Quotient reflective subcategories of the category of bounded uniform filter spaces

1.   Introduction

The Keller and Segel model in ^[22] was introduced in 1970, and the mathematical study of this system has extensively developed the parabolic-parabolic equations in ^{[13,24,28,36,39]} and the parabolic-elliptic equations in ^{[2,3,7,14,15,37]}. This model is used to describe the chemotaxis-aggregation phenomena in nature.

Cells and microorganisms usually live in fluid, so it is particularly important to consider the interaction of fluids with them. In view of this idea, Tuval et al. considered the experiment of the collective behavior of \emph{Bacillus subtilis} in ^[49]. Then, a large number of related results of global solvability for chemotaxis-fluid were investigated in recent years. For example, we can see the researches of introducing the Keller-Segel equations in ^{[1,20,34,46,55,78]}, the Keller-Segel-Navier-Stokes equations in ^{[5,6,9,10,21,25,27,31,41,42,43,47,51,52,53,54,56,57,58,62,63,64,66,67,68,69,70,71,73,76,77,79]}, the rotational flux term in ^{[5,21,31,51,58,59,64,79]}, the nonlinear diffusion in ^{[8,11,41,48,73]}, the logistic source in ^{[12,47,54,62,78]}, the singular sensitivity in ^{[13,14,15,24,52,65,75]} etc. These papers on global existence and boundedness analysis gave a good theoretical and guiding significance for our understanding of biological growth of cells. Due to the global existence of the solution, we do not have to worry about the occurrence of sudden change and other unexpected results, and can achieve the purpose of guiding experiments with theory.

Recently, a macroscopic model called the spatial Solow-Swan was proposed by Juchem Neto et al. in ^[16,17,18] for describing economic growth phenomena under capital induction and labor migration. Very recently, Li-Li ^[26] investigated global boundedness of the following model

Assuming that the dynamic behavior of microscopic particles also meets the above macroscopic model, it is necessary to consider the Keller-Segel-Solow-Swan model. For the above model, there are two difficulties: the first equation contains cross diffusion term $\nabla\cdot(n\nabla c)$, and the second contains the Cobb-Douglas function $\mu_3c^\alpha n^{1-\alpha}$. Therefore, it becomes very interesting to use the corresponding mathematical theory to deal with this problem. Recently, more results in ^{[29,30,32,33,60,72,74]} have turned their attention to the indirect signal production model under multi-signal, and the researches on the global solvability of this model have become very important.

Compared with the chemical substance concentration term of the indirect signal model, we found that the system became more difficult to control after adding Cobb-Douglas term. We can explain it by Sturm's comparison theorem in ^[44] as follows:

where $y = \|c\|_{L^1(\Omega)}$ and $w$ are the concentrations of another chemical involved in the reaction, which is given in the following model (1.1). Let

If $\alpha = 0$, the above system degenerates into an indirect signal model, and if $\alpha > 0$ increase, then the corresponding solution will be raised. When we assume that the differential equation of the indirect signal model $c$ is

and assume that they have the same initial data and velocity, namely, $y(0) = \tilde{y}(0), \, \dot{y}(0) = \dot{\tilde{y}}(0)$, as wall as suppose that $y(a) = y(b) = y(0)$, then we have $a\leq b$ and

This shows that the distance between the two solutions increases gradually during the evolution. Motivated by the above works, we think that the relationship between cells and chemicals also meets the operating mechanism in the Solow-Swan model. In this paper, we let $\Omega\subset\mathbb{R}^N\, (N = 2, 3)$ be a bounded domain smooth boundary with outer norm vector $\nu$ and investigate the following chemotaxis-fluid-Solow-Swan system:

Here, the unknowns $n = n(t, x), c = c(t, x)$ and $w = w(t, x)$ denote the cell density and the two concentrations of chemical substance, respectively. $u = u(t, x)$ represents the fluid velocity field, and $P = P(t, x)$ denotes the associated pressure. The scalar valued function $\Phi = \Phi(x)$ is given and it accounts the effects of external forces such as gravity or centrifugal forces. The parameters satisfy $\chi > 0, k\geq N, \mu_1\in\mathbb{R}, \mu_2\geq0, \mu_3 > 0, \alpha\in(0, 1), \, \, \kappa\in\{0, 1\}$. Moser-Trudinger inequality ^[4,38,50] has natural advantage as a priori estimate for dealing with two-dimensional critical cases, and Winkler ^[68] has promoted it and provided a better version. For the three-dimensional case, we control it with help of the order of logistic source and the estimate of heat semigroup. Based on these results, we describe the work of this paper. For the convenience of this paper, we let

We assume that potential function $\Phi$ fulfills

and that the initial data $n_0, c_0, w_0, u_0$ satisfies

where $W^{1, 2}_{0, \sigma}: = W^{1, 2}_0(\Omega; \mathbb{R}^N)\cap L^2_{\sigma}(\Omega)$, with $L^2_{\sigma}: = \Big\{\varphi\in L^2(\Omega; \mathbb{R}^N)\, \big|\, \, \nabla\cdot\varphi = 0\, \, \mathrm{in}\, \, \mathcal{D}(\Omega)\Big\}$ denoting the space of all solenoidal vector fields in $L^2(\Omega; \mathbb{R}^N)$.

Under this assumption, our main results on global boundedness and asymptotic behavior of the initial-boundary value problems (1.1) and (1.3) can be formulated as follows.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^N\, (N = 2, 3)$ be a bounded domain with smooth boundary and $\Phi$ comply with (1.2), and suppose that $n_0, c_0, w_0,$ and $u_0$ satisfy (1.3), and if $N = 2, \, \mu_1\in\mathbb{R}, \mu_2 > 0$ or $\mu_1 = 0, \mu_2\geq0$ and if $N = 3, \mu_1\in\mathbb{R}, \, \mu_2 > 0$, then there exist functions $(n, c, w, u, P)$ satisfying

and fulfill $n > 0, c > 0$ and $w > 0$ in $\bar\Omega\times[0, \infty)$.

Theorem 1.2. Let $\Omega\subset\mathbb{R}^N\, (N = 2, 3)$ be a bounded domain with smooth boundary, and let $(n, c, w, u, \Phi)$ satisfy the conditions of Theorem 1.1.

(I) If $\mu_1 < 0, \, \mu_2\geq0$, then there exist $C > 0$, suitable small $\delta > 0$, and $t_\star > 1$ satisfying

and

as well as

If $\mu_1 = 0, \, \mu_2 < 0$, then there exist $C > 0$, suitable small $\delta > 0$, and $t_\star > 1$ fulfilling

and

as well as

Remark 1.1. For notational convenience, we do not explain the constants of $\, C_i, \, i = 1, 2, \cdots, 40$ and $C_{GN}$ in the following. Here, $C_{GN}$ is Gagliardo-Nirenberg constant.

2.   Local existence of $N=2$ and $N=3$

First of all, we give the local existence result. This proof is based on the Banach's fixed point theorem in a bounded closed set in $L^\infty((0, T);C^0(\bar\Omega)\times\big(W^{1, q}(\Omega)\big)^2\times D(A^\gamma))$ for all $\gamma\in(\frac{1}{2}, 1)$ and suitably small $T$, where $A$ is the realization of the stokes operator in the solenoidal subspace. Additionally, here we omit the details of the proof, which can be found in ^[1,20,63]. For the positive solutions, we can obtain them using the principle of comparison. Because $\underline{n}\equiv0$ is a sub-solution of the first equation in (1.1) and $n(x, 0)\geq0$, we have $n(x, t)\geq 0$. Furthermore, we can obtain $n(x, t) > 0$ due to $n_0(x)\not \equiv 0$. Therefore, we can get $w(x, t) > 0$ and $c(x, t) > 0$, respectively.

Lemma 2.1. Let $\Omega\subset\mathbb{R}^N\, (N = 2, 3)$ be a bounded domain with smooth boundary and $\Phi$ comply with (1.2), and suppose that $n_0, c_0, w_0,$ and $u_0$ satisfy (1.3), then there exist functions

and fulfill $n > 0, c > 0$ and $w > 0$ in $\bar\Omega\times[0, T_{\max})$. Moreover, if $T_{\max} < \infty$, then for all $q > N, \, \gamma\in(\frac{1}{2}, 1)$ we have

3.   Global existence of $N=2, \, \, \mu_1=\mu_2=0$

For the treatment of the global existence for two-dimensional Keller-Segel-Navier-Stokes-Solow-Swan system, we adopt the following Moser-Trudinger inequalities.

Lemma 3.1. (^[68]) Suppose that $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. Then for all $\epsilon > 0$ there exists $M = M(\epsilon, \Omega) > 0$ such that if $0\not \equiv \phi\in C^0(\bar\Omega)$ is nonnegative and $\psi\in W^{1, 2}(\Omega)$, then for each $a > 0$,

where $\bar\phi: = \frac{1}{|\Omega|}\int_\Omega\phi dx$.

Lemma 3.2. (^[68]) Suppose that $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary, and let $0\not \equiv \phi\in C^0(\bar\Omega)$ is nonnegative. Then for any choice of $\epsilon > 0$,

where $M = M(\epsilon, \Omega) > 0$ is as in Lemma 3.1.

Next, we give the required a prior estimates.

Lemma 3.3. Assume that (1.3) holds. Then we have

and

as well as

Proof. Since $\mu_1 = \mu_2 = 0$, we integrate the first equation of (1.1) to get (3.2) and integrate the third equation of (1.1) and use the ODE argument to obtain (3.3). Then, using the similar method for the second equation of (1.1), we can complete the proof of the Lemma 3.3. □

Lemma 3.4. Suppose that (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there exists $C(T) > 0$ such that

and

as well as

Proof. We first integrate by parts in the first equation from (1.1) and use $\nabla\cdot u = 0$ and the Young's inequality to deduce that

Multiplying the second equation of (1.1) by $c$, we have

Multiplying (3.8) by $\chi^2$ and then substituting it into (3.7), we have

For the right hand side of (3.9), using the Gagliardo-Nirenberg inequality and Young's inequality, we have

where $\epsilon_1 > 0$ is small enough and to be determined.

Multiplying the third equation of (1.1) by $w$, one has

In order to control the last term at the right end of (3.11), using Lemma 3.1, we obtain

For the first term at the right end of (3.12), using Lemma 3.2, we can get

Multiplying (3.13) by $\frac{1}{a}$, that is

We now substituting (3.12) and (3.14) into (3.11) to deduce that

Let $\lambda_0: = \frac{4(1+\epsilon)m_0}{\pi a} > 0$. Multiplying (3.10) by $\lambda_0$ and adding it to (3.15), we can see that

Therefore, we only need to select the appropriate positive numbers $\epsilon, \epsilon_1$ and $a$ such that $2-\frac{(1+\epsilon)m_0a}{4\pi }-\epsilon_1\lambda_0 > 0$. If $\epsilon$ is fixed, we can take $a = \frac{2\pi}{(1+\epsilon)m_0}$ and $\epsilon_1 = \frac{\pi a}{4(1+\epsilon)m_0}$, which can meet the conditions we need. Then we use the inequality $\int_\Omega\ln(n+1)dx\leq\int_\Omega ndx = m_0$ to get (3.4) and (3.5). Finally, we use (3.5), (3.13) and the fact that $n\ln n\geq-e^{-1}$ to arrive at (3.6). □

Lemma 3.5. Assume (1.3) is satisfied. Then, for all ${T\in(0, T_{\max})}$ there exists $C(T) > 0$ such that

and

Proof. We test the fourth equation of (1.1) by $u$ and use the Hölder's inequality and Moser-Trudinger inequality to get

where

and $\kappa_1 > 0$ is to be determined, it will be given by the following Poincaré's inequality.\\ On the other hand, using Poincaré's inequality and Hölder's inequality we have

Therefore, (3.18) together with (3.19) shows that

So, using Gronwall's inequality and (3.6), we have the descried results. □

Lemma 3.6. If (1.3) holds, then for all ${T\in(0, T_{\max})}$ there exists $C(T) > 0$ such that

Moreover, we have

Proof. We multiply the Eq $(1.1)_2$ with $-\Delta c$ and use the integration by parts and Hölder's inequality to obtain

Applying the Gagliardo-Nirenberg inequality and Young's inequality, we have

and

We plug (3.22) into (3.21) to obtain

where $C_{41} > 0$ is a constant.

On the other hand, note that the identities $|\Delta c|^2 = \nabla\cdot(\Delta c\nabla c)-\nabla c\cdot\nabla\Delta c$ and $\Delta|\nabla c|^2 = 2\nabla c\cdot\nabla\Delta c+2|D^2c|^2$, we deduce that

Thanks to the fact $\frac{\partial|\nabla c|^2}{\partial\nu}\leq2\kappa_2|\nabla c|^2$, where $\kappa_2: = \kappa_2(\Omega) > 0$ is an upper bound for the curvatures of $\partial\Omega$ in (^[35], Lemma 4.2), the trace theorem and (3.25), we can see that

where $\tilde{C}_{41}, \tilde{C}_{42}, \tilde{C}_{43}$ and $s\in(0, 1)$ are positive constants.

That is

Similarly, we have

Then, we apply Gagliardo-Nirenberg inequality, Lemma 3.5 and Poincaré's inequality to get

where $C_{42}, C_{43}$ are two positive constants.

Therefore, (3.24) together with (3.26) and (3.28) shows that

where $C_4 = \max\left\{C_{41}C_{43}, \, 2C_{GN}\sqrt{C_{43}}\right\}$.

So, we use Gronwall inequality, and use Lemmas 3.4 and 3.5 and Hölder's inequality to arrive at the Lemma 3.6. □

Lemma 3.7. Suppose that (1.3) holds and that $T\in(0, T_{\max})$. Then there exists $C(T) > 0$ such that

and

Proof. Multiplying the Eq $(1.1)_3$ with $-\Delta w$ and using Hölder's inequality, (3.23), (3.27) and (3.28), one has

where $C_5 > 0$ is a constant.

For the term of $\|n\|^2_{L^2(\Omega)}$, we apply the Gagliardo-Nirenberg inequality and the mass conservation of $\|n\|_{L^1(\Omega)}$ to deduce that

Multiplying the Eq $(1.1)_1$ with $(1+\ln n)$ and using Hölder's inequality and Young's inequality, we have

Then, we add (3.31) into (3.32) to obtain

We integrate the two ends of (3.33) with respect to $t$, and use Lemma 3.6 to get

Finally, we use Gronwall's inequality to (3.30) and note that $n\ln n\geq-e^{-1}$ and (3.34) to complete the Lemma 3.7. □

Lemma 3.8. Assume (1.3), and let $T\in(0, T_{\max})$. Then there exists $C(T) > 0$ such that

Proof. Testing the first equation in $(1.1)$ against $n$ and integrating by parts show that

Applying the identity $n\nabla\cdot(n\nabla c) = n\nabla n\cdot\nabla c+n^2\Delta c$, we show that

Using the Gagliardo-Nirenberg inequality again, we have

Combining (3.35) with (3.36) and using the Young' s inequality, one has

Applying Gronwall's inequality and the Lemma 3.7, we can obtain

Thus, we complete the proof of the Lemma 3.8. □

Lemma 3.9. Suppose that (1.3) holds and that ${T\in(0, T_{\max})}$. Then there exists $C(T) > 0$ such that

and

Proof. Testing $(1.1)_4$ by $Au$ and using Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality and (3.16), one has

Applying the variation of constant formula and (3.17), we have

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

Integrating the two sides of (3.37) and applying (3.38), we complete the proof. □

Lemma 3.10. Assume that (1.3) holds and let $\gamma_0\in(\frac{1}{2}, \gamma]\subset(\frac{1}{2}, 1)$. Then for all ${T\in(0, T_{\max})}$. there exists $C(T) > 0$ such that

and

Proof. We fix $\gamma_0$ and let $p > \frac{1}{1-\gamma_0}$, then use the Helmholtz projection operator to the fourth equation of (1.1) and the variation of constant formula to deduce that

Due to $p > \frac{1}{1-\gamma_0}$, we have $\frac{p\gamma_0}{p-1}\in(0, 1)$. So, $J_1\in(0, \infty)$.

For $J_2$, we apply the Hölder's inequality, Sobolev embedding, Poincaré's inequality and Gagliardo-Nirenberg inequality to obtain

Applying Lemma 3.9, we can get (3.39). Then we apply the embedding of $D(A^{\gamma_0})\hookrightarrow {C}^\theta(\Omega)$ for all $\theta\in(0, 2\gamma_0-1)$ to complete the proof of Lemma 3.10. □

Lemma 3.11. If (1.3) holds, there for all ${T\in(0, T_{\max})}$. there exists $C(T) > 0$ such that

Proof. Without loss of generality, we assume that $q > 2$. Using the Duhamel principle for $c$ and using standard semigroup estimates for the Neumann heat semigroup in (^[61], Lemma 1.3) and embedding in (^[19], Lemma 1.6.1) and the estimate in (^[20] Lemma 2.1 or ^[15], Lemma 2.2), and using the Lemmas 3.4, 3.6 and 3.7, we can see that

Lemma 3.12. Suppose that (1.3) holds and that ${T\in(0, T_{\max})}$. Then there exists $C(T) > 0$ such that

Proof. Let $M(T^\star): = \sup\limits_{t\in(0, T^\star)}\|n(\cdot, t)\|_{L^\infty(\Omega)}$ for all $T^\star\in(0, T)$ and let $t_0 = (t-1)_+$. We use the Duhamel principle for $n$ and use the semigroup estimate, Interpolation inequality and Young's inequality to deduce that

We take the supremum of time for both sides of (3.40) to obtain the Lemma 3.12. □

Lemma 3.13. Assume $(1.3)$, and let ${T\in(0, T_{\max})}$. Then there exists $C(T) > 0$ such that

Proof. Since the estimate of $\|n\|_{L^\infty(\Omega)}$ in Lemma 3.12 has been obtained, we only need to use the Duhamel principle and the processing techniques similar to Lemma 3.11. □

Proof of Theorem 1.1. For the two-dimensional Navier-Stokes case, applying the Lemmas 2.1 and 3.10–3.13, if $T$ is finite, then using the extendability criterion, we can see that $n, c, w$ and $u$ are unbounded of their respective norms, which contradict the boundedness of our a prior estimates. Next, we will give the asymptotic behavior of the system (1.1) with logistic source. Finally, we give a priori estimates of the corresponding solution in the three-dimensional case.

4.   Boundedness and asymptotic behavior of $n.c, w$ and $u$ with logistic source

For $\mu_1 < 0$, we can obtain the decay estimates of the following.

4.1.   $\mu_1 < 0$ and $\mu_2\geq0$

Since $\mu_1 < 0$ and $\mu_2$ are nonnegative, we can easily obtain the corresponding global boundedness results of the system (1.1) by using the previous processing ways. Next, we give the corresponding large time behavior.

Lemma 4.1. Under the assumption of Lemma 3.10, there exist $\theta\in(0, 1)$ and $C = C(\chi, \mu_1, \mu_2, \mu_3, \alpha) > 0$, independent of $t$, such that

Proof. Applying the estimates obtained by Lemmas 3.10 and 3.12, and then combining with the standard Schauder estimate in ^[45], we arrive the proof. □

Lemma 4.2. Under the assumption of Lemma 3.12, there is an $C$, independent of time $t$ such that

Proof. Let ${\bf{p}}: = \nabla n, \, {\bf{q}}: = \nabla{c}$. We rewrite the first equation of (1.1) to obtain

where $a(x, t, {\bf{p}}) = {\bf{p}}-n\big(\chi\, {\bf q}-u\big)$ and $b = \mu_1 n-\mu_2 n^k$.

Using Lemmas 3.10–3.12 and 4.1, there exists $C_{18} > 0$ satisfying

and

as well as

Thanks to ${\bf q}\in L^{\infty}(0, T; L^\infty(\Omega))$, it evident that $\frac{1}{\infty}+\frac{N}{2\cdot\infty} = 0 < 1$. Apply the standard result on Hölder's regularity in scalar parabolic equation in (^[40], Theorem 1.3) to get $\|n\|_{C^{\theta, \frac{\theta}{2}}(\Omega\times(0, T))}$ bounded. Then the Lemma 4.2 now follows from (^[23], Theorem IV. 5.3). □

Next, we adapt the similar methods to obtain the following:

Lemma 4.3. Under the assumption of Lemmas 3.11 and 3.13, there is an $C$, independent of time $t$ such that

Lemma 4.4. Assume that (1.3) holds. If $\mu_1 < 0, \, \mu_2\geq0$, then there exist a constant $c_1$, independent of time $t$ such that

Proof. We integrate the first equation of (1.1) to obtain

Using the Gronwall's inequality for the Eq (4.1), we can see that

Applying the Gagliardo-Nirenberg inequality, the Lemma 4.2 and the estimate (4.2), we have

Thus, we complete the proof of the Lemma 4.4. □

Lemma 4.5. Suppose that (1.3) holds. If $\mu_1 < 0, \, \mu_2\geq0$, then there exist a constant $c_2$, independent of time $t$ such that

Proof. We integrate the first equation of $(1.1)$ and (4.2) to deduce that

Thus, using the Gronwall's inequality, we can obtain

Similarly, using Hölder's inequality and Young's inequality, there exist a suitable small $0 < \delta_2\ll1$ such that

Thus, we use ODE argument to get

Then, for all $q > 1$ we apply the Gagliardo-Nirenberg inequality to see that

and

Using the above two estimates and (4.4), (4.5) proves that the Lemma 4.5. □

Lemma 4.6. Suppose (1.3) and $\mu_1 < 0, \, \mu_2\geq0$ hold, then there exist a constant $c_3$, independent of time $t$ such that

Proof. Testing the Eq $(1.1)_4$ with $u$ and using Poincaré's inequality and Young's inequality, we have

And using Poincaré's inequality once more, there is a constant $\tilde C_{23} > 0$ such that

Using Gronwall's inequality and the Lemma 4.4, there exists a constant $C_{24} > 0$ fulfilling

Then, applying the Gagliardo-Nirenberg inequality, this shows that

4.2.   $\mu_1=0$ and $\mu_2 > 0$

Lemma 4.7. Assume that (1.3) holds. If $\mu_1 = 0, \, \mu_2 > 0$, then there exist a constant $c_4$, independent of time $t$ such that

Proof. We integrate the first equation of (1.1) to obtain

We use Hölder's inequality to deduce that

We apply ODE argument to get

Similarly, using the inequality (4.3), we complete the proof of the Lemma 4.5. □

Lemma 4.8. Suppose that (1.3) holds. If $\mu_1 = 0, \, \mu_2 > 0$, then there exist a constant $c_5$, independent of time $t$ such that

and

as well as

Proof. The proof is completely similar to Lemmas 4.5 and 4.6, so we omit the details. □

4.3.   $\mu_1 > 0$ and $\mu_2 > 0$

Next, we will give a priori estimates when $\mu_1 > 0, \, \mu_2 > 0$.

Lemma 4.9. Assume that (1.3) holds. Then for all $T > 0$ there exist $C(T) > 0$ such that

and

Proof. We integrate the first equation of (1.1) to get

Applying ODE comparison, we have

or

Combining (4.10) with (4.11), this entails (4.7). Then, we integrate the two sides of Eq (4.9) to get (4.8). □

Lemma 4.10. Suppose that (1.3) holds. Then for all $T > 0$ there exist $C(T) > 0$ such that

and

Proof. Using the inequality (3.8) and (3.11), and using Hölder's inequality and Young's inequality we have

and

We can get (4.12) and (4.13) by integrating (4.14) and (4.15) and using Lemma 4.9. □

Lemma 4.11. If (1.3) holds, then for all $T$ there exist $C(T) > 0$ such that

and

Proof. Applying the estimate of (4.6), we have

Integrating both sides of (4.18) and applying the estimate of (4.8), we obtain (4.16) and (4.17). □

4.4.   For the two-dimensional Navier-Stokes case

The proof of the remaining part is completely similar to the processing of Lemmas 3.6–3.13, so we omit the details.

4.5.   For the three-dimensional Stokes case

Next, we can use semigroup estimation to obtain the following prior estimates for the three-dimensional case.

Lemma 4.12. Suppose that (1.3) holds and let $\gamma_0\in(\frac{1}{2}, \gamma]\subset(\frac{1}{2}, 1)$. Then for all ${T\in(0, T_{\max})}$ there exist $C(T) > 0$ and $\theta > 0$ such that

and

Proof. Let $\delta_0 = 0.1, \, \gamma_0 = 0.501\, , r_0 = 3, \, r_1 = 3.7, \, r_2 = 4.4$. We have $2\delta_0 > \frac{3}{2}(\frac{1}{r_0}-\frac{1}{r_1})$ and $\gamma_1: = \gamma_0+\delta_0+\frac{3}{2}(\frac{1}{r_1}-\frac{1}{r_2}) < \frac{2}{3}$. Therefore, we use standard semigroup estimates, Hölder's inequality and (4.8) to deduce that

Then, we apply the embedding $D(A_{r_2}^{\gamma_0})\hookrightarrow C^\theta, \, 0 < \theta < 2\gamma_0-\frac{3}{r_2}$ to obtain the Lemma 4.12. □

Lemma 4.13. Assume that (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there exist $C(T) > 0$ such that

and

Proof. We multiply the Eq $(1.1)_2$ with $-\Delta c$ and use the integration by parts and Hölder's inequality to obtain

That is

Integrating the two sides of the inequality (4.19) and applying the Lemmas 4.10 and 4.12, we completely the proof of the Lemma 4.13. □

Lemma 4.14. If (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there exist $C(T) > 0$ such that

Proof. We integrate the first equation of (1.1) and use the Höder's inequality and Young's inequality to get

For $k = 3$, using the same method, we can get

By integrating the expressions of (4.21) or (4.22) and using the Lemma 4.13, the proof is complete. □

Lemma 4.15. Assume that (1.3) holds. Then for all $T > 0$ there exist $C(T) > 0$ such that

and

Proof. Multiplying the Eq $(1.1)_3$ with $-\Delta w$ and using Hölder's inequality, one has

Integrating the two sides of (4.23) and applying the estimates (4.12) and (4.20), we complete the proof of the Lemma 4.15. □

Lemma 4.16. If (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there exist $C(T) > 0$ such that

Proof. Applying the variation of constant formula of $n$, we have

Then, we use the similar method of Lemma 3.11 to deduce that

Next, we give the estimates of $n$, and then apply them to obtain the estimate of $w$.

Lemma 4.17. Suppose that (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there is $C(T) > 0$ such that

Proof. Let $M(T^\star): = \sup\limits_{t\in(0, T^\star)}\|n(\cdot, t)\|_{L^\infty(\Omega)}$ for all $T^\star\in(0, T)$ and let $t_0 = (t-1)_+$. Applying the variation of constant formula of $n$, we can see that

Thus, using the Young's inequality, we obtain the result. □

Lemma 4.18. Assume that (1.3) holds. Then for all ${T\in(0, T_{\max})}$ there exist $C(T) > 0$ such that

Proof. Using the variation of constant formula of $w$ and taking $\delta_6 > 0$ suitable small, we have

for all $q > 1, \, \kappa_5 > \frac{1}{2}-\frac{3}{2q}, \, \delta_6 < \frac{3}{2q}$.

Similar to Lemma 4.16, we get the proof of Lemma 4.18. □

Proof of Theorem 1.1 for the three-dimensional case. Finally, we arrive at the proof of Theorem 1.1, using the estimates we obtained in Lemmas 4.16–4.18 and then using the extendability criterion. □

Proof of Theorem 1.2. Based on the estimates collected in Lemmas 4.4–4.8, and the three-dimensional case is similar. We finish the proof. □

Use of AI tools declaration

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

Acknowledgments

The first author was supposed by Scientific Research Funds of Chengdu University under grant No. 2081921030. The second author was supposed by the NSFC Youth Fund under grant No. 12001384.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.