Global boundedness of a higher-dimensional chemotaxis system on alopecia areata

1.   Introduction

In this paper, we consider the spatio-temporal dynamics of a three-component chemotaxis system

where $\Omega\subset\mathbb{R}^n$ $(n\geq 4)$ is a bounded convex domain with smooth boundary $\partial\Omega$ and $\nu$ denotes the outer normal vector to $\partial\Omega$. Here, the parameters $\chi_i$, $\mu_i\; (i = 1, 2)$ and $r$ are positive constants. The initial data $(u_0, v_0, w_0)$ satisfies

This system was originally proposed by Dobreva et al. ^[1] to describe the complex dynamic behavior of alopecia areata (AA). Alopecia areata is mainly manifested as hair loss, which is caused by the attack of the immune system on the hair follicle. Previous investigations ^[2,3] have shown that the development of AA is usually initiated by abnormally high production of pro-inflammatory cytokines, such as interferon-gamma (IFN-$\gamma$) which is the most influential inducer of hair follicles immune privilege (HF IP) collapse. IFN-$\gamma$ is secreted by the two types of immune cells which are CD4$^+$ T cells and CD8$^+$ T cells, and it diffuses and degrades, moreover, it is also the chemoattractant for CD4$^+$ T cells and CD8$^+$ T cells; CD4$^+$ T cells which move randomly are triggered by IFN-$\gamma$ and decrease on density-dependent death; CD8$^+$ T cells which also move randomly, are triggered by IFN-$\gamma$, proliferate with the help of CD4$^+$ T cells, and undergo density-dependent death. Based on the above biological mechanism, in the system (1.1), the unknown functions $u(x, t)$, $v(x, t)$ and $w(x, t)$ respectively denote the density of CD4$^+$ T cells, the density of CD8$^+$ T cells and the concentration of IFN-$\gamma$.

From the mathematical perspective, Lou and Tao ^[4] recently studied global boundedness and the asymptotic stability of the solution for (1.1), when $n = 2$ or $n = 3$ and

this system admitted a global boundedness classical solution. Furthermore, as

where $\chi_0$ was a positive constant, the classical solution was globally asymptotic stability. Subsequently, Tao and Xu ^[5] showed the spatio-temporal evolution of IFN-$\gamma$ describing a quasi-steady-state approximation of the equation, and got the dimensionless parabolic-parabolic-elliptic version of (1.1) by replacing the third equation in (1.1) with $0 = \Delta{w}+u+v-w$, they proved the global boundedness of its solution when

then the large-time behavior of (1.1) was also gained as

To better understand the system (1.1), the previously existing two-component chemotactic system should be mentioned

Many known results about the system (1.3) have been obtained in past studies. For instance, as $f(u) = 0$, system (1.3) corresponds to the classical KS system ^[6], its global boundedness has been testified in ^[7], and the blow-up behavior has also been constructed in a finite or infinite time (cf. ^[8,9,10], for instance) which strongly relied on the spatial dimension and the domain $\Omega$. However, the logical damping term $f(u) = \mu u(1-u)$ plays an important role in preventing the blow-up phenomenon. Then, as $n \leq 2$, for an arbitrarily small coefficient $\mu > 0$, the system (1.3) can admit a uniformly bounded classical solution and prevent the blow-up phenomenon ^[11,12,13]. Whereas, $n\geq 3$, the global classical solution of (1.3) exists and remains uniformly bounded if the coefficient $\mu$ is large enough ^{[14,15,16,17]}. Moreover, it is worth mentioning that a variant of system (1.3) describes the movement of cells driven in an incompressible fluid under the influence of chemoattractant which is a different but interesting system called chemotaxis-Navier-Stokes system. The analysis of the solution of this system and its variants have attracted wide attention in recent years, such as Winkler ^[18,19] and Zheng ^{[21,22,23,24]}. For a more detailed discussion on (1.3) or its variants, we refer the reader to ^[8,25] and the references therein.

Main results and ideas.

Compared with (1.3) or other pre-existing chemotaxis systems ^[26,27,28], the main differences come from the terms $w$ and $ruv$ in the system (1.1), it is worth emphasizing that the nonlinear generation term $ruv$ is obviously different from the nonlinear proliferation term appearing in ^[29] or ^[30], which is composed of the product of signal concentration and cell density. Inspired by Winkler ^[17] and Xiang ^[16], we can deal with the three-component chemotactic system (1.1) by some appropriate improvements for the functional $y(t): = \int_{\Omega}u^p +\int_{\Omega}|\nabla v|^{2q}$ for any $p > 1$, $q > 1$ and $t > 0$. Motivated by the above ideas, we shall study the evolution of the combined integral

to address the difficulties from the activated term $w$ and nonlinear production term $ruv$. We find that the behavior of solution can be impacted by nonlinear diffusion, the nonlinear zero-order production term $ruv$ and logistic damping. Our main results are started as follows.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^n$ $(n = 4, 5)$ be a bounded convex domain with smooth boundary, $\chi_i, \mu_i > 0\; (i = 1, 2)$ and $r > 0$. For any $\epsilon\in(0, 1)$, if

and

then for any initial data $(u_0, v_0, w_0)$ satisfying (1.2), then system (1.1) admits a globally uniformly bounded solution, there also exists a constant $K > 0$ such that

Remark 1.1. It is worth noting that we add the convexity of $\Omega$ to simplify the total process. However, this assumption can be dropped by the methods in ^[31]. In fact, we first need to establish the estimates on $\int_\Omega{u^2}$, $\int_\Omega{v^2}$, $\int_\Omega{w^2}$ and $\int_\Omega{|\nabla{w}|^4}$ on non-convex domain $\Omega$, this will lead to stronger restrictions for the parameters $\mu_i\; (i = 1, 2)$.

Remark 1.2. This method can also be used to deal with the higher-dimensional case, however, the estimates of lower bound for $\mu_i\; (i = 1, 2)$ will be more complex, we omit here.

Compared with the asymptotic stability of some classical KS system in ^[32] or ^[33], it seems that the nonlinear term $ruv$ greatly changed the large-time behavior of the smooth solutions of (1.1). To see this more intuitively, we first write

where $a: = \frac{r+\sqrt{r^2+4\mu_1\mu_2}}{2\mu_2} > 0$. Then we have the following theorem.

Theorem 1.2. Assume the conditions in Theorem 1.1 hold. Let

and

If

holds, where $M = M(\mu_1, \mu_2, u_0, v_0, w_0)$ is a positive constant, then for any global classical solution $(u, v, w)$ of (1.1) with the initial data fulfilling (1.2), it satisfies the following property

Remark 1.3. According to (1.6) and in view of (1.4), we obverse that

The structure of this paper is as follows: In Section 2, we provide some crucial lemmas which will be used in the following context. In Section 3, we make some basal estimates which will help us to deal with the boundary integrals on $\Omega$ in the next section. Then we show some priori estimates in Section 4 and prove the global boundedness of the classical solution of (1.1) in Section 5. In Section 6, we mainly analyze the large-time behavior of the solution for (1.1).

2.   Preliminaries

In this section, we start from local in time existence which is a crucial lemma for the existence of globally bounded classical solutions.

Lemma 2.1. ^[34] Let $\Omega\subset\mathbb{R}^n$ $(n\ge1)$ be a bounded domain with smooth boundary, $\chi_i$, $\mu_i > 0\; (i = 1, 2)$ and $r > 0$. Then for any initial data $(u_0, v_0, w_0)$ fulfilling (1.2), there exists $T_{max}\in(0, \infty]$ and a unique nonnegative solution $(u, v, w)$ of (1.1) which satisfies $u, v\in C^0\left(\overline{\Omega}\times[0, T_{max})\right)\cap C^{2, 1}\left(\overline{\Omega}\times(0, T_{max})\right)$ and $w\in C^0\left([0, T_{max}); W^{1, p}{(\Omega)}\right)\cap C^{2, 1}\left(\overline{\Omega}\times(0, T_{max})\right)$ for any $p > n$. Furthermore, if $T_{max} < \infty$, then

Proof. The local in time existence of the classical solution to (1.1) follows from well-known standard methods in ^[34,35], we omit it here.

Next, we present a estimate of the Neumann heat semigroup which will be used in section 5.

Lemma 2.2. ^[36] Let $(e^{t\Delta})_{t\ge0}$ be the Neumann heat semigroup in $\Omega$, and let $p\in(0, \infty]$. Then there exists $C > 0$ such that for all $\varphi\in C^1\left(\overline{\Omega}; \mathbb{R}^n\right)$ satisfying $\varphi\cdot\nu = 0$ on $\partial\Omega$, we have

Then, we give some basic estimates for our following work.

Lemma 2.3. There exist $m > 0$ and $h > 0$ such that the solution $(u, v, w)$ of (1.1) fulfills

Proof. This lemma can be proved by integrating three equations in (1.1) over $\Omega$, then applying the ODE comparison can complete it easily. The interested readers can get detailed proof from Lemma 2.2 of ^[4].

3.   Estimates on $\int_\Omega{u^3}$, $\int_\Omega{v^3}$, $\int_\Omega{w^3}$ and $\int_\Omega{|\nabla{w}|^6}$

Refer to some previous results in ^[25] or ^[16], when $n = 4 \mbox{ or } 5$, the global existence of classical solution for (1.1) will be obtained by a priori bounds on $\int_\Omega{u^3}$, $\int_\Omega{v^3}$, $\int_\Omega{w^3}$ and $\int_\Omega{\lvert\nabla{w}\rvert}^6$. In order to establish those estimates, we begin with the following energy inequalities.

Lemma 3.1. Let $(u, v, w)$ be a solution of (1.1), then for any $\epsilon \in(0, 1)$ there holds that

for all $t\in(0, T_{max})$.

Proof. Multiplying the first equation in (1.1) by $u^2$, integrating by parts, we obtain

for all $t\in(0, T_{max})$. Applying Young's inequality with any $\epsilon\in(0, 1)$, we estimate

for all $t\in(0, T_{max})$. Substituting (3.3) into (3.2) yields that

for all $t\in(0, T_{max})$. Similarly, testing the second equation of (1.1) by $v^2$, we also have

for all $t\in(0, T_{max})$. Finally, integrating the third equation of (1.1) behind multiplying $w^2$, we know

for all $t\in(0, T_{max})$. Combining with (3.4)–(3.6), we immediately deduce (3.1).

Next, we make a priori estimate for $\int_\Omega{\lvert\nabla{w}\rvert}^6$.

Lemma 3.2. Let $\Omega$ be convex, then we have

Proof. Using the third equation of (1.1) and $2\nabla{w}\cdot\nabla\Delta{w} = \Delta{\lvert\nabla{w}\rvert}^2-2{\lvert{D^2w}\rvert}^2$, we have

$\mbox{for all } t\in(0, T_{max})$. Thanks to the convexity assumption on $\Omega$ and $\frac{\partial{w}}{\partial{\nu}}\bigg{\arrowvert}_{\partial\Omega} = 0$ ^[37], we ensure that

Moreover, due to ${\lvert \Delta w\rvert}^2 \le n{\lvert D^2 w\rvert}^2$, we know

for all $t\in(0, T_{max})$. Similarly, we get

$\mbox{for all } t\in(0, T_{max})$. At the end, collecting the above results yields (3.7).

The integrals $\int_\Omega{u^2{\lvert\nabla{w}\rvert}^4}$ and $\int_\Omega{v^2{\lvert\nabla{w}\rvert}^4}$ appearing on the right-hand side of (3.7) can be treated by utilizing the logical damping terms in (1.1) properly. To obtain this end, we shall establish the following similar integrals $\int_\Omega{u{\lvert\nabla{w}\rvert}^4}$ and $\int_\Omega{v{\lvert\nabla{w}\rvert}^4}$.

Lemma 3.3. Let $\Omega$ be convex, then there exist two positive constants $\epsilon_1$ and $\epsilon_2$ independent of system parameters $\mu_i$, $\chi_i$ $(i = 1, 2)$ and $r$ such that

and

for all $t \in (0, T_{max})$.

Proof. Applying the $u$ and $w$-equations of (1.1), then integrating it over $\Omega$, we know

for all $t\in(0, T_{max})$. Here, similar to (3.9), we get

Moreover, we observe that

and

for all $t\in(0, T_{max})$ with some $\epsilon_1 > 0$ and $\epsilon_2 > 0$ independent of system parameters $\mu_i$, $\chi_i$ $(i = 1, 2)$ and $r$. Using the similar computation in (3.10), we have

for all $t\in(0, T_{max})$. Combining with (3.14)–(3.18), we obtain (3.12). In the same way, by the combination of $v$ and $w$-equations in (1.1), we can also prove (3.13) easily.

In order to deal with the terms $\int_{\Omega}|\nabla u|^2 |\nabla w| ^2$ and $\int_{\Omega}|\nabla v|^2 |\nabla w| ^2$ in the right-hand sides of (3.12) and (3.13), let us establish estimates for $\int_{\Omega}u^2|\nabla w|^2$ and $\int_{\Omega} v^2 |\nabla w|^2$, respectively.

Lemma 3.4. Let $\Omega$ be convex. Then for any $\epsilon\in(0, 1)$, there exist two constants $\epsilon_3 > 0$ and $\epsilon_4 > 0$ independent of system parameters $\mu_i$, $\chi_i$ $(i = 1, 2)$ and $r$ such that

and

for all $t\in(0, T_{max})$.

Proof. By directly computing the coupling of $u^2$ and ${\lvert\nabla{w}\rvert}^2$, we obtain

for all $t\in(0, T_{max})$. As before, we have

and by applying Young's inequality, we derive

and

for all $t\in(0, T_{max})$ with some $\epsilon_3 > 0$ and $\epsilon_4 > 0$ independent of system parameters $\mu_i$, $\chi_i$ $(i = 1, 2)$ and $r$. Moreover, we also have

and

for all $t\in(0, T_{max})$ with $\epsilon\in(0, 1)$. In addition, it is obvious that

Then (3.19) can be proved by taking (3.22)–(3.27) into (3.21). For the similarity of (3.20) and (3.19), the same operation can be done to testify (3.20).

As usual, we rely on Gronwall's inequality to get $L^3$–boundedness of $u$, $v$ and $w$ and $L^6$–boundedness of $\nabla{w}$. For this purpose, set

for all $t\in[0, T_{max})$ with $\delta_i > 0\; (i = 1, 2, 3, 4)$ independent of system parameters $\mu_i$, $\chi_i$ $(i = 1, 2)$ and $r$. Then we combine Lemmas 3.1–3.4 and choose appropriate parameters to get the following lemma.

Lemma 3.5. Let $\Omega$ be convex and and the initial data $(u_0, v_0, w_0)$ fulfill (1.2). Then for any $\epsilon\in(0, 1)$, assume that $\mu_1$, $\mu_2$ satisfy

and

then we find a positive constant $K_1$ to ensure that

And there also exists a constant $K_2 > 0$ such that

Proof. For the convenience of readers, we divide the proof into the following two steps:

Step 1. In this step, we deal with the coupling terms appearing on the right-hand side of Lemma 3.1–Lemma 3.4 to approach Gronwall's inequality.

A direct linear combination Lemma 3.1–Lemma 3.4 after multiplying $\delta_i > 0$ ($i = 1, 2, 3, 4$), we have

for all $t\in(0, T_{max})$, where

Then applying Young's inequality, we estimate

and

for all $t\in(0, T_{max})$ with some positive constants $C_i\; (i = 1, 2, 3, 4, 5, 6)$. Apart from those, we also have

and

for all $t\in(0, T_{max})$. Together with (3.33)–(3.38), we can find a positive constant $C$ to ensure that

where

Step 2. To deal with the extra terms on the left of (3.39), we choose to ignore them if the coefficient in front of them is positive. This requires that the parameters $\mu_i\; (i = 1, 2)$ satisfy certain conditions. {Here} we need $B_i\ge0\; (i = 1, 2, \cdot\cdot\cdot, 7)$ and $B^*_j\ge0\; (j = 1, 2, 3)$, and from (3.40), we find that the fourth constraint $B_4$ has something special in common with the sixth constraint $B_6$, then we have

We use the fact that

then the inequality manipulations from (3.41) and $B_i\geq0\; (i = 1, 2, 3, 7)$ show that

and the following inequality comes from (3.41), $B^*_j\ge0\; (j = 1, 2, 3)$ and $B_7\ge0$

After fixing $\epsilon_i\; (i = 1, 2, 3, 4)$, it is possible for us to ignore some of the terms on the left of (3.39) if $\mu_i\; (i = 1, 2)$ are sufficiently large to obtain

then we obverse that $y(t)$ fulfills

where $\xi = \min \left\{2, 3\left(1-\frac{\delta_3}{\delta_1}\right), 2\left(3-\frac{2\delta_3}{3\delta_4}\right)\right\}$. Applying Gronwall's inequality will yield

which means

Then there exists a constant $K_2 > 0$ such that

due to the parabolic regularity ^[34,38]. This completes the proof.

4.   Proof of Theorem 1.1

In Section 3, we get $L^{\infty}$-boundedness of $w$ which plays an important role in our subsequent proof. In this section, we will use heat semigroup theory to prove Theorem 1.1.

Lemma 4.1. Let $\Omega\subset\mathbb{R}^n\; (n\le5)$ be a convex domain and the initial data $(u_0, v_0, w_0)$ satisfy (1.2), then there exists $K_3 = K_3(n, m, K_1, K_2) > 0$ such that

and

as well as

$\mathit{\mbox{for all}}\;~ t\in(0, T_{max})$.

Proof. The proof is based on ^[31]. Given $T\in(0, T_{max})$, write

Since $u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2$ in $\Omega\times(0, T_{max})$, we represent $u(\cdot, t)$ for each $t\in(0, T_{max})$ according to

where $t_0: = (t-1)_+$. Here by the maximum principle, we can estimate

whereas if $t > 1$, then standard $L^p-L^q$ estimates for the Neumann heat semigroup (cf. [7,Lemma 1.3 (i)], for instance) provide $C_1 > 0$ such that

holds because of (2.3).

Next, according to (3.32), $\sup\limits_{x\in\Omega}\lvert{w(x, t)}\rvert\le K_2\mbox{ for all } t\in(0, T_{max})$ can be obtained. Again by the maximum principle, we have

At the end, to estimate $u_2$, we choose an arbitrary $p\in(n, 6)$. Then invoking known smoothing properties of $(e^{\eta\Delta})_{\eta\ge0}$ ([7,Lemma 1.3 (iv)]) and applying the H$\ddot{\mbox{o}}$lder inequality to find a constant $C_2 > 0$ such that

where $a: = \frac{7p-6}{6p}\in(0, 1)$. In view of (2.3), (3.31) and the definition of $M(T)$, this yields that

so that since $\frac{1}{2}+\frac{n}{2p} < 1$ according to our limitation $p > n$. Combining with (4.4)–(4.7), we obtain $C_3 > 0$ such that

for all $t\in(0, T_{max})$. Therefore, we know

this implies

this completes the proof of (4.1). Then we can get

With this, (4.2) can be proved by the same operations as $u$. Then we obtain (4.3) due to the parabolic regularity. Collecting (4.1)–(4.3), we can easily find a positive constant $K$ to prove Theorem 1.1.

5.   Proof of Theorem 1.2

Theorem 1.1 claims the global existence and boundedness of the classical solution of (1.1) if $\mu_i\; (i = 1, 2)$ are sufficiently large. In this section, we shall show the proof of Theorem 1.2, which relies on constructing the Lyapunov functional

where $u_*, v_*\; and\; w_*$ are given in (1.4). As a matter of fact, the above Lyapunov functional has been widely used in asymptotic analysis (cf. ^[33,39], for instance). Similar to the previous work ^[4], we begin with some basic calculations and outline the main ideas of the proof.

Lemma 5.1. Let

and

and let $L_w > 0$. Then whenever $\sqrt{\chi^2_1+\chi^2_2} < M$, where $M = M(\mu_1, \mu_2, u_0, v_0, w_0) > 0$, and $(u, v, w)$ is a positive global classical solution of (1.1) in $\overline\Omega\times(0, \infty)$ with the initial data $(u_0, v_0, w_0)$ fulfill (1.2) and

then the Lyapunov functional (5.1) holds the decay property:

which implies that

Proof. Starting from structuring the functional (5.1), we divide the proof process into the following steps:

Step 1. Using the three equations in (1.1), and computing it straightforward to obtain

Under the assumption of (5.3) and the definition of $a$ in (1.4), we obverse that

then we have

and

Let $M: = \frac{4}{L_w}$, and in view of (5.9), we get

Without losing generality, we let $0 < \chi_1, \chi_2 < 1$. Then since the $L^\infty$ of $w$ only depends on the upper bound of $\chi_i\; (i = 1, 2)$ and on the lower bound of $\mu_i\; (i = 1, 2)$ as well as on the initial data $(u_0, v_0, w_0)$, we notice that $L_w$ has the following dependence relationship:

Therefore, in view of (5.4), it is possible for us to fix some $M = M(\mu_1, \mu_2, u_0, v_0, w_0) > 0$ such that whenever $\sqrt{\chi^2_1+\chi^2_2} < \min\left\{1, M\right\}$, (5.11) holds. Then collecting (5.7)–(5.11), we obtain (5.5).

Step 2. Applying H$\ddot{\mbox{o}}$lder regularity [42,Theorem 1.3] and ^[40] to establish weak convergence of $u$ and $v$ in $L^{\infty}(\Omega)$. According to (1.1), there exist $\theta\in(0, 1)$ and $C > 0$ such that

for all $t > 1$.

Step 3. Due to the fact that $s-1-\ln{s}\ge0$ for all $s > 0$, $\Gamma$ is non-negative. In view of (5.5), we obverse that $\Gamma(1)$ is finite due to positivity of $\big{(}u(\cdot, 1), v(\cdot, 1), w(\cdot, 1)\big{)}$ in $\overline\Omega$, then integrating (5.5) in time shows that

which implies that

due to the contradiction argument (cf. ^[41], for instance).

Step 4. Multiplying the third equation in (1.1) by $w-w_*$, using Young's inequality again to obtain

which implies

This completes the proof.

We are now in a position to prove Theorem 1.2.

Proof of Theorem 1.2. Theorem 1.2 is a direct consequence of Lemma 5.1, and the interested readers may refer to [4,Lemma 6.1] for a detailed proof.

6.   Conclusions

In this work, we considered the dynamics behavior of a three-component chemotaxis system on alopecia areata in the higher-dimensional case. We mainly proved the existence of the global bounded classical solution for the discussed chemotaxis system if $\mu_{i}$ $(i = 1, 2)$ is large enough. In the further work, we will consider this chemotaxis system with the nonlinear self-diffusion, nonlinear chemotactic sensitivity and the generalized logistic sources. Moreover, the chemotaxis system with singular chemotactic sensitivity can also be discussed.

Acknowledgments

The second author is supported by Natural Science Foundation of Xinjiang Autonomous Region (Grant No: 2022D01C335), Scientific Research Program of the Higher Education Institution of XinJiang (Grant No: XJEDU2021Y043) as well as Science and Technology Project of YiLi Prefecture (No.YZ2022B038). The third author is supported by the National Natural Science Foundation of China (No.12261092) and the project of "Distinguished Professor of Academic Integrity" at Yili Normal University (YSXSJS22005)and the scientific research and innovation team at Yili Normal University (CXZK2021018).

Conflict of interest

The authors declare there is no conflict of interest.