The existence of a compact global attractor for a class of competition model

1.   Introduction and main results

In population dynamics, N. Shegesada, K. Kawasaki and E. Teromoto ^[15] proposed the following quasilinear competition model with cross-diffusion,

where the functions $u, \; v$ are the population densities of the two competing species and the initial values $u_0, \; v_0$ are nonnegative functions, which are not identically zero. $\Omega$ is a bounded smooth region in $\mathbb{R}^{n}$ with $\nu$ as its unit outward normal vector to $\partial \Omega.$ The constants $a_{j}, b_{j}, c_{j},$ $d_{j} \, (j = 1, 2)$ are all positive, and the constants $\rho_{i j}\, (i, j = 1, 2)$ are nonnegative, where $d_{1}$ and $d_{2}$ are the random diffusion rates, $\rho_{11}$, $\rho_{22}$ are the self-diffusion rates which represent intraspecific population pressures, and $\rho_{12}$, $\rho_{21}$ are the so-called cross-diffusion rates which represent the interspecific population pressures.

When $\rho_{ij} = 0\, (i, j = 1, 2)$, (1.1) is reduces to the well-known Lotka-Volterra competition-diffusion system, which has been researched intensively. When $\rho_{12}$ or $\rho_{21}$ is positive, $(1.1)$ is a strongly coupled parabolic system, which has received much attention, since it occurs frequently in biological and chemical models. H. Amann considered a general class of strongly coupled parabolic systems and established the local existence (in time) and uniqueness results in a series of papers ^[1,2,3]. Roughly speaking, H. Amann showed that if $u_{0}, \, v_{0}$ in $W^{1, p}(\Omega)$ with $p > n$, then (1.1) has a unique solution $u, v$ defined in $(0, t_0)$ with $t_0 > 0$ small.

The global existence of nonnegative solutions to (1.1) is considered under some restrictive hypotheses on the smallness of cross-diffusion pressures or on the space dimension. For the case $\rho_{12} > 0, \rho_{21} > 0$, if $\rho_{11} = \rho_{22} = 0,$ J. Kim ^[8] proved the global existence of classical solutions by energy method when $n = 1$ and $d_{1} = d_{2}.$ Later, S. Shim ^[16] improved J. Kim's results and obtained the uniform boundedness of the global solutions in time by interpolated estimates. P. Deuring ^[6] proved the global existence of classical solutions when $n \geqslant 1$ and $\rho_{12}, \, \rho_{21}$ are small enough depending on the $C^{2, \alpha}$ norm of initial values $u_{0}, v_{0}.$ If the self-diffusion rates $\rho_{11}$ and $\rho_{22}$ are not zero, A. Yagi ^[22] proved the global existence of solutions when $n = 2$ and $0 < \rho_{12} < 8\rho_{11}, \, 0 < \rho_{21} < 8 \rho_{22}$, he also proved the same results for $\rho_{22} = \rho_{21} = 0$ and $\rho_{11} > 0$. In addition, Y. Li and C. Zhao ^[13] obtained the global existence of classical solutions when $n\geqslant 1, \; d_{1} = d_{2}$ and $\frac{\rho_{12}}{\rho_{22}}+\frac{\rho_{21}}{\rho_{11}} = 2$.

For the case of $\rho_{21} = 0$, (1.1) becomes the following system

Y. Lou, W. Ni and Y. Wu ^[14] established a global existence of classical solutions to (1.2) for $n\leqslant2$ and $\rho_{11}$ is merely assumed nonnegative but $\rho_{12}$ and $\rho_{22}$ are allowed to be positive, which is the only available result for smooth solutions with $\rho_{11} = 0$. When $\rho_{11}$ is positive, Y. Choi, R. Lui and Y. Yamada ^[4,5] obtained some results on the global existence of the solutions to (1.2) with the restrictions $n < 6$ and $\rho_{22} > 0.$ P. Tuoc ^[20] showed the global existence of solutions for $n < 10$. The global existence of solutions for arbitrary $n$ under some restrictions on coefficients are investigated (see ^[7,9,11,19]). For the uniform boundedness of the global solutions, D. Le, L. Nguyen and T. Nguyen ^[12] using the semi-group techniques obtained the global attractor for $n < 6$, which implies the uniform boundedness of the global solutions. Q. Xu and Y. Zhao ^[21] obtained the global attractor for $n < 8$. And Y. Tao and M. Winkler ^[17] showed the boundedness of the solutions for $n < 10$ when $\Omega\in\mathbb{R}^{n}$ is a bounded convex domain with smooth boundary.

In this paper, we considered the following more general strongly coupled parabolic system

Due to the absence of the cross-diffusion term in the $v-$equation, the diffusion matrix of (1.3) is triangular. H. Amann ^[3] showed that if one can obtain $u\in L^\infty, \, v\in L^\infty$, then the solution of (1.3) exists globally in time. D. Le ^[10] proved that if $u\in L^n, \; v\in L^\infty$, then the system (1.3) has a global attractor with finite Hausdorff dimension, which attracts all the solutions of (1.3). D. Le, L. Nguyen and T. Nguyen ^[12] improved the results of that in ^[10]. In order to state their results, we first introduce the following definition.

Definition 1.1. (see ^[10], Definition 2.1) Assume that there exists a solution $(u, v)$ of system ${\rm(1.3)}$ defined on a subinterval I of $\mathbb{R}^{+}$. Let $\mathcal{P}$ be the set of function $\omega$ on $I$ such that there exists a positive constant $C_{0},$ which may generally depend on the parameters of the system and the $W^{1, p_{0}}$ norm of the initial value $\left(u_{0}, v_{0}\right),$ such that

Furthermore, if $I = (0, \infty),$ then there exists a positive constant $C_{\infty}$ that depends only on the parameters of the system, but does not depend on the initial value of $\left(u_{0}, v_{0}\right)$, such that

If $\omega \in \mathcal{P}$ and $I = (0, \infty),$ one says $\omega$ is ultimately uniformly bounded.

In ^[12], D. Le, L. Nguyen and T. Nguyen suppose that

(H1) There exist a continuous function $\Phi$ and positive constant $d$ such that the differentiable functions $P, \, Q, \, R$ satisfying

(H2) There exists a nonnegative continuous function $C(v)$ such that

Under the above hypotheses (H1) and (H2), the authors proved the following results.

Lemma 1.2. (see ^[12], Theorem 2.4) Assume ${\rm(H1)}$ and ${\rm(H2)}$ hold. Let $(u, v)$ be a nonnegative solution to ${\rm(1.3)}$ with its maximal existence interval I. If $\|u\|_{q, r, [t, t+1] \times \Omega} = \left(\int_{t}^{t+1}\|u(\cdot, s)\|_{q, \Omega}^{r} d s\right)^{1 / r}$ (as a function in $t$) is in $\mathcal{P}$ for some $q, r$ satisfying

then there exists an absorbing ball where all solutions will enter eventually. Thus, if the system ${\rm(1.3)}$ is autonomous then there is a compact global attractor with finite Hausdorff dimension in $\mathscr{B}$, which attracts all solutions, with

In this paper, we impose some conditions on the functions $P, \, Q, \, R, \, f, \, g$ in system (1.3) as follows.

(A1) The functions $P, \, Q, \, R$ are differentiable in there variables, and there exist constants $\beta > 0, \; b > 0$ and continuous function $\phi(v)\geqslant 0$ for $v\geqslant 0$, such that

(A2) For the reaction terms $(f, \, g)$, we assume that there exist positive constants $a, \, b, \, c, \, \alpha$ and nonnegative continuous functions $f_1(u, v), \, \varphi(v)$, such that

Remark 1. Our assumptions ${\rm(A1)–(A2)}$ on $(P, \, Q, \, R, \, f, \, g)$ in this paper satisfy ${\rm(H1)–(H2)}$ in ^[12].

Now, we state our main results.

Theorem 1.3. Suppose ${\rm(A1)–(A2)}$ hold and $(u_0, \, v_0)\in \mathscr{B}$ with some $p_{0} > n$. Then ${\rm(1.3)}$ has a compact global attractor with finite Hausdorff dimension in the space $\mathscr{B}$, which attracts all the solutions, for any given $\alpha > 0, \, \beta > 0$ and $n\leqslant2$, but we need the following corresponding assumptions in ${\rm(B1)–(B2)}$ for $n > 2$,

$\rm(B1)$ For $0 < \beta\leqslant \frac{n-2}{6}$,

$\rm(i)$ if $0 < \alpha\leqslant \frac{n+4}{n-2}\beta$, then $n^2-2(1+4\beta)n+4\beta < 0$;

$\rm(ii)$ if $\frac{n+4}{n-2}\beta < \alpha < \beta+1$, then $n < 2(\alpha+3\beta)$;

$\rm(iii)$ if $\alpha\geqslant\beta+1$, then $n < 2(2\alpha+2\beta-1)$.

$\rm(B2)$ For $\beta > \frac{n-2}{6}$, then

$\rm(i)$ if $0 < \alpha\leqslant \beta+1$, then $n^2-2(1+4\beta)n+4\beta < 0$;

$\rm(ii)$ if $\beta+1 < \alpha < \frac{n+4}{n-2}\beta$, then $n^2-2(\alpha+3\beta)n+4(\alpha-1) < 0$;

$\rm(iii)$ if $\alpha\geqslant \frac{n+4}{n-2}\beta$, then $n < 2(2\alpha+2\beta-1)$.

Theorem 1.4. Assume $n < 10$ and $(u_0, \, v_0)\in \mathscr{B}$ with some $p_{0} > n,$ then ${\rm(1.2)}$ has a compact global attractor with finite Hausdorff dimension in the space $\mathscr{B}$, which attracts all the solutions.

Remark 2. Theorem ${\rm1.3}$ and Theorem ${\rm1.4}$ imply the uniform boundedness of the global solutions to the systems ${\rm(1.3)}$ and ${\rm(1.2)}$, respectively.

This paper is organized as follows. In section 2, we shall prove the existence of a compact global attractor with finite Hausdorff dimension to system (1.3). As an application, we consider the Shegesada-Kawasaki-Teromoto competition model (1.2), and get the existence of a global attractor for $n < 10$ in section 3.

2.   The existence of a global attractor to (1.3)

We shall first give the uniform Gronwall inequality, which will be frequently used in our proof.

Lemma 2.1. (the uniform Gronwall inequality) (see^[18] Chapt. 3, Lemma 1.1). Suppose positive Lipschitz functions $y(t), \; r(t), \; h(t)$ defined on $\left[t_{0}, +\infty\right]$ satisfy

and

with $\tau, \, r_0, \, h_0$ and $c_0$ some positive constants. Then it holds that

For given initial data $u_0(x), v_0(x)\in\mathscr{B}$, it is standard to show that the solutions of (1.3) are still nonnegative. Then using comparison principle for parabolic equation on the $v-$equation of (1.3), it is easy to see

For the solution $u$, it is easy to get the following properties.

Lemma 2.2. The solution $u$ of ${\rm(1.3)}$ satisfies

and

Proof. Integrating the $u-$equation of (1.3) by parts and noting the condition of $f(u, v)$ in (A2), we get

which together with H$\ddot{o}$lder inequality $\|u\|_{L^{1}(\Omega)}\leqslant \|u\|_{L^{\alpha+1}(\Omega)}\|1\|_{L^{\frac{\alpha+1}{\alpha}}(\Omega)}$ gives

Then the comparison principle of ordinary differential equation implies (2.2) holds. Integrate (2.4) from $t$ to $t+1$ and use (2.2) to yield (2.3).

For the solution $v$, we will prove the following result, which plays an important role in the following estimates of $u$ in Theorem 2.4. In the rest of our paper, $C_i\, (i = 1, 2, \cdots)$ are some positive constants, and we will not point out them one by one.

Lemma 2.3. For $n\geqslant 1$, the solution $v$ of ${\rm(1.3)}$ satisfies

Proof. In order to prove (2.6), we first show

then we prove

Recalling the condition of $R(v)\geqslant d$ in (A1), (2.8) ensures (2.6) holds.

Now, we first deal with the proof of (2.7). For this purpose, multiplying the second equation of (1.3) by $v$ and integrating by parts, we have

Integrating the above equation over $[t, t+1]$, we obtain

by $R(v) \geqslant d$ in (A1), $g(u, v) \leqslant \varphi(v)(1-cu)^{\frac{\alpha+1}{2}}$ in (A2) and the fact (2.1). Therefore, it is known by (2.1) and (2.2) that

Next, we multiply the $v-$equation of (1.3) by $R(v)v_t$ and integrate by parts to get

here, we use H$\ddot{o}$lder inequality, the condition of $g(u, v)$ in (A2) and (2.1).

Due to $R(v)\geqslant d$, thus

In view of (2.1), (2.3), (2.9) and using the uniform Gronwall inequality on

we obtain

Moreover, integrate (2.10) over $[t, \, t+1]$ to know

By the $v-$equation of (1.3) and noting (2.1), we have

this together with (2.12) and (2.3) gives (2.7).

Next, we will prove (2.8). Denote $\xi = R(v)\nabla v$ and note $R^\prime(v)\geqslant 0$ in (A1) to get

By H$\ddot{o}$lder inequality, we can get

and

thus

Now, we will prove

Noting $\xi = R(v)\nabla v$, then we have

where we see $\nabla v$ as a row vector, $(\nabla v)^T$ is the transpose of $\nabla v$, and ${\nabla}^2 v$ is a matrix.

By (2.15) and the standard elliptic regularity $\|{\nabla}^2v\|_ {L^{2}(\Omega)}\leqslant C_5\|\triangle v\|_ {L^{2}(\Omega)}$, we can prove (2.14) holds.

Therefore, in virtue of (2.13) and (2.14), we obtain

This together with (2.7) and (2.1) indicates that (2.8) holds. This completes the proof of Lemma 2.3.

Next, we shall give the critical estimates in our paper.

Theorem 2.4. The solution $u$ of ${\rm(1.3)}$ satisfies

and

for $\bar{p}$ satisfying ${\rm(i)}$ $\bar{p} = \alpha+2\beta$ or ${\rm(ii)}$ $\bar{p} > \alpha+2\beta$ and $(n-2)\bar{p}\leqslant3n\beta$.

Proof. Multiplying the first equation in (1.3) by $u^{p-1}$ with $p > 1$, and integrating on $\Omega$ by parts, we have

Recalling the condition of $f(u, v)$ in (A2) and $|Q(u, v)| \leqslant \phi(v) u$ in (A1), we have

and

by Hölder inequality.

Combining these estimates and $P(u, v) \geqslant d u^{\beta}$ in (A1), then

Case I: $p\leqslant \alpha+2\beta.$ In this case, we have $2(p-\beta)\leqslant p+\alpha.$

Applying Hölder inequality and Young's inequality to the last term of (2.19), we have

Consequently, (2.19) becomes

Obviously, (2.20) entails

For the above inequality (2.21), if we can show

then (2.6) and the uniform Gronwall inequality yield

Furthermore, integrating (2.20) from $t$ to $t+1$, we can obtain

and

Now, we will use mathematical induction to prove that (2.22) holds for $p = \alpha+2\beta$. There exists some $k\in N_+$ such that $1\leqslant \alpha+2\beta-k\alpha\leqslant \alpha+1$. Denote $p_0 = \alpha+2\beta-k\alpha, \; p_{m} = p_{m-1}+\alpha\, (m = 1, \, 2, \, \cdot\cdot\cdot, \, k)$. On one hand, using (2.3) and Hölder inequality, it is easy to see that (2.22) holds for $p = p_0$. On the other hand, we suppose (2.22) holds for $p = p_{m-1}$, then (2.24) means that (2.22) holds for $p = p_{m-1}+\alpha = p_m$. Hence the mathematical induction ensures that (2.22) holds for $p = p_k = \alpha+2\beta$.

Therefore, (2.23)–(2.25) hold for $p = \alpha+2\beta$, which implies (2.16)–(2.18) hold for $\bar{p} = \alpha+2\beta.$

Case II: $p > \alpha+2\beta.$ In this case, we assume $(n-4)p\leqslant (3n-4)\beta$.

Let $w_p = u^{\frac{p+\beta}{2}}$ and denote $w_p$ as $w$ sometimes for simplicity, then (2.19) can be written as

by the Hölder inequality.

The conditions $p > \alpha+2\beta$ and $(n-4)p\leqslant (3n-4)\beta$ implies

here $\frac{2n}{n-2}$ can be replaced by $+\infty$ for $n = 2$.

It is known by Gagliardo-Nirenberg inequality that

with

Using (2.27) and Young's inequality, we have

with

Let $\varepsilon = { \frac{d(p-1)}{(p+\beta)^{2}}}$, then (2.26) becomes

Let $y(t) = \|w\|^{\frac{2p}{p+\beta}}_{L^\frac{2p}{p+\beta}(\Omega)}, \; h(t) = C_{11}\|w\|^{\frac{4(p-\beta)}{p+\beta}}_{L^{1}(\Omega)}+C_{11}\|\nabla v\|_{L^{4}(\Omega)}^4$, then we have

For the case of $y(t)\leqslant1,$ obviously we have

Since $p > \alpha+2\beta$, (2.30) implies $\|w\|_{L^{1}(\Omega)}\in \mathcal{P}$ by Hölder inequality. Let

then a direct calculation shows that $m_2\leqslant 4$. Consequently, (2.6) and Hölder inequality give

Furthermore, integrating (2.28) from $t$ to $t+1$ yields

and

For the case of $y(t) > 1,$ denote $r(t) = a+C_ \varepsilon\|\nabla v\|_{L^{4}(\Omega)}^{m_2},$ then

here, we used the fact

It is easy to see that (2.31) implies $\int_{t}^{t+1}r(s)d s \in \mathcal{P}$ for $(n-2)p\leqslant 3n\beta$, thus if we can show

then using the uniform Gronwall inequality to the inequality (2.34), we obtain (2.30). Similarly, integrate (2.28) over $[t, t+1]$ to obtain (2.32) and (2.33).

Now, we prove (2.35) for $(n-2)p = 3n\beta$. In order to get $\int_{t}^{t+1}h(s)d s \in \mathcal{P}$, recalling (2.6), the key step is to deal with

since the Minkowski's inequality ensures

Now, we will prove $\int_{t}^{t+1}y(s)d s \in \mathcal{P}$ and (2.36) by mathematical induction, simultaneously. For the case of $n > 2$, there exists some $\bar{k}\in N_+$ such that $\alpha+2\beta < \frac{3n\beta}{n-2}-\bar{k}\alpha\leqslant 2\alpha+2\beta$. Denote $q_0 = \frac{3n\beta}{n-2}-\bar{k}\alpha, \; q_{m} = q_{m-1}+\alpha\, (m = 1, \, 2, \, \cdot\cdot\cdot, \, \bar{k})$. Since we have proved (2.17) for $\bar{p} = \alpha+2\beta$, it is easy to get $\int_{t}^{t+1}y(s)d s \in \mathcal{P}$ for $p = q_0$ by Hölder inequality. In addition, noting that $\frac{q_0+\beta}{2}\leqslant \frac{2\alpha+3\beta}{2} < \alpha+2\beta$ and (2.16) holds for $\bar{p} = \alpha+2\beta$, using Hölder inequality we obtain $\|w_{q_0}\|_{L^{1}(\Omega)}\in \mathcal{P},$ which indicates that the result (2.36) is true for $p = q_0$. On the other hand, assume $\int_{t}^{t+1}y(s)d s \in \mathcal{P}$ and (2.36) hold for $p = q_{m-1}$, then (2.32) holds for $p = q_{m-1}$. According to the definition $w_p = u^{\frac{p+\beta}{2}}$, it is easy to see

thus $\int_{t}^{t+1}y(s)d s \in \mathcal{P}$ for $p = q_{m}$. Moreover, the assumption implies (2.30) holds for $p = q_{m-1}$ and hence

by $q_{m-1} > \alpha+2\beta$ and Hölder inequality with some $C > 0$. And thus (2.36) holds for $p = q_m$.

Above all, for the case of $n > 2$, we have proved (2.35) with $p = \frac{3n\beta}{n-2}$. Therefore, (2.30), (2.32) and (2.33) hold for $p = \frac{3n\beta}{n-2}$, which implies (2.16)–(2.18) hold for $\bar{p}\leqslant\frac{3n\beta}{n-2}$. Similarly, we can prove (2.16)–(2.18) hold for any positive constant $\bar{p} > \alpha+2\beta$ if $n\leqslant2$.

This complete the proof of Theorem 2.4.

Next, we will use Lemma 1.2 and Theorem 2.4 to give the proof of Theorem 1.3.

proof of Theorem 1.3. Let $s = \bar{p}+\alpha$, then by (2.17),

Define

By Lemma 1.2, we also need $\chi \in(0, 1),$ which is equivalent to

It is known by Gagliardo-Nirenberg inequality that

with $2^{*} = 2 n /(n-2)$.

Let $l = \frac{\bar{p}+\beta}{2}$ and $r = 2 l, q = 2^{*} l$, then $w = u^l$ and

The estimate $\|w\|_{L^{1}(\Omega)}\in \mathcal{P}$ comes from (2.16) by Hölder inequality, which together with (2.18) indicates

Let

By Lemma 1.2, we also need $\chi \in(0, 1),$ which means

Comparing (2.37) and (2.38), we choose $n < 2(\bar{p}+\alpha-1)$ if $\alpha > \beta+1$, otherwise, we choose $n < 2(\bar{p}+\beta)$. In addition, recall $\bar{p}$ satisfies ${\rm(i)}$ $\bar{p} = \alpha+2\beta$ or ${\rm(ii)}$ $\bar{p} > \alpha+2\beta$ and $(n-2)\bar{p}\leqslant3n\beta$ in Theorem 2.4, hence we can assign any positive number to $\bar{p}$ for $n\leqslant2$, but we choose $\bar{p} = \frac{3n\beta}{n-2}$ if $0 < \alpha\leqslant\frac{n+4}{n-2}\beta$, otherwise, we choose $\bar{p} = \alpha+2\beta$ for $n > 2$. Consequently, combining these analysis we can obtain Theorem 1.3.

3.   The uniform boundedness of the solutions to S-K-T model for $n < 10$

In this part, we will consider the boundedness of the global solutions to the following S-K-T model

Proof of Theorem 1.4. Comparing with the divergence form of system (1.3), we have

It is easy to see that $P, \, Q, \, R$ and $f, \, g$ satisfy the conditions in (A1) and (A2), respectively, with $\alpha = \beta = 1$.

Theorem 1.3 gives Theorem 1.4 for $n\leqslant2$ directly. Moreover, a simple computation shows (B2) (i) in Theorem 1.3 holds for $2 < n < 8$ and (B1) (i) holds for $8\leqslant n < 10$. This completes the proof of Theorem 1.4.

Remark 3. Our result implies the uniform boundedness of the global solutions to the system (3.1). This result extends the existence results of global attractor in ^[21] from $n < 8$ to $n < 10$, and extends the uniform boundedness results of the global solutions in ^[17] to the non-convex domain.

Acknowledgments

The author is greatly indebted to Professor Yaping Wu for her valuable suggestions and helpful discussions. And the author is very grateful for the anonymous referees for their valuable comments and many useful suggestions which helped to improve the exposition of the current paper. The work is supported by NSFC (No. 11801314).

Conflict of interest

The author declares no conflicts of interest in this paper.