There is growing biological and physiological evidence, see for instance [1] and the literature cited therein, that in some situations, specially when predators have to search for food and therefore have to share or compete for food, a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. This is supported by numerous field and laboratory experiments and observations, see for instance [5].

The prey-dependent food-chain models were studied in [6, 7, 9, 11, 13, 20], while mathematically interesting, inherit the mechanism that generates the factitious paradox of enrichment and fail to produce the often observed extinction dynamics resulting in the collapse of the system. Consequently, a ratio-dependent food chain model, which is an ODE system with three equations whose species are hence assumed to be spatially homogeneous, was proposed by Hsu, Hwang, and Kuang in [10] to describe the growth of plant, pest, and top predator. However, it is not enough that populations of species are considered in only time and density. To make it more realistic, different spatial locations should also be taken into consideration, which have resulted in reaction-diffusion model with ratio-dependent functional response [12, 14]. Despite the fact that much attention has been paid to studies of weakly-coupled reaction-diffusion system, few has been found on strong-coupled reaction-diffusion system.

This paper discusses the following three questions in the one-dimensional space: the existence and the uniform boundedness of the global solution to a ratio-dependent food chain model with self and cross-diffusion, and the global asymptotic stability of the positive equilibrium point.

Concretely, consider the following problem

$u_{t}=(d_1u+\alpha_{11}u^{2}+\alpha_{12}uv+\alpha_{13}uw)_{xx}+u(1-u-\frac{a_1v}{u+v}), \quad 0<x<1, t>0, \\ v_{t}=(d_2v+\alpha_{21}uv+\alpha_{22}v^{2}+\alpha_{23}vw)_{xx}+v(-b_1+\frac{m_1u}{u+v}-\frac{a_2w}{v+w}), \quad 0<x<1, t>0, \\ w_{t}=(d_3w+\alpha_{31}uw+\alpha_{32}vw+\alpha_{33}w^2)_{xx}+w(-b_2+\frac{m_2v}{v+w}), \quad 0<x<1, t>0, \\ u_{x}(x, t)=v_{x}(x, t)=w_{x}(x, t)=0, \quad x=0, 1, t>0, \\ u(x, 0)=u_{0}(x)\geq0, v(x, 0)=v_{0}(x)\geq0, w(x, 0)=w_{0}(x)\geq0, \quad 0<x<1,$

(1.1)

where $u_{0}, v_{0}, w_{0}$ are nonnegative functions which are not identically zero, $u, v$ and $w$ are the respective population densities of prey, predator, top predator. $d_{i}, \alpha_{ij}(i, j=1, 2, 3), a_i, b_i, m_i$ $(i=1, 2)$ are positive constants, $d_1, d_2, d_3$ are the the diffusion rates of $u, v, w$, respectively. $\alpha_{ii}(i=1, 2, 3)$ are referred as self-diffusion pressures, and $\alpha_{ij}(i\neq j, i, j=1, 2, 3)$ are cross-diffusion pressures. For more details on the biological background, see references [8, 15, 16, 17, 18]. $a_i, b_i, m_i$ $(i=1, 2)$ which can see more explanations for the biological background, refer to [10, 12, 14]. Furthermore, to avoid the case where predator and top-predator cannot survive, even when their food is infinitely abundant, we assume that $m_i > b_i, i = 1, 2$.

The ODE problem associated with (1.1) was proposed and studied by Hsu, Hwang, and Kuang in [10], and from [10], system (1.1) has a unique positive equilibrium $( \bar{u}, \bar{v}, \bar{w} )$ if and only if the following are satisfied:

$m_2>b_2, \quad A>1, \quad 0<a_1<\frac{A}{A-1}, $

(1.2)

where $A=m_1/(a_2(m_2-b_2)/m_2+b_1)$, and

$\bar{u}=\frac{[a_1+A(1-a_1)]}{A}, \quad\bar{v}=(A-1)\bar{u}, \quad\bar{w}=\frac{(m_2-b_2)\bar{v}}{b_2}.$

We also note that $m_2>b_2$ and $A>1$ imply $m_1>b_1$.

In particular, they obtained the extinction conditions of certain species and discussed the local asymptotical stability of $( \bar{u}, \bar{v}, \bar{w} )$ and various scenarios where distinct solutions can be attracted to the origin, the pest-free steady state, and the positive steady state $( \bar{u}, \bar{v}, \bar{w} )$. For more detail, we refer the reader to [10]. From their results, the authors pointed out that this ODE system is very rich in dynamics.

The corresponding weakly coupled reaction-diffusion system (1.1) has received a lot of attention, see [12, 14]. But up to now, the corresponding researches chiefly concentrate on the existence and nonexistence of nonconstant positive steady-state solutions of the weakly-coupled reaction-diffusion system (1.1). To the best of our knowledge, when $\alpha_{ij}(i\neq j, i, j=1, 2, 3)$ is positive, (1.1) is a strongly-coupled reaction-diffusion system which occurs frequently is biological and it is very difficult to analyze, there are very few results for the (1.1).

For simplicity, denote $\|\cdot\|_{W_{p}^{k}(0, 1)}$ by $|\cdot|_{k, p}$ and $\|\cdot\|_{L^{p}(0, 1)}$ by $|\cdot|_{p}$. For the time-dependent solutions of (1.1), the local existence is an immediate consequence of a series of important papers [2-4] by Amann. Roughly speaking, if $u_{0}(x), v_{0}(x), w_0(x)$ in $W_{p}^{1}(\Omega)$ with $p>N, $ then (1.1) has a unique nonnegative solution $u, v, w$ $\in C\left([0, T), W_{p}^{1}(\Omega)\right)$ $\bigcap$ $ C^{\infty}\left((0, T), C^{\infty}(\Omega)\right), $ where $T\in(0, \infty]$ is the maximal existence time for the local solution. If the solution $(u, v, w)$ satisfies the estimates

$\sup\{\|u(\cdot, t)\|_{W^1_p(\Omega)}, \|v(\cdot, t)\|_{W^1_p(\Omega)}, \|w(\cdot, t)\|_{W^1_p(\Omega)}: 0<t<T\}<\infty, $

then $T=+\infty.$ Moreover, if $u_{0}(x), v_{0}(x), w_0(x)\in W_{p}^{2}(\Omega)$, then $u, v, w\in C\left([0, \infty), W_{p}^{2}(\Omega)\right).$

Our main results as follows:

Theorem 1 Let $u_0(x), v_0(x), w_0(x)\in W_{2}^{2}(0, 1), (u, v, w)$ is the unique nonnegative solution of system (1.1) in the maximal existence interval $[0, T).$ Assume that

$8\alpha_{11}\alpha_{21}\alpha_{31}>\alpha_{21}\alpha_{13}^2+\alpha_{12}^2\alpha_{31}, \\ 8\alpha_{12}\alpha_{22}\alpha_{32}>\alpha_{32}\alpha_{21}^2+\alpha_{23}^2\alpha_{12}, \\ 8\alpha_{13}\alpha_{23}\alpha_{33}>\alpha_{23}\alpha_{31}^2+\alpha_{32}^2\alpha_{13}.$

(1.3)

Then there exist $t_{0}>0$ and positive constants $M, M'$ which depend only on $ d_{i}, \alpha_{ij} (i, j=1, 2, 3)$, $a_i, b_i, m_i (i=1, 2)$ such that

$\sup\{|u(\cdot, t)|_{1, 2}, |v(\cdot, t)|_{1, 2}, |w(\cdot, t)|_{1, 2}: t\in(t_{0}, T)\}\leq M',$

(1.4)

$\max\{u(x, t), v(x, t), w(x, t): (x, t)\in[0,1]\times(t_{0}, T)\}\leq M,$

(1.5)

and $T=+\infty$. Moreover, in the case that $d_{1}, d_{2}, d_{3}\geq1, \frac{d_2}{d_1}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}]$, where $\underline{d}$ and $\overline{d}$ are positive constants, $M', M$ depend on $\underline{d}, \overline{d}$ but do not on $d_{1}, d_{2}, d_{3}.$

Theorem 2 Assume that all conditions in Theorem 1 are satisfied and (1.2) holds. Assume that the following hold :

$a_1<1, \quad a_2+b_1<m_1,$

(1.6)

$a_1(A-1)/A<m_1K/(a_2+b_1),$

(1.7)

$a_2b_2m_1(m_2-b_2)(a_2+b_1)<b_1m_2K[m_1-(a_2+b_1)][b_1m_2+a_2(m_2-b_2)],$

(1.8)

$4\alpha\beta \bar{u}\bar{v}\bar{w}d_1d_2d_3 >M^2\bar{u}(\alpha\alpha_{23}\bar{v}+\beta\alpha_{32}\bar{w})^{2}(d_1+2\alpha_{11}M+\alpha_{12}M+\alpha_{13}M) \\ +\alpha M^2\bar{v}(\alpha_{13}\bar{u}+\beta\alpha_{31}\bar{w})^{2} (d_2+\alpha_{21}M+2\alpha_{22}M+\alpha_{23}M)\\ +\beta M^2\bar{w}(\alpha_{12}\bar{u}+\alpha\alpha_{21}\bar{v})^2(d_3+\alpha_{31}M+\alpha_{32}M+2\alpha_{33}M),$

(1.9)

where $\alpha=\frac{a_1\bar{u}}{m_1\bar{v}}, \beta=\frac{a_1a_2\bar{u}}{m_1m_2\bar{w}}$, $K=\frac{1}{2}\left\{2-\frac{m_1}{b_1}+\sqrt{(2-\frac{m_1}{b_1})^2+4(1-a_1)(\frac{m_1}{b_1}-1)}\right\}$, $M$ is the positive constant in (1.5). Then the positive equilibrium point $(\bar{u}, \bar{v}, \bar{w})$ is global asymptotic stable.

Remark 1 Problem (1.1) has a positive solution implies (1.2) holds. From [14], the positive equilibrium point $(\bar{u}, \bar{v}, \bar{w})$ of the corresponding weakly coupled reaction-diffusion system (1.1) is also global asymptotic stable under conditions (1.6)-(1.8) hold.

Remark 2 Problem (1.1) has no non-constant positive steady-state solution if all conditions of Theorem 2 hold.

In order to establish the uniform $W_{2}^{1}-$estimate of the solution to system (1.1), the following corollaries to Gagliardo-Nirenberg-type inequality (see [16]) play important roles.

Corollary 1 There exists a positive constant $C$ such that

$|u|_{2}\leq C(|u_{x}|_{2}^{\frac{1}{3}}|u|_{1}^{\frac{2}{3}}+|u|_{1}), \qquad\forall\, u\in W_{2}^{1}(0, 1),$

(2.1)

$|u|_{4}\leq C(|u_{x}|_{2}^{\frac{1}{2}}|u|_{1}^{\frac{1}{2}}+|u|_{1}), \qquad\forall\, u\in W_{2}^{1}(0, 1),$

(2.2)

$|u|_{\frac{5}{2}}\leq C(|u_{x}|_{2}^{\frac{2}{5}}|u|_{1}^{\frac{3}{5}}+|u|_{1}), \qquad\forall\, u\in W_{2}^{1}(0, 1),$

(2.3)

$|u_{x}|_{2}\leq C(|u_{xx}|_{2}^{\frac{3}{5}}|u|_{1}^{\frac{2}{5}}+|u|_{1}), \qquad\forall\, u\in W_{2}^{2}(0, 1).$

(2.4)

In this section we always denote that $C$ is Sobolev embedding constant or other kind of absolute constant, $A_{j}, B_{j}, C_{j}$ are the positive constants which depend only on $\alpha_{ij}(i, j=1, 2, 3)$, $a_i, b_i, m_i (i=1, 2)$ and $K_{j}$ are positive constants depending on $d_{i}$ and $\alpha_{ij} (i, j=1, 2, 3)$, $a_i, b_i, m_i (i=1, 2)$. When $d_{1}, d_{2}, d_{3}\geq1, \frac{d_{1}}{d_{2}}, \frac{d_{3}}{d_{2}}\in[\underline{d}, \overline{d}], $ $L_{j}$ depend only on $\underline{d}, \overline{d}$ but do not on $d_{1}, d_{2}, d_{3}.$

Proof of Theorem 1 First, we establish $L^{1}$-estimates of the solution $(u, v, w)$ of (1.1). Taking integrations of the first three equations in (1.1) over the domain [0,1], respectively, and then combining the three integration equalities linearly, we have

$\frac{d}{dt}\int_{0}^{1}\left[m_1u+a_1v+\frac{a_1a_2}{m_2}w\right]dx \leq -\int_{0}^{1}\left[\frac{b_1}{a_1}\left(a_1v \right)+b_2\left(\frac{a_1a_2}{m_2}w\right)\right] dx+ m_1\int_{0}^{1}(u-u^2)dx.$

Let $m_1\int_{0}^{1}udx-m_1\int_{0}^{1}u^2dx \leq C_1-C_2\int_{0}^{1}udx$, where $C_2=\min\{\frac{b_1}{a_1}, b_2\}$, by Young inequality,

$m_1(C_2+1)\int_{0}^{1}udx\leq \frac{1}{2\varepsilon} (m_1(C_2+1))^2+\frac{\varepsilon}{2}\int_{0}^{1}u^2dx.$

Let $\varepsilon=2m_1$, then $ C_1=\frac{1}{4}m_1(C_2+1)^2.$ Thus

$\frac{d}{dt}\int_{0}^{1}\left[m_1u+a_1v+\frac{a_1a_2}{m_2}w\right]dx \leq C_1-C_2\int_{0}^{1}\left[m_1u+a_1v+\frac{a_1a_2}{m_2}w\right]dx.$

(2.5)

Then there exists a positive constant $\tau_{0}$ such that

$\int_{0}^{1}udx, \int_{0}^{1}vdx, \int_{0}^{1}wdx\leq M_{0}, \qquad t\geq\tau_{0},$

(2.6)

where $M_{0}=\frac{3C_1}{2C_2}\max\{(a_1)^{-1}, (m_1)^{-1}, \frac{m_2}{a_1a_2}\}$. Moreover, there exists a positive constant $M_{0}'$ which depends on $a_{i}, b_{i}, m_{i}(i=1, 2)$ and the $L^1-$norm of $u_{0}, v_{0}$ and $w_{0}, $ such that

$\int_{0}^{1}udx, \int_{0}^{1}vdx, \int_{0}^{1}wdx\leq M'_{0}, \qquad t\geq0. $

(2.6')

Second, we will obtain $L^{2}$-estimates of $u, v, w.$ We multiply the first three equations in (1.1) by $u, v, w, $ respectively, and integrate over [0,1] to have

$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int_{0}^{1}u^{2}dx&\leq-d_{1}\int_{0}^{1}u_{x}^{2}dx -\int_{0}^{1}[(2\alpha_{11}u+\alpha_{12}v+\alpha_{13}w)u_x^{2}+\alpha_{12}uu_xv_x+ \alpha_{13}uu_xw_x]dx\\ &\quad +\int_{0}^{1}u^2dx, \\ \frac{1}{2}\frac{d}{dt}\int_{0}^{1}v^{2}dx&\le -d_{2}\int_{0}^{1}v_{x}^{2}dx -\int_{0}^{1}[(\alpha_{21}u+2\alpha_{22}v+\alpha_{23}w)v_x^{2}+\alpha_{21}vu_xv_x+ \alpha_{23}vv_xw_x]dx\\ &\quad +m_1\int_{0}^{1}v^2dx, \\ \frac{1}{2}\frac{d}{dt}\int_{0}^{1}w^{2}dx&\le -d_{3}\int_{0}^{1}w_{x}^{2}dx -\int_{0}^{1}[(\alpha_{31}u+\alpha_{32}v+2\alpha_{33}w)w_x^{2}+\alpha_{31}wu_xw_x+ \alpha_{32}wv_xw_x]dx\\ &\quad +m_2\int_{0}^{1}w^{2}dx. \end{aligned}$

Let $d^*=\min\{d_{1}, d_{2}, d_{3}\}.$ We proceed in the following two cases.

(1) $t\geq\tau_{0}. \ $

$ \begin{aligned} \frac{1}{2}\frac{d}{dt}\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx&\le -d^*\int_{0}^{1}(u_{x}^{2}+v_{x}^{2}+w_{x}^{2})dx-\int_{0}^{1}q(u_x, v_x, w_x)dx\\&\quad +A\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx, \end{aligned} $

where $A=\max\{1, m_1, m_2\}$, and

$ \begin{aligned} q(u_x, v_x, w_x)&=(2\alpha_{11}u+\alpha_{12}v+\alpha_{13}w)u_x^{2}\\ &\quad +(\alpha_{21}u+2\alpha_{22}v+\alpha_{23}w)v_x^{2} +(\alpha_{31}u+\alpha_{32}v+2\alpha_{33}w)w_x^{2}\\ &\quad +(\alpha_{12}u+\alpha_{21}v)u_xv_x+ (\alpha_{13}u+\alpha_{31}w)u_xw_x+ (\alpha_{23}v+\alpha_{32}w)v_xw_x \end{aligned}$

is positive semi-definite quadratic form of $ u_x, v_x, w_x$ if (1.3) holds. Then

$\frac{1}{2}\frac{d}{dt}\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx \le -d^*\int_{0}^{1}(u_{x}^{2}+v_{x}^{2}+w_{x}^{2})dx +A\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx.$

(2.7)

Notice by (2.1) and (2.6) that $|u|_{2}^{6}\leq C(|u_{x}|_{2}^{2}|u|_{1}^{4} +|u|_{1}^{6})\leq CM_{0}^{4}(|u_{x}|_{2}^{2}+M_{0}^{2}).$ Therefore

$-d^*\int_{0}^{1}(u_{x}^{2}+v_{x}^{2}+w_{x}^{2})dx\leq 3d^*M_0^2-C_{3}d^*\left[\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx\right]^{3}.$

(2.8)

Substituting (2.8) into (2.7), we have

$\frac{1}{2}\frac{d}{dt}\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx \leq -C_{3}d^*\left[\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx\right]^{3} +A\int_{0}^{1}(u^{2}+v^{2}+w^{2})dx+3d^*M_0^2.$

(2.9)

This means that there exist positive constants $\tau_{1}$ and $M_{1}$ depending on $d_{i}, a_{ij} (i, j=1, 2, 3), $ $a_{i}, b_{i}, m_{i} (i=1, 2)$ such that

$\int_{0}^{1}u^{2}dx, \int_{0}^{1}v^{2}dx, \int_{0}^{1}w^{2}dx\leq M_{1}, t\geq\tau_{1}.$

(2.10)

When $d^*\geq1$, $M_{1}$ is independent of $d^*$ since the zero point of the right-hand side in (2.10) can be estimated by positive constants independent on $d^*$

(2) $t\geq0 $. Replacing $M_{0}$ with $M'$ and repeating estimates (2.7)-(2.10), one can obtain a new inequality which is similar to (2.10). The coefficients of this new inequality depend not only on $d_{i}, a_{ij} (i=1, 2, 3)$, $a_{i}, b_{i}, m_{i} (i=1, 2)$ but also on initial functions $u_{0}, v_{0}$ and $w_{0}.$ Then there exists positive constant $M_{1}'$ depending on $d_{i}, a_{ij}(i, j=1, 2, 3), $ $a_{i}, b_{i}, m_{i} (i=1, 2)$ and the $L^{2}$-norm of $u_{0}, v_{0}, w_{0}$ such that

$\int_{0}^{1}u^{2}dx, \int_{0}^{1}v^{2}dx, \int_{0}^{1}w^{2}dx\leq M'_{1}, \qquad t\geq0.\qquad $

(2.10')

For $d\geq1$, $M_{1}'$ is independent of $d^*$.

Finally, $L^{2}$-estimates of $u_{x}, v_{x}$ and $w_{x}$ will be obtained. We introduce the scaling that

$\tilde{u}=\frac{u}{d_{2}}, \tilde{v}=\frac{v}{d_{2}}, \tilde{w}=\frac{w}{d_{2}}, \tilde{t}=d_{1}t,$

(2.11)

denoting $\xi=\frac{d_2}{d_1}, \eta=\frac{d_{3}}{d_1}$, and using $u, v, w, t$ instead of $\tilde{u}, \tilde{v}, \tilde{w}, \tilde{t}, $ respectively, then system (1.1) reduces to

$u_{t}=P_{xx}+uf(u, v, w), \ 0<x<1, t>0, \\ v_{t}=Q_{xx}+vg(u, v, w), \ 0<x<1, t>0, \\ w_{t}=R_{xx}+wh(u, v, w), \ 0<x<1, t>0, \\ u_{x}(x, t)=v_{x}(x, t)=w_{x}(x, t)=0, \ x=0, 1, t>0, \\ u(x, 0)=\tilde{u}_{0}(x)\geq0, v(x, 0)=\tilde{v}_{0}(x)\geq0, w(x, 0)=\tilde{w}_{0}(x)\geq0, 0<x<1,$

(2.12)

where

$ \begin{array}{lr} P=u+\alpha_{11}\xi u^2+\alpha_{12}\xi uv+\alpha_{13}\xi uw, \\[1ex] Q=\xi(v+\alpha_{21}uv+\alpha_{22}v^2+\alpha_{23}vw), \\[1ex] R=\eta w+\alpha_{31}\xi uw+\alpha_{32}\xi vw+\alpha_{33}\xi w^2, \\[1ex] f(u, v, w)=d_1^{-1}(1-d_2u-\frac{a_1v}{u+v}), \\[1ex] g(u, v, w)=d_1^{-1}(-b_1+\frac{m_1u}{u+v}-\frac{a_2w}{v+w}), \\[1ex] h(u, v, w)=d_1^{-1}(-b_2+\frac{m_2v}{v+w}). \end{array} $

We still divide the subsequent discuss into two cases.

(1) $t\geq\tau_{1}^*(=d_{2}\tau_{1})$ (namely, $t\geq\tau_{1}$ in original scale). It is clearly that

$\int_{0}^{1}udx, \int_{0}^{1}vdx, \int_{0}^{1}wdx\leq M_{0}d_{2}^{-1}, \\ \int_{0}^{1}u^{2}dx, \int_{0}^{1}v^{2}dx, \int_{0}^{1}w^{2}dx\leq M_{1}d_{2}^{-2}, \\ |P|_{1}, |Q|_{1}, |R|_{1}\leq A_{1}K_{1}d_{2}^{-1},$

(2.13)

where $K_{1}=(1+\xi+\eta)M_0+M_{1}\xi d_2^{-1}$. By Young inequality, one can obtain

$\begin{aligned}\int_{0}^{1}u^{4}dx&\leq\left(\int_{0}^{1}u^{2}dx\right)^{\frac{1}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{2}{3}} \leq M_{1}^{\frac{1}{3}}d_{1}^{-\frac{2}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{2}{3}}, \\ \int_{0}^{1}u^{2}v^{2}dx&\leq\left(\int_{0}^{1}u^{2}dx\right)^{\frac{1}{6}}\left(\int_{0}^{1}v^{2}dx\right)^{\frac{1}{6}} \left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}}\left(\int_{0}^{1}v^{5}dx\right)^{\frac{1}{3}}\\& \leq M_{1}^{\frac{1}{3}}d_{1}^{-\frac{2}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}} \left(\int_{0}^{1}v^{5}dx\right)^{\frac{1}{3}}, \\ \int_{0}^{1}u^{3}dx&\leq\left(\int_{0}^{1}u^{2}dx\right)^{\frac{2}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}} \leq M_{1}^{\frac{2}{3}}d_{1}^{-\frac{4}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}}, \\ \int_{0}^{1}uv^{2}dx&\leq\left(\int_{0}^{1}u^{2}dx\right)^{\frac{1}{2}}\left(\int_{0}^{1}v^{2}dx\right)^{\frac{1}{6}} \left(\int_{0}^{1}v^{5}dx\right)^{\frac{1}{3}} \leq M_{1}^{\frac{2}{3}}d_{1}^{-\frac{4}{3}}\left(\int_{0}^{1}v^{5}dx\right)^{\frac{1}{3}} .\end{aligned}$

(2.14)

Multiply the first three equations in (2.12) by $P_{t}, Q_{t}, R_{t}, $ and integrating them over the domain [0, 1], respectively, then adding up the three integration equalities, we have

$\begin{aligned} \frac{1}{2}\bar{y}'(t)&\le -\int_{0}^{1}u_{t}^{2}dx-\xi\int_{0}^{1}v_{t}^{2}dx-\eta\int_{0}^{1}w_{t}^{2}dx -\xi\int_{0}^{1}q(u_t, v_t, w_t)dx\\&\quad +\int_{0}^{1}[(1+2\alpha_{11}\xi u+\alpha_{12}\xi v+\alpha_{13}\xi w)uu_{t}f+\alpha_{12}\xi u^2v_{t}f+\alpha_{13}\xi u^2w_{t}f]dx\\&\quad +\xi\int_{0}^{1}[\alpha_{21}v^2u_tg+(1+\alpha_{21}u+2\alpha_{22}v+\alpha_{23}w)vv_{t}g+\alpha_{23}v^2w_{t}g]dx\\&\quad +\int_{0}^{1}[\alpha_{31}\xi w^2u_th+\alpha_{32}\xi w^2v_{t}h+(\eta+\alpha_{31}\xi u+\alpha_{32}\xi v+2\alpha_{33}\xi w)ww_{t}h]dx, \end{aligned} $

where $\bar{y}=\displaystyle\int_{0}^{1}(P_{x}^{2}+Q_{x}^{2}+R_{x}^{2})dx$. It is not hard to verify by (1.3) that there exists a positive constant $C_4$ depending only on $\alpha_{ij}$ $(i, j=1, 2, 3), $ such that

$q(u_t, v_t, w_t)\geq C_{4}(u+v+w)(u_{t}^{2}+v_{t}^{2}+w_{t}^{2}).$

Thus

$\frac{1}{2}\bar{y}'(t) \le -\int_{0}^{1}u_{t}^{2}dx-\xi\int_{0}^{1}v_{t}^{2}dx-\eta\int_{0}^{1}w_{t}^{2}dx -C_{4}\xi\int_{0}^{1}(u+v+w)(u_{t}^{2}+v_{t}^{2}+w_{t}^{2})dx\\ \;+\int_{0}^{1}(1+2\alpha_{11}\xi u+\alpha_{12}\xi v+\alpha_{13}\xi w)uu_{t}f dx +\int_{0}^{1}\xi(1+\alpha_{21}u+2\alpha_{22}v+\alpha_{23}w)v_{t}gvdx\\ \;+\int_{0}^{1}(\eta+\alpha_{31}\xi u+\alpha_{32}\xi v+2\alpha_{33}\xi w)w_{t}hw dx +\int_{0}^{1}\alpha_{12}\xi u^2v_{t}fdx+\int_{0}^{1}\alpha_{13}\xi u^2w_{t}fdx \\ \; +\int_{0}^{1}\alpha_{21}\xi v^2u_tgdx+\int_{0}^{1}\alpha_{23}\xi v^2w_{t}gdx +\int_{0}^{1}\alpha_{31}\xi w^2u_thdx+\int_{0}^{1}\alpha_{32}\xi w^2v_{t}hdx.$

(2.15)

By the estimates (2.13), (2.14), one can obtain the following estimates for the terms on the right-hand side of (2.15)

$\quad -\int_{0}^{1}u_{t}^{2}dx\leq -\frac{1}{2}\int_{0}^{1}P_{xx}^{2}dx+\int_{0}^{1}u^2f^{2}dx, \\ \quad -\xi\int_{0}^{1}v_{t}^{2}dx\leq -\frac{\xi}{2}\int_{0}^{1}Q_{xx}^{2}dx+\xi\int_{0}^{1}v^2g^{2}dx, \\ \quad -\eta\int_{0}^{1}w_{t}^{2}dx\leq -\frac{\eta}{2}\int_{0}^{1}R_{xx}^{2}dx+ \eta\int_{0}^{1}w^2h^{2}dx, \\ \quad \int_{0}^{1}u^2f^{2}dx \le d_1^{-2}(1+a_1^2)\int_{0}^{1}u^{2}dx+d_1^{-2}d_2^{2}\int_{0}^{1}u^{4}dx +a_1d_1^{-2}d_2\int_{0}^{1}u^{3}dx\\ \le (1+a_1^2)d_1^{-2}d_2^{-2}M_{1}+d_1^{-\frac{8}{3}}d_{2}^{2}M_{1}^{\frac{1}{3}} \left(\int_{0}^{1}u^{5}dx\right)^{\frac{2}{3}}+a_1d_1^{-\frac{10}{3}}d_2 M_{1}^{\frac{2}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}}, \\ \quad \xi\int_{0}^{1}v^2g^{2}dx\le \int_{0}^{1}\xi d_1^{-2}(b_1^2+a_2^2+m_1^2+a_2b_1)v^{2}dx\leq \xi M_{1}d_1^{-2}d_{2}^{-2}(b_1^2+a_2^2+m_1^2+a_2b_1), \\ \quad \eta\int_{0}^{1}w^2h^{2}dx\le d_1^{-2}(b_2^2+m_2^2)\eta\int_{0}^{1}w^{2}dx \leq\eta d_1^{-2}(b_2^2+m_2^2) M_{1}d_{2}^{-2}. $

Therefore,

$\quad -\int_{0}^{1}u_{t}^{2}dx-\xi\int_{0}^{1}v_{t}^{2}dx-\eta\int_{0}^{1}w_{t}^{2}dx\\ \le -\frac{1}{2}\int_{0}^{1}P_{xx}^{2}dx-\frac{\xi}{2}\int_{0}^{1}Q_{xx}^{2}dx-\frac{\eta}{2}\int_{0}^{1}R_{xx}^{2}dx \\ \quad +C_{5}(1+\xi+\eta)M_{1}d_{1}^{-2}d_{2}^{-2}+C_{6}\xi^{2}(1+\eta)M_{1}^{\frac{1}{3}}d_2^{-\frac{2}{3}} \left[\int_{0}^{1}u^{5}dx\right]^{\frac{2}{3}}\\ \quad +C_{7}\xi d_1^{-1}M_{1}^{\frac{2}{3}}d_2^{-\frac{4}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}}.$

(2.16)

For $\displaystyle\int_{0}^{1}uu_{t}fdx$, one can obtain

$\quad \int_{0}^{1}uu_{t}fdx\\ \le d_1^{-1}(1+a_1)\left|\int_{0}^{1}u_{t}udx\right|+\xi\left|\int_{0}^{1}u^2u_{t}dx\right|\\ \le d_1^{-1}(1+a_1)\left(\frac{1}{2\epsilon}\int_{0}^{1}udx+\frac{\epsilon}{2}\int_{0}^{1}uu_{t}^{2}dx\right) +\xi\left(\frac{1}{2\epsilon}\int_{0}^{1}u^3dx+\frac{\epsilon}{2}\int_{0}^{1}uu_{t}^{2}dx\right)\\ \le \frac{1+a_1}{2\epsilon}M_0d_1^{-1}d_2^{-1}+\frac{1}{2\epsilon}\xi M_{1}^{\frac{2}{3}}d_1^{-\frac{4}{3}}\left(\int_{0}^{1}u^{5}dx\right)^{\frac{1}{3}} +\frac{(d_1^{-1}+\xi)}{2}\epsilon\int_{0}^{1}uu_{t}^2dx. $

Similarly, we estimates the rest term on the right-hand side of (2.15), we have

$\quad \int_{0}^{1}(1+2\alpha_{11}\xi+\alpha_{12}\xi v+\alpha_{13}\xi)uu_{t}fdx +\int_{0}^{1}\xi(1+\alpha_{21}u+2\alpha_{22}v+\alpha_{23}w)vv_{t}gdx \\ \quad+\int_{0}^{1}(\eta+\alpha_{31}\xi u+\alpha_{32}\xi v+2\alpha_{33}\xi w)ww_{t}hdx+ \int_{0}^{1}\alpha_{12}\xi uv_{t}fdx+\int_{0}^{1}\alpha_{13}\xi uw_{t}fdx \\ \quad +\int_{0}^{1}\alpha_{21}\xi v^2u_tgdx+\int_{0}^{1}\alpha_{23}\xi v^2w_{t}gdx +\int_{0}^{1}\alpha_{31}\xi w^2u_thdx+\int_{0}^{1}\alpha_{32}\xi w^2v_{t}hdx\\ \le \lambda \epsilon \xi \int_{0}^{1}(u+v+w)(u_t^2+v_t^2+w_t^2)dx +\frac{C_{8}}{\epsilon}M_0d_1^{-1}d_2^{-2}(1+\xi+\eta)\\ \quad+\frac{C_9}{\epsilon}M_{1}^{\frac{2}{3}}d_2^{-\frac{4}{3}}\xi(1+d_1^{-1}+\eta) \left[\int_{0}^{1}(u^{5}+v^{5}+w^{5})dx\right]^{\frac{1}{3}} +\frac{C_{10}}{\epsilon}\xi^2\int_{0}^{1}(u^{5}+v^{5}+w^{5})dx,$

(2.17)

where $\lambda$ is a positive integer. Choose a small enough positive number $\epsilon=\epsilon(\alpha_{ij} (i=1, 2, 3), $ $a_i, b_i, $ $m_i, (i=1, 2)$ such that $\lambda\epsilon<C_{4}.$ Substituting inequalities (2.16) and (2.17) into (2.15), one can obtain

$\begin{aligned} \frac{1}{2}\bar{y}'(t)&\le -\frac{1}{2}\int_{0}^{1}P_{xx}^{2}dx-\frac{\xi}{2}\int_{0}^{1}Q_{xx}^{2}dx -\frac{\eta}{2}\int_{0}^{1}R_{xx}^{2}dx\\ &\quad+B_{1}K_{2}d_{1}^{-1}d_{2}^{-1} +B_{2}K_{3}d_{1}^{-\frac{4}{3}}z^{\frac{1}{3}}+B_{3}K_{4}d_{1}^{-\frac{2}{3}}z^{\frac{2}{3}}+B_{4}K_{5}z, \end{aligned}$

(2.18)

where $z=\int_{0}^{1}(u^{5}+v^{5}+w^{5})dx, $ $ K_{2}=(1+\xi+\eta)(M_0+M_{1}), $ $ K_{3}=M_{1}^{\frac{2}{3}}\xi(1+d_1^{-1}+\eta ), $ $K_{4}=M_{1}^{\frac{1}{3}}\xi^2(1+\eta), \; K_{5}=\xi^{2}$. Clearly,

$P\geq\alpha_{11}\xi u^{2}, \;Q\geq\alpha_{22}\xi v^{2}, \;R\geq\alpha_{33}\xi w^{2}.$

It follows from inequality (2.3) to functions $P, Q, R$ that

$z\leq B_{5}\xi^{-\frac{5}{2}}\int_{0}^{1}(P^{\frac{5}{2}}+Q^{\frac{5}{2}}+R^{\frac{5}{2}})dx \leq B_{6}\xi^{-\frac{5}{2}}K_{1}^{\frac{3}{2}}d_{1}^{-\frac{3}{2}}\bar{y}^{\frac{1}{2}} +B_{6}\xi^{-\frac{5}{2}}K_{1}^{\frac{5}{2}}d_{1}^{-\frac{5}{2}}, \\ z^{\frac{1}{3}}\leq B_{7}\xi^{-\frac{5}{6}}K_{1}^{\frac{1}{2}}d_{1}^{-\frac{1}{2}}\bar{y}^{\frac{1}{6}} +B_{7}\xi^{-\frac{5}{6}}K_{1}^{\frac{5}{6}}d_{1}^{-\frac{5}{6}}, \\ z^{\frac{2}{3}}\leq B_{8}\xi^{-\frac{5}{3}}K_{1}d_{1}^{-1}\bar{y}^{\frac{1}{3}} +B_{8}\xi^{-\frac{5}{3}}K_{1}^{\frac{5}{3}}d_{1}^{-\frac{5}{3}}.$

(2.19)

Moreover, one can obtain by (2.4) and (2.15)

$\quad -\frac{\xi}{2}\int_{0}^{1}P_{xx}^{2}dx-\frac{1}{2}\int_{0}^{1}Q_{xx}^{2}dx -\frac{\eta}{2}\int_{0}^{1}R_{xx}^{2}dx\\ \le-B_{9}\min\{1, \xi, \eta\}L_{1}^{-\frac{4}{3}}d_{1}^{\frac{4}{3}}\bar{J}^{\frac{5}{3}} +(1+\xi+\eta)L_{1}^{2}d_{1}^{-2}.$

(2.20)

Combining (2.18), (2.19) and (2.20) we have

$\quad \frac{1}{2}\bar{y}'(t)\\ \le -A_{1}\min\{1, \xi, \eta\}K_{1}^{-\frac{4}{3}}d_2^{\frac{4}{3}}\bar{y}^{\frac{5}{3}} +A_{2}\xi^{-\frac{5}{6}}K_{1}^{\frac{1}{2}}K_{3}d_2^{-\frac{11}{6}}\bar{y}^{\frac{1}{6}}+ A_{3}\xi^{-\frac{5}{3}}K_{1}K_{4}d_2^{-\frac{5}{3}}\bar{y}^{\frac{1}{3}}\\ \quad + A_{4}\xi^{-\frac{5}{2}}K_{1}^{\frac{3}{2}}K_{5}d_2^{-\frac{3}{2}}\bar{y}^{\frac{1}{2}}\\ \quad +A_{5}[K_1^2d_2^{-2}(1+\xi+\eta) +K_2d_1^{-1}d_2^{-1}+K_{1}^{\frac{5}{6}}K_{3}\xi^{-\frac{5}{6}}d_2^{-\frac{13}{6}}+ K_1^{\frac{5}{3}}K_{4}\xi^{-\frac{5}{3}}d_2^{-\frac{7}{3}}+K_{1}^{\frac{5}{2}}K_{5}\xi^{-\frac{5}{2}}d_2^{-\frac{5}{2}}].$

(2.21)

Multiplying inequality (2.21) by $d_2^{2}, $ we have

$\begin{aligned} \frac{1}{2}y'(t)&\le -A_{1}\min\{1, \xi, \eta\}K_{1}^{-\frac{4}{3}}y^{\frac{5}{3}} +A_{2}\xi^{-\frac{5}{6}}K_{1}^{\frac{1}{2}}K_{3}d_2^{-\frac{1}{6}}y^{\frac{1}{6}}\\ &\quad +A_{3}\xi^{-\frac{5}{3}}K_{1}K_{4}d_2^{-\frac{1}{3}}y^{\frac{1}{3}}+ A_{4}\xi^{-\frac{5}{2}}K_{1}^{\frac{3}{2}}K_{5}d_2^{-\frac{1}{2}}y^{\frac{1}{2}}\\&\quad +A_{5}[K_1^2(1+\xi+\eta) +K_2\xi+K_{1}^{\frac{5}{6}}K_{3}\xi^{-\frac{5}{6}}d_2^{-\frac{1}{6}}+ K_1^{\frac{5}{3}}K_{4}\xi^{-\frac{5}{3}}d_2^{-\frac{1}{3}}+K_{1}^{\frac{5}{2}}K_{5}\xi^{-\frac{5}{2}}d_2^{-\frac{1}{2}}], \end{aligned}$

(2.22)

where $y=\int_{0}^{1}[(d_2P_{x})^{2}+(d_2Q_{x})^{2}+(d_2R_{x})^{2}]dx$. Inequality (2.22) implies that there exist $\tilde{\tau_{2}}>0$ and positive constant $\tilde{M_{2}}$ depending only on $d_{i}, \alpha_{ij}(i, j=1, 2, 3)$, $a_i, b_i, m_i (i=1, 2)$ such that

$\int_{0}^{1}(d_2P_{x})^{2}dx, \int_{0}^{1}(d_2Q_{x})^{2}dx, \int_{0}^{1}(d_2R_{x})^{2}dx\leq \tilde{M_{2}}, \qquad t\geq\tilde{\tau_{2}}.$

(2.23)

In the case that $d_{1}, d_{2}, d_{3}\geq1, \frac{d_2}{d_1}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}]$, the coefficients of inequality (2.22) can be estimated by some constants depending on $\underline{d}, \overline{d}$ but not on $d_{1}, d_{2}, d_{3}.$ So $\tilde{M_{2}}$ depends on $\alpha_{ij}(i, j=1, 2, 3), a_i, b_i, m_i, (i=1, 2), \underline{d}, \overline{d}$ and is irrelevant to $d_{1}, d_{2}, d_{3}$ when $d_{1}, d_{2}, d_{3}\geq1$ and $\frac{d_{2}}{d_{1}}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}].$ Since

$\left( \begin{array}{c} u_{x} \\ v_{x} \\ w_{x} \\ \end{array} \right) =\left( \begin{array}{ccc} P_{u} P_{v} P_{w}\\ Q_{u} Q_{v} Q_{w}\\ R_{u} R_{v} R_{w}\\ \end{array} \right)^{-1} \left( \begin{array}{c} P_{x}\\ Q_{x}\\ R_{x}\\ \end{array} \right), $

we can transform the formulations of $u_{x}, v_{x}, w_{x}$ into fraction representations, then distribute the denominators of the absolute value of the fractions to the numerators term by term and enlarge the term concerning with $u_{x}, v_{x}$ or $w_{x}$ to obtain

$|d_2u_{x}|+|d_2v_{x}|+|d_2w_{x}| \leq K(|d_2P_{x}|+|d_2Q_{x}|+|d_2R_{x}|), 0<x<1, t>0, $

(2.24)

where $K$ is a constant depending only on $\xi, \eta, \alpha_{ij}(i, j=1, 2, 3)$, After scaling back and contacting estimates (2.23) and (2.24), there exist positive constant $M_{2}$ depending on $d_{i}, \;\alpha_{ii}, \;(i=1, 2, 3), \alpha_{12}, \;\alpha_{21}, \; \alpha_{23}, \;\alpha_{32}, a_{i}, \;b_{i}, m_i (i=1, 2)$ and $\tau_{2}>0, $ such that

$\int_{0}^{1}u_{x}^{2}dx, \int_{0}^{1}v_{x}^{2}dx, \int_{0}^{1}w_{x}^{2}dx\leq M_{2}, \qquad t\geq\tau_{2}.$

(2.25)

When $d_{1}, d_{2}, d_{3}\geq1$ and $\frac{d_{2}}{d_{1}}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}]$, $M_{2}$ independent on $d_{1}, d_{2}, d_{3}.$

(2) $t\geq0.$ Modifying the dependency of the coefficients in inequalities (2.15)-(2.17), namely replacing $M_{0}, M_{1}$ with $M_{0}', M_{1}', $ there exists positive constant $M_{2}'$ depending on $d_{i}, \alpha_{ij}(i, j=1, 2, 3), a_i, b_i, m_i, (i=1, 2) $ and the $W_{2}^{1}$ -norm of $u_{0}, v_{0}, w_{0}$ such that

$\int_{0}^{1}u_{x}^{2}dx, \int_{0}^{1}v_{x}^{2}dx, \int_{0}^{1}w_{x}^{2}dx\leq M'_{2}, \qquad t\geq0.$

(2.25')

Furthermore, in the case that $d_{1}, d_{2}, d_{3}\geq1, \frac{d_{2}}{d_{1}}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}], $ $M_{2}'$ depends on $\underline{d}, \overline{d}$ but not on $d_{1}, d_{2}, d_{3}.$

Summarizing estimates (2.6), (2.10), (2.25) and using Sobolev embedding theorem, there exist positive constants $M, M'$ depending only on $d_{i}, \; \alpha_{ij} (i, j=1, 2, 3), \; a_i, b_i, m_i (i=1, 2) $ such that (1.4) and (1.5) hold. In particular, $M, M'$ depend only on $\alpha_{ij} (i, j=1, 2, 3), \; a_i, b_i, m_i (i=1, 2)\; \underline{d}, \overline{d}$ but do not depend on $d_{1}, d_{2}, d_{3}$ when $d_{1}, d_{2}, d_{3}\geq1, \frac{d_{2}}{d_{1}}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}]$.

Similarly, there exist positive constant $M''$ depending on $d_{i}, \;\alpha_{ij} (i, j=1, 2, 3), \; a_i, b_i, m_i (i=1, 2) $ and the initial functions $u_{0}, v_{0}, w_{0}$ such that

$|u(\cdot, t)|_{1, 2}, |v(\cdot, t)|_{1, 2}, |w(\cdot, t)|_{1, 2}\leq M'', t\geq0.$

Further, in the case that $d_{1}, d_{2}, d_{3}\geq1, \frac{d_2}{d_1}, \frac{d_{3}}{d_1}\in[\underline{d}, \overline{d}], $ $M''$ depends only on $\underline{d}, \overline{d}$ but not on $d_{1}, d_{2}, d_{3}.$ Thus $T=+\infty.$ This is complete proof of Theorem 1.

In this section we discuss global asymptotic stability of positive equilibrium point $(\bar{u}, \bar{v}, \bar{w})$ for (1.1), namely to prove Theorem 2. Define

$H(u, v, w)=\int_{0}^{1}\left(u-\bar{u}-\bar{u}\ln\frac{u}{\bar{u}}\right)dx+\alpha\int_{0}^{1}\left(v-\bar{v}-\bar{v}\ln\frac{v}{\bar{v}}\right)dx+ \beta\int_{0}^{1}\left(w-\bar{w}-\bar{w}\ln\frac{w}{\bar{w}}\right)dx, $

where $\alpha=\frac{a_1\bar{u}}{m_1\bar{v}}, \beta=\frac{a_1a_2\bar{u}}{m_1m_2\bar{w}}$. Obviously, $H(u, v, w)$ is nonnegative and $H(u, v, w)=0$ if and only if $(u, v, w) =(\bar{u}, \bar{v}, \bar{w}).$ By Theorem 1, $H(u, v, w)$ is well-posed for $t\geq0$ if $(u, v, w)$ is a non-zero solution to system (1.1). The time derivative of $H(u, v, w)$ for system (1.1) satisfies

$\quad \frac{dH(u, v, w)}{dt}\\ =-\int_{0}^{1}\bigg[\bigg(d_{1}+2\alpha_{11}u+\alpha_{12}v+\alpha_{13}w\bigg)\frac{\bar{u}}{u^2}u_{x}^{2}+\alpha\bigg(d_{2}+\alpha_{21}u+2\alpha_{22}v +\alpha_{23}w\bigg)\frac{\bar{v}}{v^2}v_{x}^{2}\\ \quad +\beta\bigg(d_{3}+\alpha_{31}u+\alpha_{32}v+2\alpha_{33}w\bigg)\frac{\bar{w}}{w^2}w_{x}^{2} +\bigg(\alpha_{12}\frac{\bar{u}}{u}+\alpha\alpha_{21}\frac{\bar{v}}{v}\bigg)u_{x}v_{x}\\ \quad +\bigg(\alpha_{13}\frac{\bar{u}}{u}+\beta\alpha_{31}\frac{\bar{w}}{w}\bigg)u_xw_x+ \bigg(\alpha\alpha_{23}\frac{\bar{v}}{v}+\beta\alpha_{32}\frac{\bar{w}}{w}\bigg)v_{x}w_{x}\bigg]dx\\ \quad -\int_{0}^{1}\bigg[\bigg(1-\frac{a_1\bar{v}}{(\bar{u}+\bar{v})(u+v)}\bigg)(u-\bar{u})^{2}+\beta\frac{m_2\bar{v}}{(\bar{v}+\bar{w})(v+w)}(w-\bar{w})^{2} \\ \quad +\alpha\bigg(\frac{m_1\bar{u}}{(\bar{u}+\bar{v})(u+v)}-\frac{a_2\bar{w}}{(\bar{v}+\bar{w})(v+w)}\bigg)(v-\bar{v})^{2}\bigg]dx. $

(3.1)

The first integrand in above equality is positive semi-definite if

$\quad 4\alpha\beta\frac{\bar{u}}{u^2}\frac{\bar{v}}{v^2}\frac{\bar{w}}{w^2}(d_{1}+2\alpha_{11}u+\alpha_{12}v+\alpha_{13}w)(d_{2}+\alpha_{21}u+2\alpha_{22}v +\alpha_{23}w)\\ \quad \cdot(d_{3}+\alpha_{31}u+\alpha_{32}v+2\alpha_{33}w)\\ \quad +\bigg(\alpha_{12}\frac{\bar{u}}{u}+\alpha\alpha_{21}\frac{\bar{v}}{v}\bigg)\bigg(\alpha_{13}\frac{\bar{u}}{u}+\beta\alpha_{31}\frac{\bar{w}}{w}\bigg)\bigg(\alpha\alpha_{23}\frac{\bar{v}}{v}+\beta\alpha_{32}\frac{\bar{w}}{w}\bigg)\\ >\frac{\bar{u}}{u^2}\bigg(\alpha\alpha_{23}\frac{\bar{v}}{v}+\beta\alpha_{32}\frac{\bar{w}}{w}\bigg)^{2}(d_{1}+2\alpha_{11}u+\alpha_{12}v+\alpha_{13}w)\\ \quad +\alpha\frac{\bar{v}}{v^2}\bigg(\alpha_{13}\frac{\bar{u}}{u}\beta\alpha_{31}\frac{\bar{w}}{w}\bigg)^{2}(d_{2}+\alpha_{21}u+2\alpha_{22}v +\alpha_{23}w)\\ \quad +\beta\frac{\bar{w}}{w^2}\bigg(\alpha_{12}\frac{\bar{u}}{u}+\alpha\alpha_{21}\frac{\bar{v}}{v}\bigg)^2(d_{3}+\alpha_{31}u+\alpha_{32}v+2\alpha_{33}w).$

(3.2)

By the maximum-norm estimate in Theorem 1, condition (1.9) implies (3.2). Under our assumptions (1.6)-(1.8), we can claim that for $t\gg1$ the following hold:

$\frac{a_1\bar{v}}{(\bar{u}+\bar{v})(u+v)}\leq1, \quad \frac{a_2\bar{w}}{(\bar{v}+\bar{w})(v+w)}\leq\frac{m_1\bar{u}}{(\bar{u}+\bar{v})(u+v)}.$

So, the second integrand in above equality is positive semi-definite if conditions (1.6)-(1.8) hold. Therefore, when the all conditions in Theorem 2 hold, there exists a positive constant $\delta$ such that

$\frac{dH(u, v, w)}{dt}\leq -\delta\int_{0}^{1}[(u-\bar{u})^{2}+(v-\bar{v})^{2}+(w-\bar{w})^{2}]dx, \\ \frac{dH(u, v, w)}{dt}<0, (u, v, w)\neq(\bar{u}, \bar{v}, \bar{w}).$

(3.3)

Now, we recall the following lemma which can be find in [19].

Lemma 1 Let $a$ and $b$ be positive constants. Assume that $\varphi, \; \psi\in C^1[a, +\infty)$, $\psi(t)\geq0$, and $\varphi$ is bounded from below. If $\varphi'(t)\leq-b\psi(t)$ and $\psi'(t)$ is bounded from above in $[a, +\infty)$, then $\lim\limits_{t\to\infty}\psi(t)=0$.

Using partial integration, Hölder inequality and (1.5), one can easily verify that

$\frac{d}{dt}\int_{0}^{1}[(u-\bar{u})^{2}+(v-\bar{v})^{2}+(w-\bar{w})^{2}]dx$

is bounded from above. Then from Lemma 1 and (3.3) we have

$|u(\cdot, t)-\overline{u}|_{2}\to0, \; |v(\cdot, t)-\overline{v}|_{2}\to0, \; |w(\cdot, t)-\overline{w}|_{2}\to0 \quad( t\to\infty).$

Clearly, $|u(\cdot, t)|_{\infty}\leq C|u|_{1, 2}^{\frac{1}{2}}|u|_{2}^{\frac{1}{2}}$. By (1.4), we have

$|u(\cdot, t)-\overline{u}|_{\infty}\to0, \; |v(\cdot, t)-\overline{v}|_{\infty}\to0, \; |w(\cdot, t)-\overline{w}|_{\infty}\to0 \quad( t\to\infty).$

Namely, $(u, v, w)$ converges uniformly to $(\bar{u}, \bar{v}, \bar{w}).$ By the fact that $H(u, v, w)$ is decreasing for $t\geq0, $ it is obvious that $(\bar{u}, \bar{v}, \bar{w})$ is global asymptotic stable. The proof of Theorem 2 is completed.