Boundedness and stabilization in a quasilinear chemotaxis model with nonlocal growth term and indirect signal production

1.   Introduction

Chemotaxis describes the directional movement of cells or organisms in the direction of the concentration gradient of chemical signals. In order to simulate the phenomenon that cells are attracted to the high concentrations of chemical signals secreted by themselves. In 1970, Keller and Segel proposed a classical biological chemotaxis model ^[1] as follows

where $\Omega\subset\mathbb{R}^{n}, \tau\in\{0, 1\}, \nu$ denotes the outward unit normal vector on $\partial\Omega, u(x, t)$ represents the density of cells and $v(x, t)$ denotes the density of a chemical signal. $\nabla\cdot(D(u)\nabla u)$ and $-\nabla\cdot(S(u)\nabla v)$ represent self-diffusion and cross-diffusion, respectively. The function $f(u)$ describes cell proliferation and death. As we all know, chemotaxis research has important applications in both biology and medicine, so it has been one of the hottest research focuses in applied mathematics nowadays. The system (1.1) with $\tau = 0$ or $\tau = 1$ has been investigated extensively in the past few decades. For $\tau = 0, D(u)\equiv1, S(u) = \chi u$ and $f\in C^{1}([0, \infty))$ is assumed to satisfy $f(u)\leq a-\mu u^2$ for all $u\geq0$ with some positive constants $a, \mu$. The solutions to (1.1) are global and bounded for arbitrarily small $\mu > 0$ with $n\leq2$ ^[2], or $n\geq3$ and $\mu > \frac{(n-2)\chi}{n}$ sufficiently large^[3]. When $f(u) = \lambda u-\mu u^{\alpha}$ with $\alpha > 1, \lambda\geq 0$ and $\mu > 0,$ Winkler ^[4] introduced a concept of very weak solutions and proved global existence of such solutions for any nonnegative initial data $u_{0}\in L^{1}(\Omega)$ under the assumption that $\alpha > 2-\frac{1}{n}.$ In the case of $D, S\in C^{2}([0, \infty))$ and $D(u)\geq c_{0}u^{\rho}, c_{1}u^{q}\leq S(u)\leq c_{2}u^{q},$ and $f(u)$ is a smooth function fulfilling $f(0)\geq 0$ and $f(u)\leq au-\mu u^2$ for all $u > 0$ with constants $a\geq0$ and $\mu\geq 0,$ Cao in ^[5] showed that there exists a unique global bounded classical solution. For $\tau = 1, D(u) = 1, S(u) = \chi u$ and $f(u) = 0$ with $\chi > 0,$ the system (1.1) has the global solutions with $n = 1$ ^[6]; when $n = 2, \int_{\Omega}u_{0} > \frac{8\pi}{\chi},$ the solution of the system (1.1) will blow up in finite time ^[7], if $\int_{\Omega}u_{0} < \frac{8\pi}{\chi},$ the system (1.1) possesses a globally bounded classical solution ^[8]. In the case of $n\geq3,$ if $\Omega$ is a ball, then for arbitrarily small mass $m: = \int_{\Omega}u_{0} > 0,$ there exists the finite-time blow-up solutions ^[9] with proper initial coditions. Besides, for $f(u) = 0, S(u)/D(u)\leq K(u+\varepsilon)^\alpha$ with $u > 0, \alpha < \frac{2}{N}(N\in\mathbb{N}), K > 0, \varepsilon\geq0$, Ishida et al. ^[10] ruled out convexity of $\Omega,$ then established global-in-time existence and uniform-in-time boundedness of solutions. For $f(u) = au-\mu u^{2}, a\in \mathbb{R},$ Cao ^[11] used an approach based on maximal Sobolev regularity and proved that if the ratio $\frac{\mu}{\chi}$ is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by $(\frac{a_{+}}{\mu}, \frac{a_{+}}{\mu})$ is globally asymptotically stable without the restrictions $\tau = 1$ and the convexity of $\Omega$. For the case $f(u) = u(1-u^{\gamma}), \, D(0) > 0, \, D(u)\geq K_{1}u^{m_{1}}$ and $S(u)\leq K_{2}u^{m_{2}}, \, \forall u\geq 0$, $K_{i}\in\mathbb{R^{+}}, \, m_{i}\in\mathbb{R}, \, i = 1, 2$, Wang et al. ^[12] showed that the system admits global classical solutions and they are uniformly bounded in time with the parameter pair $(m_{1}, m_{2})$ lies in some specific regions and $N\geq 2$. When $D(u)\simeq a_{0}(u+1)^{-\alpha}, S(u)\simeq b_{0}u(u+1)^{\beta-1}$ and $f(u) = ru-\mu u^{1+\sigma},$ Zheng ^[13] showed the globally bounded classical solutions of the system (1.1) if $0 < \alpha+\beta < \max\{\sigma+\alpha, \frac{2}{n}\},$ or $\beta = \sigma$ with $\mu$ sufficiently large. In addition, the signal generation may be in a nonlinear form. Zhuang ^[14] established that (1.1) admits a globally bounded classical solution under $\beta < \sigma-1$ or $\beta = \sigma-1$ with $r = 1, \mu > 0$ sufficiently large, and the second equation replaced by $v_{t} = \Delta v-v+u(u+1^{\beta-1}).$ When $D(u)\geq a_{0}(u+1)^{-\alpha}, 0\leq S(u)\leq b_{0}u(u+1)^{\beta-1}$ with $a_{0}, b_{0} > 0$ and $\alpha, \beta \in \mathbb{R},$ Ding ^[15] provided a boundedness result under $\alpha+\beta+\gamma < \frac{2}{n},$ or $\beta+\gamma < 1+\sigma,$ or $\beta+\gamma = 1+\sigma$ with $\mu$ large enough and the bounded classical solution $(u, v)\rightarrow ((\frac{r}{\mu})^{\frac{1}{\sigma}}, (\frac{r}{\mu})^{\frac{1}{\sigma}})$ in $L^{\infty}(\Omega)$ exponentially under the condition of $b > b_{0}.$ More relevant works of system (1.2) can refer to (^[16,17,18]). Furthermore, many scholars consider the situation which the growth or death of cells is influenced by external factors, that is, the logistic sources contains nonlocal growth term. Negreanu and Tello ^[19] proposed the following model

where $\Omega\subset\mathbb{R}^{n}$ is a smooth bounded domain, if $m = 1$ and $a_{1} > 2\chi+|a_{2}|,$ it is shown that the solution of (1.2) satisfies $\lim\limits_{t\rightarrow \infty}\|u-\frac{a_{0}}{a_{1}+a_{2}} \|_{L^{\infty}(\Omega)} = 0.$ when the second is replaced by $v_{t} = \Delta v-v+u^{\gamma}$ and the logistic source term is $u^{\alpha}(1-\frac{\sigma}{|\Omega|}\int_{\Omega}u^{\beta}dx)$ with $\alpha\geq1, \beta > 1$ and $m = \gamma = \frac{\sigma}{|\Omega|} = 1,$ Bian ^[20] proved that the system (1.2) possesses a unique global strong solution which is uniformly bounded in whole space under either $\gamma+1\leq\sigma < 1+\frac{2\beta}{n}$ or $\sigma < \gamma+1 < \frac{2(\sigma+\beta)}{n+2}+\frac{n}{n+2}.$ In reference ^[21], when $\gamma+m\leq\alpha < 1+\frac{2\beta}{n}$ or $\frac{n+4}{2}-\beta < \alpha < \gamma+m,$ the system (1.2) admits a uniformly bounded global classical solution. For the logistic source term is replaced by $u(a_{0}-a_{1}u^{\alpha}+a_{2}\int_{\Omega}u^{\alpha}dx),$ Negreanu ^[22] obtained the global-in-time existence of classical solutions and the convergence to steady state $(a_{0}^{\frac{1}{\alpha}}(a_{1}-a_{2}|\Omega|)^{-\frac{1}{\alpha}}, (a_{0}^{\frac{1}{\alpha}}(a_{1}-a_{2}|\Omega|)^{-\frac{1}{\alpha}})^{\gamma})$ under assumptions that $\alpha, \gamma\geq1, m > 1, \alpha+1 > m+\gamma, a_{1} > 0$ and $a_{1}-a_{2}|\Omega| > 0.$ Moreover, when the logistic source term is replaced by $u^{\sigma}(a_{0}-a_{1}u-a_{2}\int_{\Omega}u^{\beta}dx),$ Ren ^[23] proved that system (1.2) possesses a unique global classical solution in three different cases, namely parabolic-elliptic, fully parabolic, and parabolic-parabolic-elliptic. For more chemotaxis systems with nonlocal terms, we can find the literature works (^{[24,25,26,27]}). The chemotactic signal is produced directly by cells in the classical Keller–Segel system, yet the signal generation undergoes intermediate stages in some realistic biological processes ^[28]. The related models can be described as follows:

where $u, v, w$ represent the density of cells, the density of chemical substances and the concentration of indirect signal, respectively. For $\tau = 0, D(s)\geq a_{0}(s+1)^\alpha, |S(s)|\leq b_{0}(1+s)^{\beta-1}$ for all $s\geq0$ with $a_{0}, b_{0} > 0, \alpha, \beta\in \mathbb{R},$ Li in ^[29] have obtained the nonnegative classical solution $(u, v, w)$ is global in time and bounded for $\beta\leq \gamma-1.$ Moreover, if $\mu$ satisfies some suitable conditions, the solution $(u, v, w)$ converges to $(1, 1, 1)$ in $L^{\infty}$-norm as $t\rightarrow \infty.$ When the signal generation is in a nonlinear form, it also has been shown that the boundedness and large time behaviors of classical solutions in ^[30]. For $\tau = 1, D(u) = 1, S(u) = u$, Zhang in ^[28] proved that if $\gamma > \frac{n}{4}+\frac{1}{2},$ the solution is globally bounded; if $\mu > 0$ is sufficiently large, $(u, v, w)$ satisfies $\|u(\cdot, t)-1\|_{L^{\infty}(\Omega)}+ \|v(\cdot, t)-1\|_{L^{\infty}(\Omega)}+\|w(\cdot, t)-1\|_{L^{\infty}(\Omega)} \rightarrow 0$ as $t\rightarrow \infty.$ For $D(u)\geq D_{1}u^{\alpha}, S(u) = u$ with $D_{1} > 0, \alpha\in \mathbb{R}, \gamma\geq 2$ and $\mu > 0,$ Wu in ^[31] proved the global existence and boundedness of solutions if the assumption $\frac{\alpha}{n}+\frac{\gamma}{2} > \frac{1}{2}$ holds with $n\geq 3.$ When $D, S\in C^{2}([0, \infty))$ satisfying $D(s)\geq a_{0}(s+1)^{-\alpha}, 0 \leq S(s) \leq b_{0}(s+1)^{\beta}$ for $a_{0}, b_{0} > 0$ and logistic source term is replaced by $b-\mu s^{\gamma}$ for all $s, b\geq0, \gamma \geq1$, Wang ^[32] obtained the global boundedness of solutions in four cases: the self-diffusion dominates the cross-diffusion; the logistic source suppresses the cross-diffusion for $\mu > 0$ sufficiently large; the logistic dampening balances the cross-diffusion; the self-diffusion and the logistic source both balance the cross-diffusion with $\mu > 0$ suitably large. If $D, S$ are smooth functions satisfying $D(s)\geq a_{0}(s+1)^{\alpha}, |S(s)|\leq b_{0}s(s+1)^{\beta-1}$ for all $s > 0$ and the logistic source term is $\mu(s-s^{\gamma})$ with $s\geq0, \gamma > 1,$ Zhang in ^[33] showed that the system (1.3) possesses a globally bounded classical solution $(u, v, w).$

The above systems only discuss the forms of direct or indirect generation of chemical signals, but do not discuss systems with nonlocal source terms and indirect signal production.Thereafter, inspired by reference ^[22], considering the growth or death of cells is influenced by external factors, this article added the production of chemical signals goes through intermediate stages and studied the following fully parabolic chemotaxis system with a nonlocal growth term and indirect signal production

in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq3)$ with smooth boundary $\partial\Omega.$ The diffusion function $D(u)$ satisfies

where $D_{1}$ is a positive constant and $\gamma \in \mathbb{R}.$ And $f(u)$ is the logistic function, which satisfies

with

the initial data fulfill

Under these assumptions, our main results on the global boundedness and large time behavior of solutions to system (1.4) are as follows:

Theorem 1. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. Suppose that functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 0, 1, 2)$ satisfy (1.5)-(1.7) with $\frac{4}{n}+\frac{\gamma}{\sigma} > \frac{3-\sigma}{1+\sigma}.$ For any nonnegative initial data $(u_{0}, v_{0}, w_{0})$ evolve from (1.8), the system (1.4) possesses a global classical solution

which is bounded in $\Omega\times(0, \infty)$.

Remark 1. When $a_{2} = 0$, the result in Theorem 1 is consistent with that in ^[31] with $S(u) = u$ in system (1.3), which needs $\mu > 0$. If $a_{2} > 0$, system (1.4) does not require any restrictions on $a_{0}$. These findings suggest that the nonlocal term $a_{2}\int_{\Omega} u^{\sigma}dx$ could play a crucial role in ensuring the global existence and boundedness of solutions in (1.4).

Remark 2. This work extends the study in ^[22] to the system with nonlinear diffusion and indirect signal production. Our results show that the nonlinear diffusion mechanism and nonlocal logistic sources have an inhibitory effect on the blow-up solution.

Theorem 2. Assume the conditions in Theorem 1 hold. Let $(u, v, w)$ be the solution of (1.4) obtained in Theorem 1, then there exist some $\kappa > 0$ and $C = C(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|) > 0$ such that

for all $t > 0,$ where $\kappa = \frac{1}{4(n+2)}, u_{*} = v_{*} = w_{*} = (\frac{a_{0}}{a_{1}-a_{2}|\Omega|})^{\frac{1}{\sigma}}.$

The framework structure of this article is as follows. In Sect.2, we give the local existence of the solution in (1.4) and show several related inequalities. In Sect.3, we consider the global boundedness of solutions for problem (1.4) under some suitable conditions and prove Theorem 1. In Sect.4, we give a lower bound for $u$ via comparison ^[34]. In Sect.5, we obtain the asymptotic behavior of (1.4) by constructing Lyapunov functions, thus completing the proof of Theorem 2.

2.   Local existence and preliminaries

In this section, we present several important lemmas that will be used in the following sections. First, we will state the local-in-time existence result of solutions for problem (1.4), which can be proved by adapting well-established approaches for parabolic-parabolic chemotaxis models (see ^[35]).

Lemma 1. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary, $D(u), f(u)$, and initial data $(u_{0}, v_{0}, w_{0})$ satisfy (1.5), (1.6), and (1.8), respectively. Then there exist $T_{max}\in(0, \infty]$ and a tripe $(u, v, w)$ of nonnegative functions

which solves (1.4) in the classical sense. Moreover, if $T_{max} < \infty,$ we can see that

We next state a lemma which guarantees $L^{1}$-boundedness of $u, v$ and $w$.

Lemma 2. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and the parameters $\sigma, a_{i}(i = 0, 1, 2)$ satisfy (1.5)-(1.7). Assume that $(u, v, w)$ is the solution of (1.4). Then there exist constants $M_{i}(i = 1, 2, 3, 4)$ such that

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

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

Proof. Integrating the first equation of the system (1.4) over $\Omega,$ we have

By applying Hölder's inequality, we derive

therefore

Assuming (1.7) holds, applying the Hölder's inequality again yields

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

On an ordinary differential equation(ODE) comparison, this implies that

where $M_{1}: = \max \Big \{\int_{\Omega}u_{0}dx, (\frac{a_{0}}{a_{1}-a_{2}|\Omega|})^{\frac{1}{\sigma}}|\Omega| \Big \}.$ Integrating the third equation in (1.4) over $\Omega$ yields

then

So (2.2) and (2.11) imply (2.4). By the same method, using the second equation of the system (1.4) and (2.4), we obtain (2.3). Moreover, integrating (2.8) upon $(t, t+\tau)$ for $t\in (0, T_{max}-\tau),$ and using (2.9), we have

where $M_{4} = (a_{1}-a_{2}|\Omega|)^{-1}M_{1}(1+a_{0}\tau).$ □

We provide the well-known Neumann heat semigroup theory without proof for our subsequent work (see Ref. ^[36,37]).

Lemma 3. Assume that $\lambda \in \{0, 1\}, p \in[1, \infty], q \in(1, \infty)$ and $\theta\in(0, 1)$. Then there exists positive constant $c_{1}$ such that for all $u \in D(A^{\theta}), A: = -\Delta + \lambda$,

if

If in addition $q\geq p,$ then there exist some constants $c_{2} > 0$ and $\alpha > 0$ such that for all $u \in L^{p}(\Omega),$

where the associated diffusion semigroup $\{e^{-tA}\}_{t\geq 0}$ maps $L^{p}(\Omega)$ into $D(A^\theta).$ Moreover, for any $p \in (1, \infty)$ and $\varepsilon > 0,$ there exist $c_{3} > 0$ and $\mu > 0$ such that

is valid for all $R^{n}$-Valued $u \in L^{p}(\Omega).$

We recall the Gagliardo–Nirenberg interpolation inequality (see Ref. ^[38] for detail), which will be used frequently in the proof of our main results.

Lemma 4. Let $u \in W^{1, q}(\Omega) \cap L^{p}(\Omega)$ and $h\geq1, p \in (0, h),$ where $\Omega \subset \mathbb{R}^{n}(n\geq3)$ is a smooth bounded domain. Then there exists a constant $c_{4} > 0$ such that

where

and $\lambda \in (0, 1)$ satisfies

Finally, we give the following lemma from ^[39], which is also important for our proof.

Lemma 5. Let $T > 0, c_{5} > 0, c_{6} > 0$ and $\tau \in(0, T).$ Assume that the function $z:[0, T)\rightarrow [0, \infty)$ is absolutely continuous and such that the following inequality holds:

for a.e $t\in (0, T)$, where $c \in L^{1}_{loc}([0, T))$ is a nonnegative function satisfying

for all $t\in [0, T-\tau)$.

Then

for a.e $t\in (0, T)$.

3.   Glocal existence and boundedness

In this section, we will give the proof of global existence and boundedness of solutions to system (1.4). Some necessary estimations are needed. We first give an inequality involving $\int_{\Omega}w^{p+1}dx$ and then use this inequality to establish the $L^{p}$-estimate for $w$.

Lemma 6. Assume that $(u, v, w)$ is the solution of (1.4) on $[0, T_{\max})$ as in Lemma 2.1 and $p > 1$. Then we have the following inequality

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

Proof. Multiplying the third equation of (1.4) by $w^{p},$ integrating over $\Omega$ and with the help of Young's inequality, we obtain

□

In addition, in order to obtain the proof of Theorem 1, a key step is to derive the upper bounds of $\|w(\cdot, t)\|_{L^{\sigma+1}(\Omega)}$ and $\|\nabla v(\cdot, t)\|_{L^{\beta}(\Omega)}.$ We will expand the proof from the following lemmas.

Lemma 7. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Assume that $(u, v, w)$ is the solution of (1.4). Then for each $\beta \in(1, \frac{n(\sigma+1)}{n-(\sigma+1)}),$ there exist positive constants $M_{5}$ and $M_{6}$ such that

Proof. Using Lemma 6 for $p = \sigma,$ we have

Taking $z(t): = \int_{\Omega} w^{\sigma+1}dx, t \in(0, T_{max}),$ and $c(t): = \sigma^{\sigma}\int_\Omega u^{\sigma+1}(\cdot, t)dx, t \in(0, T_{max}),$ we obtain

According to (2.5), one implies

where $C_{1} = \sigma^{\sigma}M_{4} > 0.$

In view of (3.5) and Lemma 5, this yields that

thus, we have

where $M_{5} = \max\Big\{\int_{\Omega}w_{0}^{\sigma+1}+C_{1}, \frac{C_{1}} {\sigma\tau}+2C_{1}\Big\}^{\frac{1}{\sigma+1}}.$

Next, we obtain the $L^{\beta}$-bound of $\nabla v$ by applying semigroup arguments (see, for example, ^[36,37]). First, using the variation of constants formula for the second equation of system (1.4) indicates

Choosing $\tau < 1$ and setting $A: = -\Delta+1, \lambda = 1, m = 1, q = \sigma+1, \beta \in(1, \frac{n(\sigma+1)}{n-(\sigma+1)})$ in Lemma 3, which makes $\theta \in (\frac{1}{2}+\frac{n}{2(\sigma+1)}-\frac{n}{2\beta}, 1).$ Then there exist positive constants $\varepsilon, \gamma$ such that

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

where $C_{2}, C_{3}$ represent different positive constants, and when $(1-\theta) > 0, \Gamma(1-\theta) > 0$. Therefore, we obtain that $\|\nabla v(\cdot, t)\|_{L^{\beta}(\Omega)}$ is bounded for all $t\in (0, T_{max}).$ □

Next we will calculate the first equation of system (1.4) to obtain an inequality for $\int_{\Omega}u^{p}dx$ by applying the classical Green's formula, Young's inequality, and Hölder's inequality.

Lemma 8. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Assume that $(u, v, w)$ is the solution of (1.4). Then for any $p > \max\{2, \gamma\}, \sigma > -\gamma$ there exist $C_{4}, C_{5} > 0$ depend on $p,$ such that the following inequality

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

Proof. Multiplying the first equation in (1.4) by $u^{p-1}$ and integrating over $\Omega$ by parts, we have

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

Using Young's inequality to the second term on the right side of (3.11) yields

According to Young's inequality, the combination of (3.11) and (3.12) leads to

where we note that our assumption $a_{1}-a_{2}|\Omega| > 0$ and $p > 2$ warrant that

Therefore, the following inequality

holds by Young's inequality. Then (3.10) is obtained. □

In order to obtain the $L^{p}$ boundedness of $u$, we need to estimate the second term on the right side of (3.10), thus an inequality for $\int_{\Omega}|\nabla v|^{2q}dx$ has to be required.

Lemma 9. Let $(u, v, w)$ be a solution of (1.4), then for any $q > 2$ there exist positive constants $C_{6}$ and $C_{7}$ such that

Proof. According to the second equation of (1.4), we can obtain

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

Now we estimate the first and third terms of (3.15), where an important inequality (3.10) in reference ^[10] as follows is needed

with some $C_{8} > 0.$

Using (3.16), we can infer that

for all $t\in (0, T_{max}),$ where we have used $\Delta|\nabla v|^{2} = 2\nabla v \cdot \nabla(\Delta v)+2|D^{2}v|^{2}$. Applying Young's inequality and $|\Delta v|^{2}\leq n|D^{2}v|^{2},$ we get

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

And we also have

for all $t\in (0, T_{max}).$ Substituting (3.17), (3.18), and (3.19) into (3.15), we have

for all $t\in (0, T_{max})$. Then we readily derive (3.14), where $C_{7}: = 2qC_{8}$. □

Combining Lemma 6 and Lemma 8 with Lemma 9 and using Young's inequality, the following inequality will be gained. And next we will establish the boundedness of $\|u(\cdot, t)\|_{L^{p}(\Omega)},$ $\|\nabla v(\cdot, t)\|_{L^{2p}(\Omega)}$ and $\|w(\cdot, t)\|_{L^{p+1}(\Omega)}$.

Lemma 10. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and the parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Assume that $(u, v, w)$ is the solution of (1.4). Then for any $p > max\{2, \gamma\}, q > 2,$ there exist positive constants $M_{7}, M_{8}$ and $M_{9}$ such that

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

Proof. If $\sigma > 1$ or $\sigma\geq-\gamma,$ we combine (3.1), (3.10) with (3.14) and apply Young's inequality to obtain

If $\sigma = 1,$ we have the similar inequality and omit it here. In order to obtain the boundedness above, we need to estimate the right two terms of (3.22). By employing reference ^[38] and Gagliardo–Nirenberg inequality, we deduce that

and

where

And for large $p > 1,$ it will be shown that there exists constant $q > 1$ such that $\lambda_{1}, \lambda_{2}\in (0, 1)$ and $h_{1}, h_{2} < 2$ hold in (3.25).

To ensure that $\lambda_{1}, \lambda_{2}\in (0, 1)$ and $h_{1}, h_{2} < 2,$ we can choose suitable parameters such that

According to continuity argument, we discuss the case $\beta \in(1, \frac{n(\sigma+1)}{n-(\sigma+1)}),$ which is inserted into (3.26) to have

and

Thus if

holds, then (3.28) is true. And $\frac{4}{n}+\frac{\gamma}{\sigma} > \frac{3-\sigma}{1+\sigma}$ implies (3.29). Then for any $p > p_{0},$ we can choose suitable $q$ such that (3.27) and (3.28) are fulfilled, where $p_{0}: = \max\{2, \gamma, \frac{n(\sigma+1)}{2(n-(\sigma+1))}(\sigma+\gamma)-\sigma\}$.

Hence, a combination of (3.22)-(3.24) entails that

for all $t\in(0, T_{max}).$ By employing the Grönwall's inequality for (3.30), the desired result (3.21) is gained. □

Lemma 11. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Assume $(u, v, w)$ is the solution of (1.4), then there exists a positive constant $M_{10}$ such that

Proof. (3.31) can be deduced by applying the semigroup arguments (see ^[36,37] for details) to the third and second equations of (1.4).□

Next, by using Lemma 11 and the standard Alikakos–Moser iteration, we establish the $L^{\infty}$ bound of $u$.

Lemma 12. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Assume that $(u, v, w)$ is the solution of (1.4). Then for any $p > \max\{2, \gamma\}, \sigma > -\gamma,$ there exists a positive constant $M_{11} > 0$ such that

Proof. For any $p > \max\{2, \gamma\},$ multiplying the first equation of (1.4) by $u^{p-1},$ integrating over $\Omega,$ and using Hölder's inequality, one obtains

Once more employing Young's inequality, we have

Integrating (3.34) yields that

Then, we can prove the following inequality by using Alikakos–Moser iteration (see ^[16,40] for details)

Now, we complete the proof of Theorem 1. □

Proof of Theorem 1. Along with Lemma 1 part 2, this proves that $T_{max} = \infty$ and the standard parabolic regularity makes sure that $(u, v, w)$ is bounded for $(x, t)\in \Omega\times(0, \infty).$ Hence the desired result of Theorem 1 is obtained. □

4.   A lower bound for $u$

To further prove the asymptotic behavior of the solution, we are going to estimate the lower bound for $u$ and provide some lemmas as follows.

Lemma 13. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$ and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). If $(u, v, w)$ is the solution of (1.4) and$(u_{0}, v_{0}, w_{0})$ satisfy (1.8), then there exists a constant positive $C_{17}$ such that

Proof. We split the proof into two steps:

Step 1. Fix $\theta_{0}\in(0, 1)$ and set $m = 2, p = \infty$ in Lemma 3, we verify $\lim\limits_{t\rightarrow \infty} \sup\|A^{\theta_{0}}w(\cdot, t)\|_{L^{p}(\Omega)}\leq C_{17}$ for any $p > \max\{2, \gamma, \frac{n(\sigma+1)}{2(n-(\sigma+1))}(\sigma+\gamma)-\sigma\}$ with some $C_{17} = C_{17}(\theta_{0}, p, \lambda, \gamma, \sigma) > 0.$

Review of (3.21), we know that there exists suitably large $t_{0} > 0$ satisfying

In view of the variation-of-constants representation, we get

for all $t\geq t_{0}.$ Recalling that $A = -\Delta+\lambda$ in reference ^[36] and applying the (4.2), then there exist constants $C_{18}, C_{19},$ we can derive

for all $t\geq t_{0},$ where $C_{20}: = M_{7}C_{19}\int^{\infty}_{0}e^{-(1-\lambda)\tau}\tau^{-\theta_{0}}d\tau < \infty$ with $\theta_{0}\in(0, 1).$ Thereafter

and there exists $t_{0} > 0$ large enough fulfilling

Step 2. The solution $v$ satisfies (4.1).

Let us fix any $\theta_{1}\in(1, 2)$ on the condition that $\theta_{1}-1 < \theta_{0} < 1$ and choose $p > \max\{2, \gamma, \frac{n}{n-2}(\sigma+\gamma)-\sigma\}$ satisfying $p > \frac{n}{2(\theta_{1}-1)},$ thus

Then following the same procedure as Step 1 and invoking (2.11), (4.3), and reference ^[36], we estimate

for all $t\geq t_{0},$ where $C_{24}: = 2C_{21}C_{22}'\int^{\infty}_{0}e^{-(1-\lambda)\tau} \tau^{-(\theta_{1}-\theta_{0})}d\tau < \infty$ with $\theta_{1}-1 < \theta_{0} < 1.$ This implies (4.1) for some $M_{12} > 0.$ In accordance with (4.1), we can pick $t_{0} > 0$ suitably large satisfying

With the help of the comparison principle ^[34], a positive lower bound for $u$ is hereafter addressed after some suitable waiting time. □

Lemma 14. Let $\Omega\subset\mathbb{R}^{n}(n\geq3)$ be a bounded domain with smooth boundary. The functions $D(u), f(u)$, and parameters $\sigma, a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7) with the positive number $p$ being as defined in (4.4). Assume that $(u, v, w)$ is the solution of (1.4) and the initial data fulfills (1.8). Then there exists a constant $M_{13}$ such that

Proof. Dealing with the first equation in (1.4), a combination of (3.34), and (4.5) yields

for all $x\in\Omega$ and $t\geq t_{0}.$ Moreover, let the function $y(t)\in C^{1}([t_{0}, \infty))$ be defined by

Applying the comparison argument, we gain

and

Which implies that

□

5.   Asymptotic behavior

The key to proving Theorem 2 relies on seeking so-called Lyapunov functions. Thus in this section, we will construct the appropriate Lyapunov functions in the following lemmas to obtain the large-time behavior of the solution $(u_{*}, v_{*}, w_{*})$ of the system (1.4).

Lemma 15. (Lemma 3.1 in ^[41]) Let $f:(1, \infty)\rightarrow [0, \infty)$ be uniformly continuous such that $\int^{\infty}_{1}f(t)dt < \infty$, then

Lemma 16. Let functions $D(u)$ and $f(u)$ and parameters $\sigma$ and $a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7). Then for any classical solution $(u, v, w)$ of (1.4) in $\Omega \times (t_{0}, \infty)$ verifying

with some $t_{0}\geq0$ and $M_{14} > 0.$

Then there exist $\bar{\theta}\in (0, 1)$ and $C_{27} = C_{27}(M_{14}, D_{1}, \sigma, \gamma, |\Omega|, \|\nabla u(\cdot, t_{0}+2)\|_{C(\bar{\Omega})}) > 0$ such that

Proof. First, using the Neumann heat semigroup estimate ^[34] to the third and the second equations in system (1.4) we can derive that $\|\nabla v\|_{L^{\infty}(\Omega)}$ and $\|\nabla w\|_{L^{\infty}(\Omega)}$ is bounded in $t\in (0, T_{max}).$ According to Theorem 1, the system (1.4) possesses a global bounded classical solution. Thus, we can find $t\in [t_{0}, \infty)$ such that

And because of $(u, v, w)\in(C^{0}(\bar{\Omega}\times[0, T_{max}))\cap C^{2, 1}(\bar{\Omega}\times(0, T_{max})))^{3},$ applying a proof method similar to proposition 3.4 in Reference ^[15], we can obtain that there exists $\bar{\theta}\in (0, 1)$ such that $\|\nabla u(\cdot, t)\|_{C(\bar{\Omega})}\leq C_{27}((t-t_{0})^{\bar{\theta}}+1)$ holds. □

Lemma 17. Assume functions $D(u)$ and $f(u)$ and parameters $\sigma$ and $a_{i}(i = 1, 2)$ satisfy (1.5)-(1.7) and $(u, v, w)$ is the solution of (1.4). The initial data $(u_{0}, v_{0}, w_{0})$ evolves from (1.8). Then there exists a positive constant $H_{0}\in [M_{13}, M_{11}]$ such that functions $E$ and $F$ are defined by

where $t > 0, u_{*} = v_{*} = w_{*} = (\frac{a_{1}}{a_{1}-a_{2}|\Omega|})^{\frac{1}{\sigma}}$, and we have

Proof. Note that

By system (1.4) and the assumption in Theorem 2, applying Young's inequality, we deduce that

where $a_0 = a_1u_*^\sigma-a_2\int_\Omega u_*^\sigma dx$, $H_{0}$ is obtained by $u$ with a lower bound.

A simple calculation yields

and hence

We treat the last integral in (5.6) by estimating the integrated function in a pointwise way. Suppose (1.5) is valid, according to reference ^[15], and the Mean value theorem ensure

where $l > 0$, together with (5.7)-(5.9), we can conclude that

Then we use Young's equality to estimate $E_2(t)$ and $E_3(t)$,

and

Combing $E'_1(t)$, $E'_2(t)$ and $E'_3(t)$, we obtain

Since

there is $\widetilde{a_1} > 0$ such that

provided $a_1 > \widetilde{a_1}.$ This in conjunction with (5.13) entails that

therefore

where $a_1-a_2|\Omega| > 0.$ Lemma 5.3 is proved. □

We are now in a positive to prove our main result.

Proof of Theorem 2. Integrating (5.13) from $t_{0}$ to $\infty$, we have

According to the standard parabolic regularity theory^[42], with the global boundedness of $(u, v, w)$, we can see that there exists $\vartheta\in(0, 1)$ and $C_{28} > 0$ such that

for all $t > 1$. This clearly implies that $\int_\Omega(u(\cdot, t)-u_*)^{2}dx+\int_\Omega (v(\cdot, t)-v_*)^2dx+\int_\Omega (w(\cdot, t)-w_*)^2dx$ is uniformly continuous with respect to $t\geq t_{0}$ provided $t_{0} > 1.$ Therefore, we infer from Lemma 15 that the following inequality

holds.

The Gagliardo–Nirenberg inequality says

Using $\varpi$ as $u-u_{*}, v-v_{*}$ and $w-w_{*}$ in (5.16), respectively, we have from (5.14) and (5.15) that

We next defined $\varphi:(0, \infty)\rightarrow \mathbb{R}$ and

By the Taylor expansion with $s\in(0, \infty)$, there is $\xi(s)\in(0, 1)$ such that

Obviously, $\varphi(s)\geq 0$, and use L'Hôpital's rule to see

Furthermore, we can pick $t_{1} > 0$ such that

namely,

In view of (5.13) and (5.19), we have

thus, we get by the Gronwall inequality that

Together with (5.18), we obtain the estimate

for all $t\geq t_{1},$ with $C_{29} = C_{29}(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|) > 0.$ By using the Gagliardo–Nirenberg inequality with Lemma 16 and (5.14)-(5.15),

with $C_{30} = C_{30}(|\Omega|), C_{31} = C_{31}(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|, \|\nabla u(\cdot, t_{1})\|_{L^{\infty}(\Omega)}) > 0, C_{32} = C_{32}(C_{31}, \bar{\theta}).$ Similarly, an application of the Gagliardo–Nirenberg inequality with combining (3.31) and (5.20) indicates that

for some $C_{33} = C_{33}(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|, \|\nabla v(\cdot, t_{1})\|_{L^{\infty}(\Omega)}) > 0,$ and $C_{34} = C_{34}(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|, \|\nabla w(\cdot, t_{1})\|_{L^{\infty}(\Omega)}) > 0.$

A combination of (5.21)-(5.23), gives us

with $t\geq t_{1}$ and $C = C(a_{0}, a_{1}, a_{2}, \sigma, \gamma, |\Omega|) > 0$, which completes the proof of Theorem 2. □

6.   Conclusion

In this paper, we considered that the growth or death of cells is influenced by external factors and the production of chemical signals goes through intermediate stages, and thus we investigate a fully parabolic chemotaxis system with a nonlocal growth term and indirect signal production. The work is carried out under the condition of spatial dimension $n\geq 3$, when the initial data, the diffusion function, the logistic source term and related parameters satisfy certain conditions, the global boundedness of solutions to system (1.4) is proved by applying the maximum principle, variation-of constants formula, Neumann heat semigroup estimation, Young's inequality, Gagliardo–Nirenberg inequality and so on. In addition, by constructing appropriate Lyapunov functions, we obtained the asymptotic behavior of solutions.

Author contributions

Min Jiang, Dandan Liu and Rengang Huang: Methodology; Min Jiang and Dandan Liu: Writing-original draft; Dandan Liu and Rengang Huang: Writing-review and editing.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are indebted to the editor and anonymous reviewers for their kind assistance in providing insightful comments, suggestions, and valuable references. Additionally, the authors extend appreciation to reviewers for thoroughly reviewing the manuscript, engaging in fruitful discussions, and bringing to attention certain errors throughout the course of this study. The research of the first author was supported by the Project of Guizhou Minzu University under Grant No.16yjrcxm002 and the Science Research Foundation Project of Guizhou Minzu University (GZMUZK[2023]YB12).

Conflict of interest

The authors declare there is no conflicts of interest.