On the global well-posedness and exponential stability of 3D heat conducting incompressible Navier-Stokes equations with temperature-dependent coefficients and vacuum

1.   Introduction

In this paper, we consider the nonhomogeneous incompressible heat conducting Navier-Stokes equations that can be written as

supplemented with the initial data:

and the boundary condition:

Here $\Omega\subset \mathbb{R}^3$ is a bounded smooth domain, and $\mathbf{n}$ is the unit outward normal to $\partial\Omega$. $\rho, u, \theta$, and $P$ stand for the density, velocity, absolute temperature, and pressure of the fluid, respectively. $D(u) = \frac{1}{2}\big(\nabla u+(\nabla u)^T\big)$ is the deformation tensor. The coefficients $\mu$, $c_v$, and $\kappa$ denote the viscosity, specific heat at constant volume, and heat conductivity, respectively.

From a physical perspective, viscosity and heat conductivity coefficients depend on temperature. In fact, when deriving Navier-Stokes equations from the Chapman-Enskog expansion based on the Boltzmann equation, Chapman and Cowling ^[1], Liu, Xin and Yang ^[2] confirmed that viscosity and heat conductivity coefficients depend on temperature. For more details, one can refer to ^[3,4,5,6]. This played a guiding role in later mathematical research, and scholars began to focus on investigating the case of variable coefficients. Specifically, there is a special case

where $c_1, c_2, b$ are positive constants. For this situation, if the intermolecular potential of the cut-off reverse power force model varies with $r^{-a}$($a\in(0, +\infty]$, $r$ represents the intermolecular distance), then there is the following interesting discovery in ^[1]

Furthermore, $b = \frac{1}{2}$ corresponds to rigid elastic spherical molecules; $b = 1$ corresponds to Maxwellian molecules; $b = \frac{5}{2}$ corresponds to ionized gas. For these reasons, we mainly concentrate on the case that $\mu, \kappa$ satisfy the following physical restrictions:

The incompressible heat conduction Navier-Stokes equations with constant coefficients have been widely studied, and a large amount of literature can be found on their well-posedness. The weak solvability of homogeneous systems was proved by Lions ^[7]. For inhomogeneous fluids, Zhang and Tan ^[8] obtained the global well-posedness of a strong solution for the initial density with a positive lower bound in the whole 3D space. When the initial density includes vacuum, the unique local strong solution was guaranteed by ^[9] with the help of compatibility conditions. With it, Zhong ^[10] extended this local solution to the global solution under the condition that $\mu^{-4}\Vert\sqrt{\rho_0}u_0\Vert_{L^2}^2\Vert\nabla u_0\Vert_{L^2}^2$ is small in 3D bounded domain. Wang, Yu and Zhang ^[11] studied problems with general external force and also obtained a similar result. The existence and uniqueness of the global strong solution to the Cauchy problem were proved by Zhong ^[12,13], that is, 2D allows for large initial data, while 3D required $\Vert\rho_0\Vert_{L^\infty}\Vert\sqrt{\rho_0}u_0\Vert_{L^2}^2\Vert\nabla u_0\Vert_{L^2}^2$ to be sufficiently small. Furthermore, Zhong ^[14] also studied the 3D initial-boundary value problem with special boundaries, namely the Navier-slip boundary with velocity and Neumann boundary with temperature, and obtained a globally strong solution under the assumption that $\Vert\sqrt\rho_0 u_0\Vert_{L^2}^2\Vert u_0\Vert_{L^2}^2$ is small. Without thermal conductivity, it is the classic incompressible Navier-Stokes equations. Abidi et al. ^[15,16], and Craig et al. ^[17] studied the well-posedness and decay behavior in critical space. There were many important research results on the well-posedness of solutions to nonhomogeneous incompressible Navier-Stokes equations, which can be referred to ^{[18,19,20,21,22]}, and their references.

In general, the Navier-Stokes equations with variable coefficients can more accurately describe the motion of fluids in practice. When considering that viscosity and heat conduction depend on temperature or density, the strong coupling makes the study of system (1.1) more difficult. For 3D homogeneous incompressible heat conduction Navier-Stokes equations with $\mu = \mu(\theta), \, \kappa = \kappa(\theta)$, weak solutions were established by Naumann ^[23] in bounded domains and by Feireisl and Málek ^[24] in periodic domains, respectively. Amann ^[25] considered the initial-boundary value problem with coefficients depending on $D(u)$ and $\theta$, and showed the unique solvability under the assumption of small data through the linearization method. Later, Frehse et al. ^[26] proved the existence of a global weak solution for shear-thickening heat conducting incompressible fluids with $\mu = \mu(\rho, \theta, |D(u)|^2), \, \kappa = \kappa(\rho, \theta)$ in bounded regions. When the initial vacuum is taken into account, one needs to put more effort into the structure of the equation. In the 3D domain (both bounded or unbounded), Cho and Kim ^[9] considered the cases $\mu = \mu(\rho, \rho\theta), \, \kappa = \kappa(\rho, \rho\theta)$ and obtained the unique local strong solution under the compatibility conditions

for some $P_0\in H^1, \, (g_1, g_2)\in L^2$. On this basis, for the case $\mu = \mu(\rho, \theta), \, \kappa = const.$, Xu and Yu ^[27,28] extended this strong solution globally under the small initial energy. Specifically, the global strong solution to the initial-boundary value problem was established in ^[28], and the global existence and algebraic decay of the strong solution to the Cauchy problem were established in ^[27]. For the 2D case, if $\mu = \mu(\rho)\geq\underline{\mu} > 0, \, \kappa = \kappa(\rho)\geq\underline{\kappa} > 0$, the global strong solution was established by Zhong ^[29] under the smallness on $\Vert\nabla\mu(\rho_0)\Vert_{L^q}(2 < q < \infty)$. Although these research results considered variable coefficients, it is necessary that the viscosity or heat conductivity does not degenerate. Recently, for the degenerate compressible Naiver-Stokes equations for non-isentropic flow, Xin and his collaborators ^[30,31] obtained several important progresses on the local-in-time well-posedness of regular solutions with vacuum. Meanwhile, Zhang and Fang ^[32] also established the unique local strong solution and blow-up criterion for the 3D case. In the absence of vacuum, Guo and Li ^[33] showed the existence and uniqueness of a global strong solution to the problem (1.1)–(1.4) and also obtained the large-time behavior.

Based on the above research results, we are very interested in the problem of nonhomogeneous heat conduction incompressible Navier-Stokes equations with temperature-dependent viscosity and heat conductivity. Little is known about the global solvability of the initial-boundary value problem (1.1)–(1.4) with vacuum. The problem becomes quite difficult when vacuum is taken into account. To study this issue, we must deal with the following main obstacles:

(i) Strong coupling caused by temperature-dependent viscosity. Compared with the constant coefficients equation, the temperature-dependent equations have a stronger coupling between the momentum equations and the temperature equations, which makes the structure of the equations more complex. The temperature-dependent viscosity makes it difficult to estimate the velocity and temperature in the higher order, and even the lower order estimates cannot be obtained directly. Based on the regularity theory of the Stokes equations (see ^[21]), to get an $H^2$-estimate of $u$, we need to first deal with $\|\nabla \mu(\theta)\|_{L^q}$. Similarly, to get the estimate of $\theta$, we need to deal with $\|\kappa(\theta)\|_{L^\infty}$ first. Dealing with $\|\nabla \mu(\theta)\|_{L^q}$ and $\|\kappa(\theta)\|_{L^\infty}$ is essentially a matter of getting a consistent bound on temperature. It is precisely because we have no restrictions on the power index of viscosity $\mu$ and thermal conductivity $\kappa$ that this creates a certain hindrance to energy estimation. To overcome this difficulty, we need to simultaneously estimate the energy of both velocity and temperature, and we need to make very detailed estimates. Meanwhile, we make full use of the bootstrap argument.

(ii) Strong nonlinearity due to temperature-dependent heat conductivity. For the case where the heat conductivity is a constant, the key is to obtain a consistent estimate of temperature. If the heat conductivity depends on temperature, the estimation of temperature is a huge challenge. In the process of controlling the additional terms generated by nonlinear terms in the energy equation, we do time-weighted energy estimates. We first obtain the upper bound of $t^{\frac{1}{2}}\|\nabla^2 \theta\|_{L^2}$, and then combine it with the local existence theorem to obtain a consistent estimate of $\|\nabla^2 \theta\|_{L^2}$.

(iii) Degradation caused by vacuum. Vacuum leads to the degeneracy of time evolution in momentum equations and temperature equations. When it comes to the initial vacuum, it is usually natural to introduce the compatibility condition.

These obstacles require us to make full use of its structural characteristics and make precise estimates starting from the equations. The main objective of this article is to investigate the global well-posedness and large time behavior of strong solutions to the (1.1)–(1.4) under the assumption of small initial data.

Without loss of generality, we abbreviate

For $1\leq r\leq\infty, k\geq1$, the standard Sobolev spaces are abbreviated as follows:

The definition of strong solutions to be established in this article is as follows:

Definition 1.1. Given $T\in(0, \infty)$ and $q\in(3, 6]$, $(\rho, u, \theta, P)$ is called a strong solution to the initial-boundary value problem (1.1)–(1.4) on $[0, T]\times\Omega$, if

and $(\rho, u, \theta, P)$ satisfies the system (1.1)–(1.4) a.e. on $[0, T]\times\Omega$.

We prove the global existence of a strong solution, which allows for an initial density containing vacuum. The specific conclusion is as follows:

Theorem 1.1. For some positive constants $\underline{\theta}$ and $q\in(3, 6]$, suppose that the initial data $(\rho_0, u_0, \theta_0)$ satisfies

and the compatibility conditions

for some $P_0\in H^1$ and $(g_1, g_2)\in L^2$. Then there exist two positive constants $\epsilon_0$ and $\epsilon_1$ depending on $\Omega, \, \underline{\theta}, \, \alpha, \, \beta$ and the initial data, such that if

where $V = \{x\in\Omega | \rho_0(x) = 0\}$, the initial-boundary value problem (1.1)–(1.4) admits a unique global strong solution $(\rho, u, \theta, P)$, and the following large-time behavior holds:

where

Remark 1.1. The authors in ^[34] studied the well-posedness of strong solutions to the system (1.1)–(1.4) in 2D and also proposed the condition $(1.9)_1$. In addition, for the initial-boundary value problem of temperature-dependent non-isentropic compressible Navier-Stokes equations with vacuum, the unique local strong solution was established by Cao and his collaborators ^[35,36] under the assumption that the size of the initial vacuum domain is sufficiently small and compatibility conditions. From this perspective, the conditions (1.8) and $(1.9)_1$ we proposed are also reasonable. Establishing strong solutions without compatibility conditions as in ^[37] is meaningful and is left for further research.

Remark 1.2. When the initial density is far away from vacuum and $\Vert\sqrt{\rho_0}u_0\Vert_{L^2}+\Vert\sqrt{\rho_0}\theta_0\Vert_{L^2}$ is sufficiently small, the authors in ^[33] established the global existence of a strong solution to the system (1.1)–(1.4) and obtained the exponential decay-in-time results. Compared to their results, our results do not require the initial density to be far away from the vacuum, which is also the highlight of this article.

Here, we would like to express some opinions on the analysis of this article. The local existence and uniqueness of strong solutions are guaranteed by Lemma 1. Our main idea is to reasonably combine the bootstrap argument with time-weighted estimates. We are committed to establishing consistent a prior estimates that are independent of time. As pointed out in ^[33], the core is to obtain the bound of $\theta$. But the method they use is aimed at the initial density with a positive lower bound and is not suitable for situations with vacuum. In order to overcome the difficulties caused by vacuum, we borrow the technique from ^[36] and divide the domain $\Omega$ into two parts: $\{x\in\Omega|\rho(x)\leq c_0\}$ and $\{x\in\Omega|\rho(x) > c_0\}$. Further utilize the smallness of the initial data and measure of the initial vacuum region to obtain consistent estimates. With global a priori estimates, we can successfully extend the local solution to the global one. Finally, we also obtain exponential decay.

The remaining content of this article is arranged as follows: in the second section, we reviewed the existence of local strong solutions, basic inequalities and lemmas. In the third section, we mainly establish a priori estimates and prove Theorem 1.1.

2.   Preliminaries

In this section, we review the known results and basic inequalities, which are crucial for our subsequent calculations.

We first elaborate on the conclusion of the local existence of a strong solution, which is guaranteed by ^[9].

Lemma 2.1. Assume that the initial data $(\rho_0, u_0, \theta_0)$ satisfy (1.7) and (1.8). Then there exists a small time $T_0 > 0$ and a unique strong solution $(\rho, u, \theta, P)$ to the system (1.1)–(1.4) on $\Omega\times(0, T_0)$.

The following Poincaré-type inequality is extremely helpful for studying problems involving vacuum (see ^[24]).

Lemma 2.2. Let $f\in H^1, \, 0\leq g\leq C_1$, and $\int g dx\geq (C_1)^{-1}$. Then for $p\in [1, 6]$, one has

where $C$ depends only on $p, C_1, \Omega$.

The direct application of Lemma 2.2 is to control term $\Vert\theta_t\Vert_{L^6}$. Specifically, we can take $g = \rho$ and combine $\int\rho dx = \int\rho_0 dx$ to easily derive

where $C$ only depends on $p, \Vert\rho_0\Vert_{L^1}, \Omega$. This inequality is frequently used in the a priori estimates in Section 3.1.

The following estimation of high-order derivatives is based on the regularity theory of the Stokes equations with variable viscosity coefficient; please refer to ^[38] for details.

Lemma 2.3. Assume that $\mu(\theta)$ satisfies

Let $(u, P)\in H^1_{0, \sigma}\times L^2$ be the unique weak solution to the following system

There are the following conclusions:

(1) If $F\in L^r$ with $2\leq r < q$, then $(u, P)\in W^{2, r}\times W^{1, r}$ and

(2)If $F\in H^1, \, \nabla^2\mu(\theta)\in L^2$, then $(u, P)\in H^{3}\times H^{2}$ and

Here $C$ depends only on $\underline{\mu}, \overline{\mu}, q, r, \Omega$.

Proof. The proof of (2.2) has been given in ^[38], but for simplicity, we have omitted it here. We now provide a detailed proof process for (2.3). We rewrite (2.1) as

According to the regularity theory of the Stokes system, Gagliardo-Nirenberg inequality, we have

where we have used the following inequality

By virtue of (2.2),

Hence,

which implies that (2.3) is valid. We finish the proof of Lemma 2.3. □

3.   Global well-posedness

This section is divided into two subsections. The first subsection is to establish time-weighted a priori estimates. The second subsection extends the unique local strong solution to the global strong solution using the bootstrap argument based on the obtained estimates. It also proves the conclusion of exponential decay.

3.1.   A priori estimates

In this subsection, we will establish the necessary a priori estimates, which can extend the local solution to the global solution. In order to get consistent estimates that do not depend on time, we do some time-weighted estimates. Let $(\rho, u, \theta, P)$ be the strong solution of Navier-Stokes equations (1.1)–(1.4) with initial data $(\rho_0, u_0, \theta_0)$ on $\Omega\times(0, T]$, and the initial data satisfies conditions (1.7)–(1.9). For simplicity, let us take $c_v = 1$. For a general positive constant $c_v$, the calculation is the same. The letter $C$ represents a generic positive constant that depends on $\tilde{\rho}\triangleq\Vert\rho_0\Vert_{L^\infty}, \underline{\theta}$, and $\Omega$ but does not depend on $T$.

Our main idea is to apply the bootstrap argument. To this end, we first propose the following a priori assumptions:

Here $M$ is a positive constant that depends only on the initial data.

We first use the standard method to find the upper and lower bounds of density and the lower bound of temperature.

Lemma 3.1. It holds that $\forall \, (x, t)\in\Omega\times[0, T]$,

Proof. The proof of $(3.2)_1$ is provided by ^[7,24,33]. Then, applying the standard maximum principle (see ^[36] or [24, p.43]) to $(1.1)_3$ along with $\theta_0\geq \underline{\theta}$ shows $(3.2)_2$. Finally, multiplying $(1.1)_2$ by $u$ and then integrating over $\Omega$, using integration by parts, one has

By integrating the above equation on $[0, t]$ and with the help of $\theta\geq \underline{\theta}$ and $2\Vert D(u)\Vert_{L^2}^2 = \Vert\nabla u\Vert_{L^2}^2$, it can be inferred that

Furthermore, multiplying (3.4) by $t$, integrating with respect to $t$, and taking advantage of Poincaré inequality and (3.5) yield

Similarly, for $i = 2, ..., 8$, the following formula can also be obtained

which, combined with (3.5) and (3.6), can prove that $(3.3)$ holds. □

Lemma 3.2. Suppose $(\rho, u, \theta, P)$ satisfies the assumptions (3.1). There exists a small positive constant $\epsilon_1$ depending on $\Omega, \, \underline{\theta}, \, \alpha, \, \beta$ and the initial data, such that for any $i = 1, 2, 3, 4$,

Proof. Notice that Lemma 2.2 can directly provide

Multiplying $(1.1)_2$ by $u_t$ and integrating it over $\Omega$, according to (3.10) and the Gagliardo-Nirenberg inequality, we can easily show

Then, integrating the last identity over $[0, t]$, by virtue of Holder's inequality, (3.1) and Lemma 3.1, we can establish

provided that $\epsilon_1\ll 1$.

Similarly, multiplying (3.11) by $t^i (i\geq1)$, then integrating it over $[0, t]$, one has from (3.1) and Lemma 3.1 that

The proof of Lemma 3.2 is completed. □

Lemma 3.3. Suppose $(\rho, u, \theta, P)$ satisfies the assumptions (3.1). There exist two small positive constants $\epsilon_0$ and $\epsilon_1$ depending on $\Omega, \, \underline{\theta}, \, \alpha, \, \beta$ and the initial data, such that

Proof. Differentiating $(1.1)_2$ with respect to $t$, multiplying the result identity by $u_t$, then integrating over $\Omega$ yields

It follows from Hölder's inequality, the Gagliardo-Nirenberg inequality, Lemmas 3.1–3.2, and (3.1) that

The estimate of the last term in inequality (3.14) is more subtle. In view of $\rho_0\in W^{1, q}\hookrightarrow C$, we know that there exists a positive constant $c_0$ such that

where $V_{c_0} = \{x\in\Omega | \rho_0(x)\leq c_0\}$ and $V = \{x\in\Omega | \rho_0(x) = 0\}$. Then

owing to $|\{x\in\Omega|\rho(x, t)\leq c_0\}| = |V_{c_0}|$ (see [7, Theorem 2.1]), (3.1), (3.10) and the Gagliardo-Nirenberg inequality. Moreover, since

then according to the Gagliardo-Nirenberg inequality, we find

Now (3.14) implies

Next, multiplying (3.17) by $1+t$ and integrating it over $[0, t]$, we obtain after using Grönwall's inequality, Lemmas 3.1–3.2, and (3.1) that

provided that $\epsilon_0\ll 1, \ \epsilon_1\ll 1$.

Furthermore, we calculate from $(1.1)_1$ that

Then, we multiply the last identity by $q|\nabla\rho|^{q-2}\partial_i\rho$ and integrate it over $\Omega$ to obtain

Combining (3.16) and Lemmas 3.1 and 3.2, we ascertain

and

Now according to (3.18) and Grönwall's inequality, one has

The proof of this lemma is finished. □

Next, we will make estimates for each order regarding $\theta$. This is the key to all a priori estimates.

Lemma 3.4. Suppose $(\rho, u, \theta, P)$ satisfies the assumptions (3.1). It holds that

Proof. Firstly, multiplying $(1.1)_3$ by $\theta$ and integrating it over $\Omega$ yields

By integrating the above equation on $[0, t]$, we obtain from Lemma 3.1 that

Similarly, multiplying $(1.1)_3$ by $\theta^{\beta+1}$ and integrating it over $\Omega\times[0, t]$, it can be inferred that

Next, taking $(1.1)_1$ and (1.3) into account, multiplying $(1.1)_3$ by $\theta^{\beta}\theta_t$, and integrating by parts over $\Omega$, we obtain

by the Gagliardo-Nirenberg inequality and (3.1). On the one hand, integrating (3.27) with respect to $t$ leads to

where in the next to last inequality one has used Lemma 3.1 and (3.1). On the other hand, it follows from (3.27), (3.1), (3.26), and Lemma 3.1 that

By applying Lemma 2.2, (3.26) and (3.28), one can derive

Therefore, we finish the proof of Lemma 3.4. □

Lemma 3.5. Suppose $(\rho, u, \theta, P)$ satisfies the assumptions (3.1). It has

Proof. Differentiating $(1.1)_3$ with respect to $t$ yields

Multiplying (3.30) by $\theta_t$, then integrating it over $\Omega$, we obtain

Now, we will use (1.1), the Gagliardo-Nirenberg inequality, (3.1), (3.10), Lemmas 3.1–3.4, and (3.15) to estimate each term on the right hand of (3.31) as follows:

and

Substituting all the estimates of $J_i(i = 1, ..., 6)$ into (3.31), then taking $\epsilon_0$ and $\epsilon_1$ satisfy

we deduce

In view of Poincaré's inequality, Lemma 3.1 and (3.1), one has

Next, similar to (3.19), we infer that

Furthermore, we employ Lemma 3.4 to obtain

Combining (3.32)–(3.35), Lemmas 3.3, 3.4, and Grönwall's inequality gives

We then rewrite $(1.1)_3$ as

It follows from the standard $L^2$-theory of elliptic equations that

where in the last inequality we have used (3.1). Note that

thus

On the one hand, according to (3.38), (3.1), Lemma 3.2, (3.36), and Lemma 3.4, we compute

According to the local existence theorem of the solution (see Lemma 2.1), there exists a small positive time $T_0$ such that

This, together with (3.40), yields

Moreover, Lemma 3.4, the Gagliardo-Nirenberg inequality, and (3.41) imply

On the other hand, by (3.1), (3.38), and (3.39), we discover

Consequently, combining (3.36), (3.41) with (3.44), we complete the proof of this lemma. □

By synthesizing Lemmas 3.1–3.5, the following corollary can be obtained.

Corollary 3.1. There exist two small positive constants $\epsilon_0$ and $\epsilon_1$ as described in Theorem 1.1, such that if

then it holds for any $(x, t)\in\Omega\times[0, T]$ that

Proof. Now, we consider the following Stokes equations:

According to Lemma 2.3, we have the following regularity results:

and

which indicates that

Using Lemmas 3.1–3.3, it can be inferred that

Hence,

On the other hand, by virtue of (3.37) and the regularity theory to the elliptic equation, one has

Combining (3.47), (3.52) and estimates we have obtained, it can be inferred that

Considering the estimates obtained in Lemmas 3.1–3.5, combined with (3.49)–(3.51) and (3.53), we have completed the proof of the Corollary 3.1. In this way, we close the a priori assumptions (3.1). □

3.2.   Proof of Theorem 1.1

With all the a priori estimates established in Corollary 3.1 at hand, the global well-posedness part of Theorem 1.1 then follows by standard procedures.

From Lemma 2.1, we know that there exists a $T_0$ such that the system (1.1)–(1.4) has a unique local strong solution $(\rho, \, u, \, \theta, \, P)$ on $\Omega\times(0, T_0]$. We now extend this local solution to the global solution by contradiction. Therefore, from now on, we assume that $|V|\leq\epsilon_0, \, \Vert\sqrt{\rho_0}u_0\Vert_{L^2}+\beta\Vert\sqrt{\rho_0}\theta_0\Vert_{L^2}\leq\epsilon_1$ holds, where $\epsilon_0$ and $\epsilon_1$ are the same as in Corollary 3.1. Lemma 2.1 and (1.7) indicate that there is a $T_1\in (0, T_0)$ such that (3.1) holds at $T = T_1$. Denote

Obviously, $T^{\ast}\geq T_{1} > 0$. With the help of Corollary 3.1 and the standard embedding, one can deduce that for any $0 < T\leq T^{\ast}$,

Now we prove

Otherwise, we assume that $T^{\ast} < \infty$. Since $|V|\leq\epsilon_0, \, \Vert\sqrt{\rho_0}u_0\Vert_{L^2}+\beta\Vert\sqrt{\rho_0}\theta_0\Vert_{L^2}\leq\epsilon_1$, Lemmas 3.1–3.5 and the continuity argument show that the global estimates stated in Corollary 3.1 hold on $[0, T^{\ast}]$. Let $\dot{v}\triangleq v_t+u\cdot \nabla v$, it can be inferred from (3.55) that

satisfies regularity condition

Thus

with

satisfying $g_1, \, g_2\in L^2$, which can be guaranteed by Corollary 3.1. Therefore, taking $(\rho^\ast, u^\ast, \theta^\ast)$ as the initial data, according to Lemma 2.1, the local strong solution can be extended beyond $T^\ast$. This contradicts the definition of $T^\ast$. So, (3.54) has been proven.

To complete the entire proof of Theorem 1.1, we only need to prove (1.10) and (1.11), i.e., the following proposition.

Proposition 3.1. It holds that

where the positive constants $\overline{\rho_0}$ and $E_0$ are defined in Theorem 1.1.

Proof. The proof is divided into the following two steps.

Step 1. The proof of (3.58).

It follows from the Poincaré inequality that

Substituting (3.60) into (3.4) yields

Then multiplying (3.61) by $e^{C^{-1}t}$ and integrating it over $[0, t]$, we arrive at

Next, substitute (3.46) into (3.11), then (3.11) becomes

owing to Young's inequality, Lemma 3.1 and Corollary 3.1. Multiply the last inequality by $e^{C^{-1}t}$, using Grönwall's inequality, Lemma 3.4, (3.36), (3.62) and

imply

Similarly, in view of the proof of Lemma 3.3, (3.48) and

we thereby obtain

Multiplying the last inequality by $e^{C^{-1}t}$, applying (3.62), (3.63), Grönwall's inequality, and Corollary 3.1, we deduce

Thus, collecting (3.62)–(3.65), we get (3.58) immediately.

Step 2. The proof of (3.59).

The method in ^[34] is used in this process. Integrating $(1.1)_3$ over $\Omega$, we arrive at

Combining (3.4) and (3.66), it can be inferred that

Hence

Multiplying $(1.1)_3$ by $\theta-\frac{1}{\overline{\rho_0}|\Omega|}E_0$ and integrating it over $\Omega$, one obtains

It follows from Poincaré's inequality that

where $\bar{\theta} = \frac{1}{|\Omega|}\int\theta dx$. Thus, we have

and this combined with (3.62) gives

Now, we re-estimate (3.27) as follows:

Multiplying it by $e^{C^{-1}t}$, according to (3.58), and Corollary 3.1, we have

Consequently,

Recalling (3.31), (3.64) and Corollary 3.1 leads to

Then, multiplying the last inequality by $e^{C^{-1}t}$, combining the result with Grönwall's inequality, (3.58), (3.70) and Corollary 3.1 gives

Therefore, we complete the proof of (3.59) by (3.68), (3.70), (3.71), and the following estimate

The proof of Proposition 3.1 is finished. □

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 would like to thank the editor and reviewers for careful reading, helpful comments and suggestions. The work is supported by the Fundamental Research Funds for the Central Universities, CHD (No.300102122115).

Conflict of interest

The authors declare there is no conflicts of interest.