 M 1 4 / 5 d 1 - 8 / 5 0 1 u 7 d x 1 / 5 , 0 1 u 4 d x 0 1 u 7 d x 2 / 5 0 1 u 2 d x 3 / 5 M 1 3 / 5 d 1 - 6 / 5 0 1 u 7 d x 2 / 5 , 0 1 u 5 d x 0 1 u 7 d x 3 / 5 0 1 u 2 d x 2 / 5 M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 , 0 1 u 6 d x 0 1 u 7 d x 4 / 5 0 1 u 2 d x 1 / 5 M 1 1 / 5 d 1 - 2 / 5 0 1 u 7 d x 4 / 5 , 0 1 u 2 v d x 0 1 u 7 d x 1 / 5 0 1 u 2 d x 3 / 10 0 1 v 2 d x 1 / 2 M 1 4 / 5 d 1 - 8 / 5 0 1 u 7 d x 1 / 5 , 0 1 u 3 v d x 0 1 u 7 d x 2 / 5 0 1 u 2 d x 1 / 10 0 1 v 2 d x 1 / 2 M 1 3 / 5 d 1 - 6 / 5 0 1 u 7 d x 2 / 5 , 0 1 u 4 v d x 4 5 0 1 u 5 d x + 1 5 0 1 v 5 d x 4 5 M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 , 0 1 u 2 v 2 d x 1 2 0 1 u 4 d x + 0 1 v 4 d x 1 2 M 1 3 / 5 d 1 - 4 / 5 0 1 u 7 d x 2 / 5 + 0 1 v 7 d x 2 / 5 , 0 1 u 3 v 2 d x 3 5 0 1 u 5 d x + 2 5 0 1 v 5 d x 3 5 M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 , 0 1 u 4 v 2 d x 2 3 0 1 u 6 d x + 1 3 0 1 v 6 d x 2 3 M 1 1 / 5 d 1 - 4 / 5 0 1 u 7 d x 4 / 5 + 0 1 v 7 d x 4 / 5 , 0 1 u 4 v 3 d x 4 7 0 1 u 7 d x + 3 7 0 1 v 7 d x 4 7 0 1 u 7 d x + 0 1 v 7 d x , 0 1 u 5 v 2 d x 5 7 0 1 u 7 d x + 2 7 0 1 v 7 d x 5 7 0 1 u 7 d x + 0 1 v 7 d x .
(2.14)
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of (2.13), we have
- 0 1 u t 2 d x - 1 2 0 1 P x x 2 d x + 0 1 f 2 d x , - ξ 0 1 v t 2 d x - ξ 2 0 1 Q x x 2 d x + ξ 0 1 g 2 d x ,
0 1 f 2 d x 0 1 ( a 1 2 d 1 - 2 u 2 + b 11 2 u 4 + b 12 2 u 2 v 2 + e 1 2 d 1 2 u 4 v 2 + 2 b 11 b 12 u 3 v + 2 b 11 e 1 d 1 u 4 v + 2 b 12 e 1 d 1 u 3 v 2 ) d x a 1 2 M 1 d 1 - 4 + b 11 2 + 1 2 b 12 2 + 2 b 11 b 12 M 1 3 / 5 d 1 - 6 / 5 0 1 u 7 d x 2 / 5 + 0 1 v 7 d x 2 / 5 + 8 5 b 11 + 6 5 b 12 e 1 M 1 2 / 5 d 1 1 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 3 e 1 2 M 1 1 / 5 d 1 8 / 5 0 1 u 7 d x 4 / 5 + 0 1 v 7 d x 4 / 5 ,
ξ 0 1 g 2 d x ξ 0 1 a 2 2 d 1 - 2 v 2 + b 21 2 u 2 v 2 + b 22 2 v 4 + e 2 2 d 1 2 u 2 v 4 + 2 b 21 b 22 u v 3 + 2 b 21 e 2 d 1 u 2 v 3 + 2 b 22 e 2 d 1 u v 4 d x ξ a 2 2 M 1 d 1 - 4 + ξ b 22 2 + 1 2 b 21 2 + 2 b 21 b 22 M 1 3 / 5 d 1 - 6 / 5 0 1 u 7 d x 2 / 5 + 0 1 v 7 d x 2 / 5 + 2 ξ 3 5 b 21 + 4 5 b 22 e 2 M 1 2 / 5 d 1 1 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 3 ξ e 2 2 M 1 1 / 5 d 1 8 / 5 0 1 u 7 d x 4 / 5 + 0 1 v 7 d x 4 / 5 ,
and
- 0 1 u t 2 d x - ξ 0 1 v t 2 d x - 1 2 0 1 P x x 2 d x - ξ 2 0 1 Q x x 2 d x + a 1 2 + ξ a 2 2 M 1 d 1 - 4 + b 11 2 + 1 2 b 12 2 + 2 b 11 b 12 + ξ b 22 2 + 1 2 b 21 2 + 2 b 21 b 22 M 1 3 / 5 d 1 - 6 / 5 0 1 u 7 d x 2 / 5 + 0 1 v 7 d x 2 / 5 + 8 5 b 11 + 6 5 b 12 e 1 + 2 ξ 3 5 b 21 + 4 5 b 22 e 2 M 1 2 / 5 d 1 1 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 3 ( e 1 2 + ξ e 2 2 ) M 1 1 / 5 d 1 8 / 5 0 1 u 7 d x 4 / 5 + 0 1 v 7 d x 4 / 5 .
(2.15)
Similarly, we can obtain
0 1 u t f d x 0 1 u t ( a 1 d 1 - 1 u + b 11 u 2 + b 12 u v + e 1 d 1 u 2 v ) d x a 1 2 d 1 - 2 2 ε 0 1 u d x + ε 2 0 1 u u t 2 d x + b 11 2 2 ε 0 1 u 3 d x + ε 2 0 1 u u t 2 d x + b 12 2 2 ε 0 1 u v 2 d x + ε 2 0 1 u u t 2 d x + ( e 1 d 1 ) 2 2 ε 0 1 u 3 v 2 d x + ε 2 0 1 u u t 2 d x a 1 2 2 ε M 0 d 1 - 3 + b 11 2 + b 12 2 2 ε M 1 4 / 5 d 1 - 8 / 5 0 1 u 7 d x 1 / 5 + 0 1 v 7 d x 1 / 5 + 3 e 1 2 10 ε M 1 2 / 5 d 1 6 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 ε 0 1 u u t 2 d x ,
2 α 11 0 1 u u t f d x 2 α 11 0 1 u u t ( a 1 d 1 - 1 u + b 11 u 2 + b 12 u v + e 1 d 1 u 2 v ) d x α 11 2 a 1 2 d 1 - 2 ε 0 1 u 3 d x + ε 0 1 u u t 2 d x + α 11 2 b 11 2 ε 0 1 u 5 d x + ε 0 1 u u t 2 d x + α 11 2 b 12 2 ε 0 1 u 3 v 2 d x + ε 0 1 u u t 2 d x + α 11 2 e 1 2 d 1 2 ε 0 1 u 5 v 2 d x + ε 0 1 u u t 2 d x α 11 2 a 1 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 u 7 d x 1 / 5 + α 11 2 ( b 11 2 + b 12 2 ) ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 5 α 11 2 e 1 2 7 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 u u t 2 d x ,
α 12 0 1 v u t f d x α 12 0 1 v u t ( a 1 d 1 - 1 u + b 11 u 2 + b 12 u v + e 1 d 1 u 2 v ) d x α 12 2 a 1 2 d 1 - 2 2 ε 0 1 u v 2 d x + ε 2 0 1 u u t 2 d x + α 12 2 b 11 2 2 ε 0 1 u 3 v 2 d x + ε 2 0 1 u u t 2 d x + α 12 2 b 12 2 2 ε 0 1 u v 4 d x + ε 2 0 1 u u t 2 d x + α 12 2 e 1 2 d 1 2 2 ε 0 1 u 3 v 4 d x + ε 2 0 1 u u t 2 d x α 12 2 a 1 2 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 v 7 d x 1 / 5 + α 12 2 ( b 11 2 + b 12 2 ) 2 ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 α 12 2 e 1 2 7 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 u u t 2 d x ,
α 12 0 1 u v t f d x α 12 0 1 u v t ( a 1 d 1 - 1 u + b 11 u 2 + b 12 u v + e 1 d 1 u 2 v ) d x α 12 2 a 1 2 d 1 - 2 2 ε 0 1 u 3 d x + ε 2 0 1 u v t 2 d x + α 12 2 b 11 2 2 ε 0 1 u 5 d x + ε 2 0 1 u v t 2 d x + α 12 2 b 12 2 2 ε 0 1 u 3 v 2 d x + ε 2 0 1 u v t 2 d x + α 12 2 e 1 2 d 1 2 2 ε 0 1 u 5 v 2 d x + ε 2 0 1 u v t 2 d x α 12 2 a 1 2 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 u 7 d x 1 / 5 + α 12 2 ( b 11 2 + b 12 2 ) 2 ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 5 α 12 2 e 1 2 14 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 u v t 2 d x ,
ξ 0 1 v t g d x ξ 0 1 v t ( a 2 d 1 - 1 v + b 21 u v + b 22 v 2 + e 2 d 1 u v 2 ) d x ξ 2 a 2 2 d 1 - 2 2 ε 0 1 v d x + ε 2 0 1 v v t 2 d x + ξ 2 b 21 2 2 ε 0 1 u 2 v d x + ε 2 0 1 v v t 2 d x + ξ 2 b 22 2 2 ε 0 1 v 3 d x + ε 2 0 1 v v t 2 d x + ξ 2 e 2 2 d 1 2 2 ε 0 1 u 2 v 3 d x + ε 2 0 1 v v t 2 d x ξ 2 a 2 2 2 ε M 0 d 1 - 3 + ξ 2 ( b 21 2 + b 22 2 ) 2 ε M 1 4 / 5 d 1 - 8 / 5 0 1 u 7 d x 1 / 5 + 0 1 v 7 d x 1 / 5 + 3 ξ 2 e 2 2 10 ε M 1 2 / 5 d 1 6 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 2 ε 0 1 v v t 2 d x ,
α 21 0 1 u v t g d x α 21 0 1 u v t ( a 2 d 1 - 1 v + b 21 u v + b 22 v 2 + e 2 d 1 u v 2 ) d x α 21 2 a 2 2 d 1 - 2 2 ε 0 1 u 2 v d x + ε 2 0 1 v v t 2 d x + α 21 2 b 21 2 2 ε 0 1 u 4 v d x + ε 2 0 1 v v t 2 d x + α 21 2 b 22 2 2 ε 0 1 u 2 v 3 d x + ε 2 0 1 v v t 2 d x + α 21 2 e 2 2 d 1 2 2 ε 0 1 u 4 v 3 d x + ε 2 0 1 v v t 2 d x α 21 2 a 2 2 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 u 7 d x 1 / 5 + α 21 2 ( b 21 2 + b 22 2 ) 2 ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 5 α 21 2 e 2 2 14 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 v v t 2 d x ,
2 α 22 0 1 v v t g d x 2 α 22 0 1 v v t ( a 2 d 1 - 1 v + b 21 u v + b 22 v 2 + e 2 d 1 u v 2 ) d x α 22 2 a 2 2 d 1 - 2 ε 0 1 v 3 d x + ε 0 1 v v t 2 d x + α 22 2 b 21 2 ε 0 1 u 2 v 3 d x + ε 0 1 v v t 2 d x + α 22 2 b 22 2 ε 0 1 v 5 d x + ε 0 1 v v t 2 d x + α 22 2 e 2 2 d 1 2 ε 0 1 u 2 v 5 d x + ε 0 1 v v t 2 d x α 22 2 a 2 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 v 7 d x 1 / 5 + α 22 2 ( b 21 2 + b 22 2 ) ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 5 α 22 2 e 2 2 7 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 v v t 2 d x ,
α 21 0 1 v u t g d x α 21 0 1 v u t ( a 2 d 1 - 1 v + b 21 u v + b 22 v 2 + e 2 d 1 u v 2 ) d x α 21 2 a 2 2 d 1 - 2 2 ε 0 1 v 3 d x + ε 2 0 1 v u t 2 d x + α 21 2 b 21 2 2 ε 0 1 u 2 v 3 d x + ε 2 0 1 v u t 2 d x + α 21 2 b 22 2 2 ε 0 1 v 5 d x + ε 2 0 1 v u t 2 d x + α 21 2 e 2 2 d 1 2 2 ε 0 1 u 2 v 5 d x + ε 2 0 1 v u t 2 d x α 21 2 a 2 2 2 ε M 1 4 / 5 d 1 - 18 / 5 0 1 v 7 d x 1 / 5 + α 21 2 ( b 21 2 + b 22 2 ) 2 ε M 1 2 / 5 d 1 - 4 / 5 0 1 u 7 d x 3 / 5 + 0 1 v 7 d x 3 / 5 + 5 α 21 2 e 2 2 14 ε d 1 2 0 1 u 7 d x + 0 1 v 7 d x + 2 ε 0 1 v v t 2 d x .
By the above inequalities and the condition (1.9), we have
0 1 [ ( 1 + 2 α 11 u + α 12 v ) u t f + α 12 u v t f ] d x + 0 1 [ ( ξ + α 21 u + 2 α 22 v ) v t g + α 21 v u t g ] d x λ ̄ ε 0 1 ( u + v ) ( u t 2 + v t 2 ) d x + C 4 ε ( 1 + ξ 2 ) M 0 d 1 - 3 + C 5 ε ( 1 + ξ 2 + d 1 - 2 ) M 1 4 / 5 d 1 - 8 / 5 0 1 ( u 7 + v 7 ) d x 1 / 5 + C 5 ε ( 1 + ξ 2 + d 1 - 2 ) M 1 2 / 5 d 1 6 / 5 0 1 ( u 7 + v 7 ) d x 3 / 5 + C 6 ε d 1 2 0 1 ( u 7 + v 7 ) d x ,
(2.16)

where λ ̄ is a constant. Choose a small enough positive number ε(α ij , a i , b ij , e i )(i, j = 1, 2), such that λ ̄ ε < C 3