A catalogue of human secreted proteins and its implications

Shivakumar Keerthikumar; Shivakumar Keerthikumar

doi:10.3934/biophy.2016.4.563

AIMS Biophysics

2016, Volume 3, Issue 4: 563-570. doi: 10.3934/biophy.2016.4.563

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Research article Topical Sections

A catalogue of human secreted proteins and its implications

Shivakumar Keerthikumar ^,

Department of Biochemistry and Genetics, La Trobe Institute for Molecular Science, La Trobe University, Melbourne, Victoria 3086, Australia

Received: 27 September 2016 Accepted: 17 November 2016 Published: 24 November 2016

Under both normal and pathological conditions, cells secrete variety of proteins through classical and non-classical secretory pathways into the extracellular space. Majority of these proteins represent pathophysiology of the cell from which it is secreted. Recently, though more than 92% of the protein coding genes has been mapped by human proteome map project, but number of those proteins that constitutes secretome of the cell still remains elusive. Secreted proteins or the secretome can be accessible in bodily fluids and hence are considered as potential biomarkers to discriminate between healthy and diseased individuals. In order to facilitate the biomarker discovery and to further aid clinicians and scientists working in these arenas, we have compiled and catalogued secreted proteins from the human proteome using integrated bioinformatics approach. In this study, nearly 14% of the human proteome is likely to be secreted through classical and non-classical secretory pathways. Out of which, ~38% of these secreted proteins were found in extracellular vesicles including exosomes and shedding microvesicles. Among these secreted proteins, 94% were detected in human bodily fluids including blood, plasma, serum, saliva, semen, tear and urine. We anticipate that this high confidence list of secreted proteins could serve as a compendium of candidate biomarkers. In addition, the catalogue may provide functional insights in understanding the molecular mechanisms involved in various physiological and pathophysiological conditions of the cell.

Keywords:

secretory; bioinformatics; integrative; proteomics; extracellular vesicles

Citation: Shivakumar Keerthikumar. A catalogue of human secreted proteins and its implications[J]. AIMS Biophysics, 2016, 3(4): 563-570. doi: 10.3934/biophy.2016.4.563

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Abstract

1. Introduction

The model of micropolar fluids which respond to micro-rotational motions and spin inertia was first introduced by Eringen [16] in 1966. The mathematical theory of micropolar fluids has been developing in two directions. One explores incompressible and the other compressible flows. For more physical background, we can refer to [2], [3], [39]. In this paper we consider the compressible cylinder symmetric flow of the isotropic, viscous and heat-conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic. The mathematical model of the described fluid is stated for example in the book of G. Lukaszewicz [32] and reads

$\begin{eqnarray} \dot{\rho} & = & -\rho\nabla\cdot\textbf{v}, \end{eqnarray}$

(1)

$\begin{eqnarray} \rho\dot{\textbf{v}} & = & \nabla\cdot\textbf{T}+\rho \textbf{f}, \end{eqnarray}$

(2)

$\begin{eqnarray} \rho j_{I}\dot{\textbf{w}} & = & \nabla\cdot\textbf{C}+\textbf{T}_x+\rho \textbf{g}, \end{eqnarray}$

(3)

$\begin{eqnarray} \rho\dot{E} & = & -\nabla\dot{\textbf{q}}+\textbf{T}:\nabla\textbf{v} +\textbf{C}:\nabla\omega-\textbf{T}_x \cdot\textbf{w}, \end{eqnarray}$

(4)

$\begin{eqnarray} \textbf{T}_{ij} & = & (-p+\lambda v_{k,k})\delta_{i,j}+\mu(v_{j,i}-v_{i,j})-2\mu_r\varepsilon_{mij}w_{m}, \end{eqnarray}$

(5)

$\begin{eqnarray} \textbf{C} & = & c_0w_{k,k}\delta_{i,j}+c_d(w_{i,j}+w_{j,i})+c_a(w_{j,i}-w_{i,j}), \end{eqnarray}$

(6)

$\begin{eqnarray} \textbf{q} & = & -k_{\theta}\nabla\theta, \end{eqnarray}$

(7)

$\begin{eqnarray} p & = & R\rho\theta, \end{eqnarray}$

(8)

$\begin{eqnarray} \textbf{E} & = & c_v\theta. \end{eqnarray}$

(9)

with notation:

$\begin{eqnarray*} \begin{array}{ll} \rho - mass\;\; density\\ \textbf{v} - velocity\\ \textbf{w} - microrotation\;\; velocity\\ E - internal\;\; energy\;\; density \\ \theta - absolute \;\;temperature\\ \textbf{T} - stress\;\; tensor\\ \textbf{C } - couple\;\; stress\;\; tensor\\ \textbf{q} - heat\;\; flux \;\;density \;\;vector \\ \textbf{f} - body\;\; force \;\;density\\ \textbf{g} - body\;\; couple\;\; density\\ p - pressure\\ j_I - microinertia\;\; density\;\; (a\;\; positive\;\; constant)\\ k_{\theta} - heat\;\; conduction \;\;coefficien\\ \mu_r,\;\; c_0,\;\; c_a\;\; and\;\; c_d - coefficients\;\; of\;\; microviscosity\\ R - specific \;\;gas \;\;constant\\ c_v - specific \;\;heat\;\; (positive\;\; constant)\\ \textbf{T}_x - axial\;\; vector\;\; ((\textbf{T}_x)_i = \varepsilon_{ijk}\cdot T_{jk})\\ \varepsilon_{ijk} - Levi \;\; Civita \;\; symbol\\ \delta_{ij} - Kronecker\;\; delta. \end{array} \end{eqnarray*}$

Equations (1)-(4) are, respectively, local forms of the conservation laws for the mass, momentum, momentum moment and energy. Equations (5)-(6) are constitutive equations for the micropolar continuum. Equation (7) is the Fourier law and equations (8)-(9) present the assumptions that our fluid is perfect and polytropic. On account of the Clausius-Duhamel inequalities, they must have the following properties:

$\begin{eqnarray} &&\mu\geq 0,\quad 3\lambda+2\mu\geq 0,\quad \mu_r > 0, \end{eqnarray}$

(10)

$\begin{eqnarray} &&c_d\geq 0,\quad 3c_0+2c_d\geq 0,\quad|c_d-c_a|\leq c_d+c_a. \end{eqnarray}$

(11)

For simplicity reasons, we assume that

$f = g = 0.$

In this paper, we consider the three dimensional case of (1)-(9) with the assumption of cylindrical symmetry, and we study the problem with homogeneous boundary conditions as in [13]:

$\begin{eqnarray} &&t = 0: (\rho,\textbf{v},\textbf{w},\theta)(r,0) = (\rho_0(r),\textbf{v}_0(r),\textbf{w}_0(r),\theta_0(r)),\;r\in G, \end{eqnarray}$

(12)

$\begin{eqnarray} &&\textbf{v}|_{\partial G} = \textbf{0},\;\textbf{w}|_{\partial G} = \textbf{0},\;\; \frac{\partial\theta}{\partial r}\bigg|_{\partial G} = 0,\quad t > 0, \end{eqnarray}$

(13)

where $G = \{(x_1,x_2,x_3)\in R^3,\;0<a<r<b<+\infty,\;x_3\in R,\;r = \sqrt{x_1^2+x_2^2}\,\}$ is the spatial domain of our problem and $\textbf{v} = (\upsilon_{1},\upsilon_{2},\upsilon_{3})$ , $\textbf{w} = (w_{1},w_{2},w_{3})$ denote the velocity vector and microrotation velocity respectively. In the following work we give the mathematical model with cylindrical symmetry, first in the Eulerian description, which is then transformed to the Lagrangian description. The reduced system of the three-dimensional equations in the Eulerian coordinate is now of the form [11] and [22]:

$\begin{eqnarray} &\frac{\partial\rho}{\partial t}+v_1\frac{\partial\rho}{\partial r}+\rho\left(\frac{v_1}{r}+\frac{\partial v_1}{\partial r}\right) = 0, \end{eqnarray}$

(14)

$\begin{eqnarray} &\rho\left(\frac{\partial v_1}{\partial t}+v_1\frac{\partial v_1}{\partial r}\right) = -R\frac{\partial}{\partial r}(\rho\theta)+(\lambda+2\mu)\frac{\partial}{\partial r}\left(\frac{\partial v_1}{\partial r}+\frac{v_1}{r}\right)+\rho\frac{v_2^2}{r}, \end{eqnarray}$

(15)

$\begin{eqnarray} &\rho\left(\frac{\partial v_2}{\partial t}+v_1\frac{\partial v_2}{\partial r}\right) = (\mu+\mu_r)\frac{\partial}{\partial r}\left(\frac{\partial v_2}{\partial r}+\frac{v_2}{r}\right)-\rho\frac{v_1v_2}{r}-2\mu_r\frac{\partial w_3}{\partial r}, \end{eqnarray}$

(16)

$\begin{eqnarray} &\rho\left(\frac{\partial v_3}{\partial t}+v_1\frac{\partial v_3}{\partial r}\right) = (\mu+\mu_r)\left(\frac{\partial^2 v_3}{\partial r^2}+\frac{1}{r}\frac{\partial v_3}{\partial r}\right) +2\mu_r\left(\frac{\partial w_2}{\partial r}+\frac{w_2}{r}\right), \end{eqnarray}$

(17)

$\begin{eqnarray} &\rho j_I\left(\frac{\partial w_1}{\partial t}+v_1\frac{\partial w_1}{\partial r}\right) = (c_0+2c_d)\frac{\partial}{\partial r}\left(\frac{\partial w_1}{\partial r}+\frac{w_1}{r}\right)+\rho j_I\frac{w_2v_2}{r}-4\mu_r w_1, \end{eqnarray}$

(18)

$\begin{eqnarray} &\rho j_I\left(\frac{\partial w_2}{\partial t}+v_1\frac{\partial w_2}{\partial r}\right) = (c_a+c_d)\frac{\partial}{\partial r}\left(\frac{\partial w_2}{\partial r}+\frac{w_2}{r}\right)-\rho j_I\frac{w_1v_2}{r}-2\mu_r\frac{\partial v_3}{\partial r}-4\mu_rw_2, \end{eqnarray}$

(19)

$\begin{eqnarray} &\rho j_I\left(\frac{\partial w_3}{\partial t}+v_1\frac{\partial w_3}{\partial r}\right) = (c_a+c_d)\left(\frac{\partial^2 w_3}{\partial r^2}+\frac{1}{r}\frac{\partial w_3}{\partial r}\right) +2\mu_r\left(\frac{\partial v_2}{\partial r}+\frac{v_2}{r}\right)-4\mu_rw_3, \end{eqnarray}$

(20)

$\begin{eqnarray} &c_V\rho\left(\frac{\partial \theta}{\partial t}+v_1\frac{\partial \theta}{\partial r}\right) = \kappa\left(\frac{\partial^2 \theta}{\partial r^2}+\frac{1}{r}\frac{\partial \theta}{\partial r}\right)-R\rho\theta\left(\frac{\partial v_1}{\partial r}+\frac{v_1}{r}\right)+(\lambda+2\mu)\left(\frac{\partial v_1}{\partial r}+\frac{v_1}{r}\right)^2\\ &-4\mu\frac{v_1}{r}\frac{\partial v_1}{\partial r}+(\mu+\mu_r)\left(\frac{\partial v_2}{\partial r}+\frac{v_2}{r}\right)^2-4\mu\frac{v_2}{r}\frac{\partial v_2}{\partial r}+(\mu+\mu_r)\left(\frac{\partial v_3}{\partial r}\right)^2\\ &+(c_0+2c_d)\left(\frac{\partial w_1}{\partial r}+\frac{w_1}{r}\right)^2-4c_d\frac{w_1}{r}\frac{\partial w_1}{\partial r}+(c_d+c_a)\left(\frac{\partial w_2}{\partial r}+\frac{w_2}{r}\right)^2\\ &-4c_d\frac{w_2}{r}\frac{\partial w_2}{\partial r}+(c_d+c_a)\left(\frac{\partial w_3}{\partial r}\right)^2+4\mu_r w_1^2+4\mu_r w_2^2+4\mu_r w_3^2\\ &+4\mu_r w_2\frac{\partial v_3}{\partial r}-4\mu_r\left(\frac{\partial v_2}{\partial r}+\frac{v_2}{r}\right)w_3. \end{eqnarray}$

(21)

To analyze the system and draw the desired results, it is convenient to transform the system (14)-(21) to Lagrangian coordinates. The Eulerian coordinates $(r,t)$ are connected to the Lagrangian coordinates $(\xi,t)$ by the relation

$\begin{eqnarray} r(\xi,t) = r_0(\xi)+\int_0^t\tilde{v}_1(\xi,\tau)d\tau, \end{eqnarray}$

(22)

where $\tilde{v}_1(\xi,t) = v_1(r(\xi,t),t)$ and

$\begin{eqnarray} r_0(\xi) = \eta^{-1}(\xi),\quad \eta(r) = \int_a^rs\rho_0(s)ds,\quad r\in G. \end{eqnarray}$

(23)

From $(14)$ and it follows that

$\frac{\partial}{\partial t}\left(\int_a^rs\rho(s,t)ds\right) = 0,$

which implies

$\begin{eqnarray} \int_a^rs\rho(s,t)ds = \int_a^{r_0}s\rho_0(s)ds = \xi. \end{eqnarray}$

(24)

Now, we have $\xi \in \Omega = [0,L]$ , where

$\begin{eqnarray} L = \int_a^bs\rho(s,t)ds = \int_a^bs\rho_0(s)ds,\quad \forall t \geq 0. \end{eqnarray}$

(25)

Moreover, differentiating (24) with respect to $\xi$ yields

$\begin{eqnarray} \frac{\partial r}{\partial {\xi}} = \frac{1}{r(\xi,t)\rho(r(\xi,t),t)}. \end{eqnarray}$

(26)

Let us introduce the temporary notation $\tilde{\phi}(\xi,t)$ for $\phi(r(\xi,t),t)$ , we obtain

$\begin{eqnarray} &&\frac{\partial\tilde{\phi}(\xi,t) }{\partial t} = \frac{\partial\phi (r,t) }{\partial t}+v_1\frac{\partial \phi (r,t)}{\partial r}, \end{eqnarray}$

(27)

$\begin{eqnarray} &&\frac{\partial\tilde{\phi}(\xi,t) }{\partial \xi} = \frac{\partial \phi (r,t)}{\partial r}\frac{\partial r(\xi,t)}{\partial \xi} = \frac{1}{r\rho(r,t)}\frac{\partial \phi (r,t)}{\partial r}, \end{eqnarray}$

(28)

$\begin{eqnarray} &&\rho\frac{\partial}{\partial\xi}(r\tilde{\phi}(\xi,t)) = \frac{\partial \phi }{\partial r}+\frac{\phi}{r}. \end{eqnarray}$

(29)

Hereafter, without danger of confusion, we will write $(x,t)$ instead of $(\xi,t)$ and omit $\sim$ . Subscripts $t$ and $x$ will denote the (partial) derivatives with respect to $t$ and $x$ , respectively, and we will use $u = \frac{1}{\rho}$ to denote the specific volume. Thus, by (27)-(29), system (14)-(21) can be rewritten using the new variables $(x,t),\;x\in\Omega,\;\; t\geq 0$ as follows:

$\begin{eqnarray} u_t& = &(rv_1)_x, \end{eqnarray}$

(30)

$\begin{eqnarray} v_{1t}& = &r\left[(\lambda+2\mu)\frac{(rv_1)_x}{u}-R\frac{\theta}{u}\right]_x+\frac{v_2^2}{r}, \end{eqnarray}$

(31)

$\begin{eqnarray} v_{2t}& = &(\mu+\mu_r)r\left(\frac{(rv_2)_x}{u}\right)_x-\frac{v_1v_2}{r}-2\mu_r r w_{3x}, \end{eqnarray}$

(32)

$\begin{eqnarray} v_{3t}& = &(\mu+\mu_r)r\left(\frac{(rv_3)_x}{u}\right)_x+(\mu+\mu_r)\frac{uv_3}{r^2}+2\mu_r(r w_{2})_x, \end{eqnarray}$

(33)

$\begin{eqnarray} j_I w_{1t}& = &(c_0+2c_d)r\left(\frac{(r w_1)_x}{u}\right)_x+j_I\frac{w_2v_2}{r}-4\mu_r u w_1, \end{eqnarray}$

(34)

$\begin{eqnarray} j_I w_{2t}& = &(c_d+c_a)r\left(\frac{(r w_2)_x}{u}\right)_x-j_I\frac{w_1v_2}{r}-2\mu_r r v_{3x}-4\mu_r u w_2, \end{eqnarray}$

(35)

$\begin{eqnarray} j_I w_{3t}& = &(c_d+c_a)r\left(\frac{(r w_3)_x}{u}\right)_x+(c_d+c_a)\frac{u w_3}{r^2} +2\mu_r(rv_{2})_x-4\mu_r u w_3, \end{eqnarray}$

(36)

$\begin{eqnarray} c_V\theta_t& = &\kappa\left(r^2\frac{\theta_x}{u}\right)_x +\frac{1}{u}\left[(\lambda+2\mu)(rv_1)_x-R\theta\right](rv_1)_x+(\mu+\mu_r)\frac{(rv_2)_x^2}{u} \\ &&\quad+(c_0+2c_d)\frac{(r w_1)_x^2}{u}+(c_d+c_a)\frac{(rw_2)_x^2}{u} +(\mu+\mu_r)\frac{r^2v_{3x}^2}{u}\\ &&\quad+(c_d+c_a)\frac{r^2w_{3x}^2}{u} -2\mu(v_1^2+v_2^2)_x-2c_d(w_1^2+w_2^2)_x\\ &&\quad+4\mu_r(u w_1^2+u w_2^2+u w_3^2)+4\mu_r r w_2v_{3x}-4\mu_r w_3(rv_2)_x , \end{eqnarray}$

(37)

together with the initial and boundary conditions

$\begin{eqnarray} u(x,0) = u_0(x),\;\; \textbf{v}(x,0) = \textbf{v}_0(x),\;\;\textbf{w}(x,0) = \textbf{w}_0(x),\;\;\theta(x,0) = \theta_0(x),\;x\in\Omega, \end{eqnarray}$

(38)

$\begin{eqnarray} \textbf{v}(0,t) = \textbf{v}(L,t) = \textbf{w}(0,t) = \textbf{w}(L,t) = \textbf{0},\;\;\theta_x(0,t) = \theta_x(L,t) = 0,\;\;t > 0. \end{eqnarray}$

(39)

A a result of (22) and (26), we can conclude that $r(x,t)$ is determined by

$\begin{eqnarray} &&r_t(x,t) = v_1(x,t),\qquad r(x,t)r_x(x,t) = u(x,t), \end{eqnarray}$

(40)

$\begin{eqnarray} &&r_0(x) = \left(a^2+2\int_0^xu_0(y)dy\right)^{\frac{1}{2}}. \end{eqnarray}$

(41)

It is easy to see from (40) and (41) that the following is satisfied:

$\begin{eqnarray} &&r_t(x,t) = \upsilon_1(x,t), \;\;r(x,t)r_x(x,t) = u(x,t). \end{eqnarray}$

(42)

Let us mention some related results in this direction. When $w = 0$ , it reduces to be classical Navier-Stokes equations, which provide a suitable model to motion of several important fluids, such as water, oil, air, etc., the existence and asymptotic behavior of Navier-Stokes equations has been regarded as an important problem in the fluid of dynamics, and has been receiving much attention for many researchers (see[1,4,19,17,18,20,28,32,30,31] and references therein). Among them, Fujita and Kato [19] obtained the global well-posedness for small initial data and the local well-posedness for any initial data in $H^s(R^n)$ with $s\geq\frac{n}{2}-1$ . Kato [28] improved results have been established in $L^n(R^n)$ . Recently, Lei and Lin [30] proved global well-posedness result in the space $\chi^{-1}$ . Li and Liang [29] proved large time behavior for one dimensional compressible Navier-Stokes equations in unbounded domains with large data.

For the micropolar fluids case (i.e., $w\neq0$ ), compared with the classical Navier-Stokes equations, the angular velocity $w$ in this model brings benefit and trouble. Benefit is the damping term - $vw$ can provides extra regularity of $w$ , while the term $vw^2$ is bad, it increases the nonlinearity of the system. In the one dimensional case, Mujaković made a series of efforts in studying the local-in-time existence and uniqueness, the global existence and regularity of solutions to an initial-boundary value problem with both homogenous [33,34,35] and non-homogenous boundary conditions [36,37,38] respectively. Later, Huang and Nie [25] proved the exponential stability. Recently, the global attractor of this system has been established in [27]. Besides, we would also like to refer to the works in [5,14,15] for the 1D micropolar fluid model.

In the three dimensional case, for the spherical symmetric model of described micropolar fluid in a bounded annular domain, the local existence, uniqueness, global existence and the large time behavior and regularity of the solution has been proved in [6,7,8,9,10], and the exponential stability and regularity of the spherically symmetric solutions with large initial data has been established in [24,23]. Recently, for the spherical symmetric model of described micropolar fluid in an exterior unbounded domain, we proved the large time behavior for spherically symmetric flow of viscous polytropic gas with large initial data in [26]. In the case of cylinder symmetry, which model described micropolar fluid in a bounded domain with two coxial cylinders that present the solid thermoinsulated walls, Dražić and Mujaković [11] established the local existence of generalized solutions, then they proved global existence [12] and the uniqueness [13], Huang and Dražić [21,22] studied the large time behavior of the cylindrically symmetric with small initial data, but the regularity is open. Besides, we would like to mention the work on the global wellposedness of the three-dimensional magnetohydrodynamic equations, Wang and Wang [41] obtained the global existence results for classical 3-D MHD $(\alpha = 1).$ Wang and Qin [40] obtained global wellposedness and analyticity results to 3-D generalized magnetohydrodynamic equations. Later, Ye [42] obtained the global existence results for classical 3-D GMHD $(\frac{1}{2}\leq\alpha\leq1).$

As mentioned above, the regularity and exponential stability of generalized (global) solutions in $H^2(\Omega)$ has never been studied for system (14)-(21) with boundary conditions (12) and initial conditions (13). Therefore, we shall continue the work by Huang and Dražić [22] and establish the regularity and exponential stability of solutions with small initial data.

Here we study the problem (30)-(37) on the spatial domain $\Omega$ . We introduce the space

$\begin{eqnarray} H_{+}^1& = &\{(u,\textbf{v},\textbf{w},\theta)\in H^1(0,L)\times H^1(0,L)\times H^1(0,L)\times H^1(0,L): v(0,t) = \\ &&v(L,T) = 0,\;\; w(0,t) = w(L,t) = 0\}, \end{eqnarray}$

(43)

$\begin{eqnarray} H_{+}^2& = &\{(u,\textbf{v},\textbf{w},\theta)\in H^2(0,L)\times H^2(0,L)\times H^2(0,L)\times H^2(0,L): v(0,t) = \\ &&v(L,T) = 0,\;\; w(0,t) = w(L,t) = 0,\;\;\theta_x(0,t) = \theta_x(L,t) = 0\}, \end{eqnarray}$

(44)

which becomes the metric space equipped with the metrics induced from the usual norms. In this paper we will denote by $L^{\bar{p}},\; 1\leq \bar{p}\leq +\infty,\; W^{m, \bar{p}},\; m\in N,\; H^1 = W^{1,2},\; H_0^1 = W_0^{1,2}$ denote the usual (Sobolev) spaces on $[0,1]$ . In addition, $\|\cdot\|_{B}$ denotes the norm in the space $B$ , we also put $\|\cdot\| = \|\cdot\|_{L^2}.$ Subscripts $t$ and $x$ denote the (partial) derivatives with respect to $t$ and $x$ , respectively. We use $C_i(i = 1,2)$ to denote the generic positive constant depending only on $H^i$ norm of initial datum $(u_0, \textbf{v}_0, \textbf{w}_0, \theta_0)$ , ${\min_{x\in [0,L]}u_0(x)}$ and ${\min_{x\in [0,L]}\theta_0(x)}$ , but independent of variable $t$ .

We assume that the initial data have the following properties

$\begin{eqnarray} E_0 = \int_0^L\left(\frac{1}{2}|\textbf{v}_0|^2 +\frac{j_I}{2}|\textbf{w}_0|^2 +c_V\theta_0\right)dx, \quad u_0,\theta_0\geq m, \end{eqnarray}$

(45)

where $m$ is a positive constant.

Now, we are in a position to state our main result.

Theorem 1.1. Suppose that initial $(u_0,\textbf{v}_0, \textbf{w}_0, \theta_0)\in H_+^2$ and (45) hold, there exists a constant $\alpha_0 = \alpha_0(C_1) > 0$ , such that if $E_0\leq\alpha_0$ , problem (30)-(39) has a unique generalized global solution $(u(t),\textbf{v}(t), \textbf{w}(t), \theta(t))\in H_+^2$ verifying that for any $t>0$ ,

$\begin{eqnarray} &&u-u^*\in L^\infty([0, +\infty), H^2(\Omega)) \cap L^2([0, +\infty), H^2(\Omega)),\\ &&\textbf{v}\in L^\infty([0, +\infty), H^2(\Omega)) \cap L^2([0, +\infty), H^3(\Omega)),\\ &&\textbf{w}\in L^\infty([0, +\infty), H^2(\Omega)) \cap L^2([0, +\infty), H^3(\Omega)),\\ &&\theta-\theta^*\in L^\infty([0, +\infty), H^2(\Omega)) \cap L^2([0, +\infty), H^3(\Omega)). \end{eqnarray}$

(46)

Moreover, there exists a positive constant $\gamma_2 = \gamma_2(C_2)$ such that for any fixed $\gamma\in(0,\gamma_2]$ and for any $t>0$ , the following estimate holds

$\begin{eqnarray} \|u-u^*, \textbf{v}, \textbf{w},\theta-\theta^*\|\leq C_2 e^{-\gamma t} \end{eqnarray}$

(47)

where $u^* = \frac{1}{L}\int_0^Lu_0(x)dx$ , $\theta^* = \frac{1}{c_VL}\int_0^L\left(\frac{1}{2}|\textbf{v}_0(x)|^2 +\frac{j_I}{2}|\textbf{w}_0(x)|^2 +c_V\theta_0(x)\right)dx$ , $r^* = (a^2+2u^*x)^{1/2}$ .

2. Proof of Theorem 1.1

In this section, we shall complete the proof of Theorem 1.1. The global existence of cylindrically symmetric solutions for system (30)- (39) was proved in [22], we shall continue the work and prove the regularity and exponential of the solution. We begin with the following Lemma.

Lemma 2.1. (See [24] and [22]) If $(u_0, \textbf{v}_0, \textbf{w}_0, \theta_0)\in H_{+}^{1}$ and $E_0\leq \alpha_0$ are true, there exists a unique global weak solution $(u, \textbf{v}, \textbf{w}, \theta)\in H_{+}^{1}$ to the problem (30)-(39) satisfies the following estimates

$\begin{eqnarray} &&0 < a\leq r(x,t)\leq b,\quad (x,t)\in \;\; \Omega\times [0,+\infty),\\ && 0 < C_1^{-1}\leq u(x,t), \theta(x,t)\leq C_1\;\; (x,t)\in\Omega\times[0,\infty),\\ &&\|u-u^*\|_{H^1}^2+\|v_1\|_{H^1}^2+\|v_2\|_{H^1}^2+\|v_3\|_{H^1}^2 +\|w_1\|_{H^1}^2+\|w_2\|_{H^1}^2 \end{eqnarray}$

$\begin{eqnarray} &&+\|w_3\|_{H^1}^2+\|\theta-\theta^*\|_{H^1}^2 +\int_0^t(\|u-u^*\|_{H^1}^2+\|v_1\|_{H^2}^2+\|v_2\|_{H^2}^2+\|v_3\|_{H^2}^2\\ &&+\|w_1\|_{H^2}^2+\|w_2\|_{H^2}^2+\|w_3\|_{H^2}^2 +\|\theta-\theta^*\|_{H^2}^2+\|v_{1t}\|^2+\|v_{2t}\|^2+\|v_{3t}\|^2\\ &&+\|w_{1t}\|^2 +\|w_{2t}\|^2+\|w_{3t}\|^2+\|\theta_{t}\|^2)ds\leq C_1. \end{eqnarray}$

(48)

Moreover, there exists constant $\gamma_1 = \gamma_1(C_1)>0$ , for any fixed $\gamma \in (0,\gamma_1]$ and $\forall t>0$

$\begin{eqnarray} &&e^{\gamma t}(\|u-u^*\|^2_{H^1}+\|v_1\|_{H^1}^2+\|v_2\|_{H^1}^2+\|v_3\|_{H^1}^2 +\|w_1\|_{H^1}^2+\|w_2\|_{H^1}^2\\ &&+\|w_3\|_{H^1}^2+\|\theta-\theta^*\|^2_{H^1}) +\int_0^te^{\gamma s}(\|u_x\|^2+\|v_{1x}\|_{H^1}^2+\|v_{2x}\|_{H^1}^2+\|v_{3x}\|_{H^1}^2\\ &&+\|w_{1x}\|_{H^1}^2+\|w_{2x}\|_{H^1}^2+\|w_{3x}\|_{H^1}^2+\|\theta_x\|_{H^1}^2 +\|v_{1t}\| ^2+\|v_{2t}\| ^2+\|v_{3t}\| ^2\\ &&+\|w_{1t}\|^2+\|w_{2t}\|^2+\|w_{3t}\|^2+\|\theta_t\|^2)ds\leq C_1, \end{eqnarray}$

(49)

where $u^*,\theta^*$ and $r^*$ are the same as in Theorem 1.1.

Lemma 2.2. Under the assumptions of Theorem 1.1, the following estimates hold for any $t>0$ :

$\begin{eqnarray} &&\|v_{1t}(t)\|^2 +\|v_{1xx}(t)\|^2+ \|v_{1x}(t)\|_{L^{\infty}}^2+\int_0^t\|v_{1tx}(s)\|^2ds \leq C_2, \end{eqnarray}$

(50)

$\begin{eqnarray} &&\|v_{2t}\|^2 +\|v_{2xx}\|^2+ \|v_{2x}\|_{L^{\infty}}^2+\int_0^t\|v_{2tx}\|^2ds \leq C_2, \end{eqnarray}$

(51)

$\begin{eqnarray} &&\|v_{3t}\|^2 +\|v_{3xx}\|^2+ \|v_{3x}(t)\|_{L^{\infty}}^2+\int_0^t\|v_{3tx}\|^2ds \leq C_2. \end{eqnarray}$

(52)

Proof. Differentiating (31) with respect to $t$ , multiplying the resulting equation by $v_{1t}$ in $L^2(0,L)$ , using an integration by parts, we have

$\begin{eqnarray} \frac {1}{2}\frac{d}{dt}\|v_{1t}\|^2& = &\int_{0}^{L}\left[r((\lambda+2\mu)\frac {(rv_1)_x}{u}-R\frac {\theta}{u})_x\right]_tv_{1t}dx+\int_{0}^{L}(\frac{v_2^2}{r})_tv_{1t}dx\\ &\leq&-(\lambda+2\mu)\int_{0}^{L}\frac{r^2v_{1tx^2}}{u}dx+C_1\|v_{1t}\|(\|v_{1x}\| +\|u_x\|+\|v_{1x}u_x\|\\ &&+\|\theta_x\|+\|v_{1xx}\|+\|v_{2t}\|+\|v_2^2\|)+C_1\|v_{1tx}\|(\|v_{1t}\|+\|v_{1x}^2\|\\ && +\|\theta_{1t}\|+\|v_{1x}\|+\|v_{1x}v_1\|)\\ &\leq&-C_{1}^{-1}\|v_{1tx}\|^2+C_1(\|v_{1x}\|_{H^{1}}^2+\|v_{1t}\|^2+\|v_{2t}\|^2 +\|\theta_{t}\|^2\\ &&+\|u_{x}\|^2+\|\theta_{x}\|^2). \end{eqnarray}$

(53)

Integrating (53) with respect to $t$ over $[0,t]$ ( $t>0$ ), and using Lemma 2.1, there holds

$\begin{eqnarray} \|v_{1t}\|^2 +\int_0^t\|v_{1tx}\|^2ds \leq C_2. \end{eqnarray}$

(54)

Moreover, integrating (31) with respect to $x$ , and using Lemma 2.1 and Young's inequality, we obtain

$\begin{eqnarray} \|v_{1xx}\|&\leq& C_1(\|u_x\|+\|\theta_x\|+\|v_{1t}\|+\|u_{x}\|\|v_{1x}\|_{L^\infty}+\|v_{2}\|^2)\\ &\leq&\frac{1}{2}\|v_{1xx}\|+C_1(\|u_x\|+\|\theta_x\|+\|v_{t}\|+\|v_x\|). \end{eqnarray}$

(55)

Now the above facts along with the Gagliardo-Nirenberg interpolation inequality yields

$\begin{eqnarray} \|v_{1x}\|_{L^\infty}\leq C_1\|v_{1x}\|^{\frac 1 2}\|v_{1xx}\|^{\frac 1 2}+C_1\|v_{1x}\|. \end{eqnarray}$

Combined (54) and (55) to arrive at

$\begin{eqnarray} \|v_{1t}\|^2+\|v_{1xx}\|^2+\|v_{1x}\|_{L^\infty}^2 +\int_0^t\|v_{1tx}\|^2ds\leq C_2. \end{eqnarray}$

Similarly, differentiating (32) with respect to $t$ , multiplying the resulting equations by $v_{2t}$ , and then integrating by parts, we obtain

$\begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|v_{2t}\|^2 = (\mu+\mu_r)\int_0^L\left[r \left(\frac{(rv_2)_x}{u}\right)_x \right]_tv_{2t} dx-\int_0^L \left(\frac{v_1v_2}{r}\right)_tv_{2t}dx\\ &&\qquad\qquad\qquad-2\mu \int_0^L (rw_{3x})_tv_{2t}\\ &&\qquad\qquad\quad\leq-(\mu+\mu_r)\int_0^L \frac{r^2v_{2xt}^2}{u}dx+C_1 \|v_{2t}\|(\|v_{2xx}\|+\|v_{2x}\|+\|u_{x}\|\\ &&\qquad\qquad\qquad+\|v_{2x}v_{1x}\| +\|v_{2t}\|+\|w_{3t}\|)+C_1\|v_{2xt}\|(\|v_{2t}\|+\|v_{2x}v_{1x}\|\\ &&\qquad\qquad\qquad+\|v_{2x}v_1\|+\|v_{1x}v_2\|+\|w_{3t}\|)\\ &&\qquad\qquad\quad\leq-C_1^{-1}\|v_{2tx}\|^2+C_1(\|v_{2x}\|_{H^1}^2+\|v_{2t}\|^2+\|u_{x}\|^2+\|w_{3t}\|^2). \end{eqnarray}$

(56)

We integrate (56) with respect to $t$ , and use Lemma 2.1 to have

$\begin{eqnarray} \|v_{2t}\|^2 +\int_0^t\|v_{2tx}\|^2ds \leq C_2 . \end{eqnarray}$

(57)

Furthermore, integrating (32) with respect to $x$ , and using the same way, we obtain

$\begin{eqnarray} \|v_{2xx}\|&\leq& C_1(\|u_x\|+\|v_{2t}\|+\|w_{3x}\|+\|u_{x}\|\|v_{2x}\|_{L^\infty})\\ &\leq&\frac{1}{2}\|v_{2xx}\|+C_1(\|u_x\|+\|v_{2t}\|+\|v_{2x}\|+\|w_{3x}\|) . \end{eqnarray}$

(58)

We use the Gagliardo-Nirenberg interpolation inequality to give

$\begin{eqnarray} \|v_{2x}\|_{L^\infty}\leq C_1\|v_{2x}\|^{\frac 1 2}\|v_{2xx}\|^{\frac 1 2}+C_1\|v_{2x}\|. \end{eqnarray}$

Combining with (57)-(58), we arrive at

$\begin{eqnarray} \|v_{2t}\|^2+\|v_{2xx}\|^2+\|v_{2x}\|_{L^\infty}^2 +\int_0^t\|v_{2tx}\|^2ds\leq C_2. \end{eqnarray}$

Likewise, differentiating (33) with respect to $t$ , multiplying by $v_{3t}$ and integrating by parts, we have

$\begin{eqnarray} \frac{1}{2}\frac{d}{dt}\|v_{3t}\|^2& = & (\mu+\mu_r)\int_0^L\left[r\left(\frac{(rv_3)_x}{u}\right)_x\right]_tv_{3t}dx +(\mu+\mu_r)\int_0^L\frac{uv_3}{r^2}v_{3t}dx\\ &\leq&-(\mu+\mu_r)\int_0^L\frac{r^2v_{3xt}^2}{u}dx+C_1\|v_{3t}\|(\|v_{3x}\|+\|u_{x}\|+\|v_{3x}u_x\|\\ &&+\|v_{3xx}\|)+C_1\|v_{3tx}\|(\|v_{3t}\|+\|v_{3x}\|+\|w_{2}\|+\|w_{2t}\|)\\ &\leq&-C_1\|v_{3tx}\|^2+C_1(\|v_{3t}\|_{H^1}^2+\|v_{3t}\|^2+\|u_x\|^2+\|w_{2t}\|^2). \end{eqnarray}$

(59)

Integrating (59) with respect to $t$ over $[0,t]$ ( $t>0$ ), and using Lemma 2.1 to have

$\begin{eqnarray} \|v_{3t}\|^2 +\int_0^t\|v_{3tx}\|^2ds \leq C_2. \end{eqnarray}$

(60)

Similarly argument, we integrate (33) with respect to $x$ , and apply Young's inequality and Lemma 2.1 to yield

$\begin{eqnarray} &&\|v_{3xx}\|\leq C_1(\|v_{3t}\|+\|u_{x}\|+\|u_x\|\|v_{3x}\|_{L^\infty})\\ &&\qquad\quad\leq\frac{1}{2}\|v_{3xx}\|+C_1(\|u_{x}\|+\|v_{3t}\|+\|v_{3x}\|). \end{eqnarray}$

(61)

Here, we have from the Gagliardo-Nirenberg interpolation inequality that

$\begin{eqnarray} \|v_{3x}\|_{L^\infty}\leq C_1\|v_{3x}\|^{\frac 1 2}\|v_{3xx}\|^{\frac 1 2}+C_1\|v_{3x}\|, \end{eqnarray}$

which, combined with (60)-(61), gives

$\begin{eqnarray} \|v_{3t}\|^2+\|v_{3xx}\|^2+\|v_{3x}\|_{L^\infty}^2 +\int_0^t\|v_{3tx}\|^2ds\leq C_2. \end{eqnarray}$

Thus, we complete the proof.

Lemma 2.3. The following estimates hold true for any $t>0$

$\begin{eqnarray} &&\|w_{1t}\|^2 +\|w_{1xx}\|^2+ \|w_{1x}\|_{L^{\infty}}^2+\int_0^t\|w_{1tx}\|^2ds \leq C_2, \end{eqnarray}$

(62)

$\begin{eqnarray} &&\|w_{2t}\|^2 +\|w_{2xx}\|^2+ \|w_{2x}\|_{L^{\infty}}^2+\int_0^t\|w_{2tx}\|^2ds \leq C_2, \end{eqnarray}$

(63)

$\begin{eqnarray} &&\|w_{3t}\|^2 +\|w_{3xx}\|^2+ \|w_{3x}\|_{L^{\infty}}^2+\int_0^t\|w_{3tx}\|^2ds \leq C_2. \end{eqnarray}$

(64)

Proof. Differentiating (34) with respect to $t$ , multiplying the resulting identity by $w_{1t}$ and integrating by parts, we obtain

$\begin{eqnarray} &&\frac{1}{2}j_I\frac{d}{dt}\|w_{1t}\|^2\\& = & (c_0+2c_d)\int_0^L\left[r\left(\frac{(rw_1)_x}{u}\right)_x\right]_t w_{1t}dx+j_I\int_0^L\left(\frac{w_2v_2}{r}\right)_tw_{1t}dx\\ &&-4\mu_r\int_0^L(uw_1)_tw_{1t}dx\\ &\leq&-(c_0+2c_d)\int_0^L\frac{r^2w_{1tx}^2}{u}dx+C_1\|w_{1t}\|(\|w_{1x}\|+\|w_{1xx}\| +\|w_{1x}u_x\|\\ &&+\|u_{x}\|+\|w_{1x}v_{1x}\|+\|w_{2t}\|+\|v_{2t}\|)+c_1\|w_{1tx}\|(\|w_{1t}\|+\|w_{1x}v_{1x}\|\\ &&+\|w_{1x}\|+\|w_{1}v_{1x}\|)\\ &\leq&-C_1^{-1}\|w_{1tx}\|^2+C_1(\|w_{1x}\|_{H^1}^2+\|w_{1t}\|^2+\|u_{x}\|^2+\|w_{2t}\|^2+\|v_{2t}\|^2\\ &&+\|v_{1x}\|^2). \end{eqnarray}$

(65)

Integrating (65) with respect to $t$ over $[0,t]$ ( $t>0$ ) and integrating (34) with respect to $x$ , and using Lemma 2.1, respectively, we arrive at

$\begin{eqnarray} &&\|w_{1t}\|^2 +\int_0^t\|w_{1tx}\|^2ds \leq C_2, \end{eqnarray}$

(66)

$\begin{eqnarray} &&\|w_{1xx}\|\leq C_1(\|w_{1t}\|+\|u_{}\|+\|w_{1x}\|_{L^\infty}\|u_{x}\|+ \|w_{2}v_{2}\|+\|w_{1}u\|)\\ &&\qquad\quad \leq\frac{1}{2}\|w_{1xx}\|+C_1(\|w_{1t}\|+\|w_{1x}\|+\|u_{x}\|). \end{eqnarray}$

(67)

We can use the Gagliardo-Nirenberg interpolation inequality to yield

$\begin{eqnarray} \|w_{1x}\|_{L^\infty}\leq C_1\|w_{1x}\|^{\frac 1 2}\|w_{1xx}\|^{\frac 1 2}+C_1\|w_{1x}\|, \end{eqnarray}$

which together with (66)-(67) further implies that

$\begin{eqnarray} \|w_{1t}\|^2+\|w_{1xx}\|^2+\|w_{1x}\|_{L^\infty}^2 +\int_0^t\|w_{1tx}\|^2ds\leq C_2. \end{eqnarray}$

Similarly, differentiating (35) with respect to $t$ , multiplying by $w_{2t}$ and applying integration by parts to find

$\begin{eqnarray} \frac{1}{2}j_I\frac{d}{dt}\|w_{2t}\|^2& = & (c_a+c_d)\int_0^L\left[r\left(\frac{(rw_2)_x}{u}\right)_x\right]_t w_{2t}dx -j_I\int_0^L\left(\frac{w_1v_2}{r}\right)_tw_{2t}dx\\ &&-2\mu_r\int_0^L(rv_{3x})_t w_{2t}dx-4\mu_r\int_0^L(u w_2)_t w_{2t}dx\\ &\leq& -(c_a+c_d)\int_0^L\frac{r^2w_{2xt}}{u}dx+C_1\|w_{2t}\|(\|w_{2x}\|+\|w_{2xx}\|+\|w_{2}u_x\|\\ &&+\|w_{2x}u_x\|+\|w_{2}v_{1x}\|+\|w_{2x}v_{1x}\|+\|w_{2}v_{1x}\| +\|v_{3x}\|+\|v_{3xt}\|)\\ &&+C_1\|w_{2tx}\|(\|w_{2t}\|+\|w_{2}v_{1x}\|+\|w_{2x}v_1\|+\|w_{1t}v_2\| +\|w_{1}v_{2t}\|)\\ &\leq& -C_1^{-1}\|w_{2tx}\|^2+C_1(\|w_{2x}\|_{H^1}^2+\|u_{x}\|^2+\|v_{1x}\|^2+\|w_{2x}\|^2\\ &&+\|v_{3x}\|^2+\|v_{3tx}\|^2+\|w_{2t}\|^2+\|w_{1t}\|^2+\|v_{2t}\|^2). \end{eqnarray}$

(68)

Integrating (68) with respect to $t$ over $[0,t]$ ( $t>0$ ), and using Lemma 2.1, we arrive at

$\begin{eqnarray} \|w_{2t}\|^2 +\int_0^t\|w_{2tx}\|^2ds \leq C_2. \end{eqnarray}$

(69)

We integrate (35) with respect to $x$ , and apply Lemma 2.1 and Young's inequality to obtain

$\begin{eqnarray} \|w_{2xx}\|&\leq& C_1(\|w_{2t}\|+\|w_{2}\|+\|w_{2}u_x\| +\|w_{2x}u_x\|+\|w_{1}v_2\|\\ &&+\|v_{3x}\|+\|w_{2}u\|)\\ &\leq& C_1(\|w_{2t}\|+\|u_{x}\|+\|w_{2x}\|_{L^\infty}\|u_{x}\|+\|v_{3x}\|\\ &&+\|w_{1}\|+\|v_{2}\|+\|u\|+\|w_{2}\|)\\ &\leq& \frac{1}{2}\|w_{2xx}\|+C_1(\|u_{x}\|+\|w_{2t}\|+\|w_{2x}\|+\|v_{3x}\|). \end{eqnarray}$

(70)

Similar argument, we deduce from the Gagliardo-Nirenberg interpolation inequality that

$\begin{eqnarray} \|w_{2x}\|_{L^\infty}\leq C_1\|w_{2x}\|^{\frac 1 2}\|w_{2xx}\|^{\frac 1 2}+C_1\|w_{2x}\|. \end{eqnarray}$

Combining with (69)-(70), we have

$\begin{eqnarray} \|w_{2t}\|^2+\|w_{2xx}\|^2+\|w_{2x}\|_{L^\infty}^2 +\int_0^t\|w_{2tx}\|^2ds\leq C_2. \end{eqnarray}$

Differentiating (36) with respect to $t$ , multiplying the resulting equation by $w_{3t}$ and using integrating by parts, we obtain

$\begin{eqnarray} &&\frac{1}{2}j_I\frac{d}{dt}\|w_{3t}\|^2\\& = & (c_a+c_d)\int_0^L\left[r\left(\frac{(rw_3)_x}{u}\right)_x\right]_tw_{3t}dx+(c_a+c_d)\int_0^L \left(\frac{uw_3}{r^2}\right)_tw_{3t}dx\\ &&+2\mu_r\int_0^L\left[(rv_2)_x\right]_tw_{3t}dx-4\mu_r\int_0^L(uw_3)_tw_{3t}dx \end{eqnarray}$

$\begin{eqnarray} &\leq& -(c_a+c_d)\int_0^L\frac{r^2w_{3tx}^2}{u}dx+C_1\|w_{3t}\|(\|w_{3xx}\| +\|w_{3x}u_x\|+\|u_{x}\|+\|v_{1x}\|\\ &&+\|w_{3t}\|+\|w_{3x}v_{1x}\|+\|v_{2t}\|+\|v_{2x}\|+\|v_{2xt}\|) +C_1\|w_{3tx}\|(\|w_{3}v_{1x}\|\\ &&+\|w_{3t}\|+\|w_{3x}\|+\|w_{3}v_{1x}\|)\\ &\leq& -C_1^{-1}\|w_{3tx}\|^2+C_1(\|w_{3x}\|_{H^1}^2+\|u_{x}\|^2+\|w_{3t}\|^2 +\|v_{1x}\|^2+\|v_{2t}\|^2\\ &&+\|v_{2x}\|^2+\|v_{2tx}\|^2). \end{eqnarray}$

(71)

Integrating (71) with respect to $t$ over $[0,t]$ ( $t>0$ ), we can obtain

$\begin{eqnarray} \|w_{3t}\|^2 +\int_0^t\|w_{3tx}\|^2ds \leq C_2. \end{eqnarray}$

(72)

Next, integrating (36) with respect to $x$ , and then applying the same way to yield

$\begin{eqnarray} \|w_{3xx}\|&\leq& C_1(\|w_{3t}\|+\|u_x\|+\|w_{3x}\|_{L^\infty}\|u_{x}(t)\|)\\ &\leq& \frac{1}{2}\|w_{3xx}\|+C_1(\|w_{3t}\|+\|u_{x}\|+\|w_{3x}\|), \end{eqnarray}$

(73)

where we have used the following simple Gagliardo-Nirenberg interpolation inequality

$\begin{eqnarray} \|w_{3x}\|_{L^\infty}\leq C_1\|w_{3x}\|^{\frac 1 2}\|w_{2xx}\|^{\frac 1 2}+C_1\|w_{3x}\|. \end{eqnarray}$

By (72) and (73), we can get

$\begin{eqnarray} \|w_{3t}\|^2+\|w_{3xx}\|^2+\|w_{3x}\|_{L^\infty}^2 +\int_0^t\|w_{3tx}\|^2ds\leq C_2. \end{eqnarray}$

The proof is complete.

Lemma 2.4. Under the assumptions of Theorem 1.1, the following estimate holds for any $t>0$ :

$\begin{eqnarray} &&\|\theta_{t}\|^2 +\|\theta_{xx}\|^2+ \|\theta_{x}\|_{L^{\infty}}^2+\int_0^t\|\theta_{tx}\|^2ds \leq C_2, \end{eqnarray}$

(74)

Proof. Differentiating (37) with respect to $t$ , multiplying by $\theta_{t}$ , and using integration by parts, we get

$\begin{eqnarray} \frac{1}{2}c_v \frac{d}{dt}\|\theta_{t}\|^2& = & \kappa \int_0^L\left(r^2\frac{\theta_x}{u}\right)_{xt}\theta_{t}dx+\int_0^L\left[\frac{1}{u}[(\lambda+2\mu)(rv_1)_x -R\theta](rv_1)_x\right]_t\theta_tdx\\ &&+(\mu+\mu_r)\int_0^L\left(\frac{(rv_2)^2_x}{u}\right)_t\theta_tdx+(c_0+2c_d)\int_0^L\left(\frac{(rw_1)_x^2}{u} \right)_t\theta_tdx\\ &&+(c_a+c_d)\int_0^L\left(\frac{(rw_2)^2_x}{u}\right)_t\theta_t dx +(\mu+\mu_r)\int_0^L\left(\frac{r^2v_{3x}^2}{u}\right)_t\theta_tdx\\ &&+(c_a+c_d)\int_0^L\left(\frac{r^2w_{3x}^2}{u}\right)_t\theta_t dx-2\mu_r\int_0^L(v_1^2+v_2^2)_{xt}\theta_tdx\\ &&-2c_d\int_0^L(w_1^2+w_2^2)_{xt}\theta_t dx+4\mu_r\int_0^L(uw_1^2+uw_2^2+uw_3^2)_{t}\theta_tdx\\ &&+4\mu_r\int_0^L(rw_2v_{3x})_{t}\theta_t dx-4\mu_r\int_0^L(w_3(rv_2)_x)_{t}\theta_tdx \end{eqnarray}$

$\begin{eqnarray} &\leq&-C_1^{-1}\|\theta_{xt}\|^2+C_1(\|\theta_{x}\|^2+\|v_{1x}\|_{H^1}^2+\|v_{1xt}\|^2 +\|v_{2x}\|_{H^1}^2+\|v_{2xt}\|^2\\ &&+\|v_{3x}\|_{H^1}^2+\|v_{3xt}\|^2+\|w_{1x}\|_{H^1}^2+\|w_{1xt}\|^2+\|w_{2x}\|_{H^1}^2 +\|w_{2xt}\|^2\\ &&+\|w_{3x}\|_{H^1}^2+\|w_{3xt}\|^2+\|u_{x}\|+\|v_{1t}\|^2+\|v_{2t}\|^2+\|v_{3t}\|^2+\|w_{1t}\|^2\\ &&+\|w_{2t}\|^2+\|w_{3t}\|^2+\|\theta_{t}\|^2), \end{eqnarray}$

which, together with (50)-(52), (62)-(64) and Lemma 2.1, gives

$\begin{eqnarray} \|\theta_{t}\|^2 +\|\theta_{xx}\|^2+\int_0^t\|\theta_{tx}\|^2ds \leq C_2. \end{eqnarray}$

(75)

By virtue of the Gagliardo-Nirenberg interpolation inequality and (75), we can obtain (74). The proof is complete.

Lemma 2.5. Under the assumptions of Theorem 1.1, the following estimates hold for any $t>0$ :

$\begin{eqnarray} &&\|u_{xx}\|^2+\int_0^t\|u_{xx}\|^2ds \leq C_2, \end{eqnarray}$

(76)

$\begin{eqnarray} &&\int_0^t(\|v_{1xxx}\|^2+\|v_{2xxx}\|^2+\|v_{3xxx}\|^2 +\|w_{1xxx}\|^2+\|w_{2xxx}\|^2\\ &&+\|w_{3xxx}\|^2+\|\theta_{xxx}\|^2)ds \leq C_2. \end{eqnarray}$

(77)

Proof. Differentiating (31) with respect to $x$ , we have

$\begin{eqnarray} &&v_{1tx} = \left[r\left((\lambda+2\mu)\frac{(rv_1)_x}{u}-R\frac{\theta}{u}\right)_x\right]_x+(\frac{v_2^2}{r})_x\\ &&\quad\quad = \frac{uv_{1t}}{r^2}-\frac{uv_2^2}{r^2}+(\lambda+2\mu)\frac{r(rv_1)_{xxx}}{u}-(\lambda+2\mu)\frac{r(rv_1)_{xx}u_x}{u^2}\\ &&\qquad\quad-(\lambda+2\mu)\frac{r(rv_1)_{xx}u_x}{u^2}-(\lambda+2\mu)\frac{r(rv_1)_{x}u_{xx}}{u^2}+2(\lambda+2\mu)\frac{r(rv_1)_{x}u_x^2}{u^3}\\ &&\qquad\quad-rR(\lambda+2\mu)\frac{\theta_{xx}}{u}+(\lambda+2\mu)rR\frac{\theta_x u_x}{u^2} +(\lambda+2\mu)rR\frac{\theta_x u_x}{u^2}\\ &&\qquad\quad+(\lambda+2\mu)rR\frac{\theta u_{xx}}{u^2}-2(\lambda+2\mu)rR\frac{\theta u_x^2}{u^3}+\frac{2v_2v_{2x}}{r}-\frac{v_2^2u}{r^3}. \end{eqnarray}$

By virtue of (30) $(u_{txx} = (rv_1)_{xxx})$ , we can rewrite the above equation as

$\begin{eqnarray} (\lambda+2\mu)\frac{\partial}{\partial t}(\frac{u_{xx}}{u})+(\lambda+2\mu)R\frac{\theta u_{xx}}{u^2} = r^{-1}v_{1tx}+E(x,t), \end{eqnarray}$

(78)

where

$\begin{eqnarray} E(x,t)& = &-r^{-3}uv_{1t}+r^{-3}uv_2^2+2(\lambda+2\mu)\frac{(rv_1)_{xx}u_x}{u^2}\\ &&-2(\lambda+2\mu)\frac{(rv_1)_{x}u_x^2}{u^3}+R(\lambda+2\mu)\frac{\theta_{xx}}{u}-2(\lambda+2\mu)R\frac{\theta_{x}u_x}{u^2}\\ &&+2(\lambda+2\mu)R\frac{\theta u_x^2}{u^3}-2r^{-2}v_2v_{2x}+r^{-4}v_2^2u. \end{eqnarray}$

Multiplying (78) by $\frac{u_{xx}}{u}$ , and using Young's inequality, by virtue of (50)-(52), (62)-(64), (74) and Lemma 2.1, we can obtain

$\begin{eqnarray} \frac{d}{dt}\|\frac{u_{xx}}{u}\|^2\leq -C_1^1\|\frac{u_{xx}}{u}\|^2+C_1\|E(x,t)\|^2, \end{eqnarray}$

(79)

where

$\begin{eqnarray} \|E(x,t)\|^2&\leq& \frac{1}{2C_1}\|\frac{u_{xx}}{u}\|^2+C_1(\|v_{1t}\|^2+\|v_{1x}\|_{H^1}^2+\|u_{x}\|^2 +\|\theta_{x}\|_{H^1}^2\\ &&+\|v_{2x}\|^2). \end{eqnarray}$

(80)

Integrating (79) with respect to $t$ , together with (50)-(52), (62)-(64), (74) and Lemma 2.1 that

$\begin{eqnarray} \|u_{x}\|^2+\int_0^L\|u_{xx}\|^2ds\leq C_2. \end{eqnarray}$

On the other hand, differentiating (31) with respect to $x$ , we have from Lemma 2.1 and (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality that

$\begin{eqnarray} &&v_{1tx} = \frac{uv_{1t}}{r^2}-\frac{uv_2^2}{r^2}+(\lambda+2\mu)r\frac{(rv_1)_{xxx}}{u} -2(\lambda+2\mu)r\frac{(rv_1)_{xx}u_x}{u^2}\\ &&\quad\qquad-(\lambda+2\mu)r\frac{(rv_1)_x u_{xx}}{u^2}+2(\lambda+2\mu)r\frac{(rv_1)_x u_x^2}{u^3}-rR(\lambda+2)\frac{\theta_{xx}}{u}\\ &&\qquad\quad +2(\lambda+2\mu)rR\frac{\theta_x u_x}{u^2}+(\lambda+2\mu)rR\frac{\theta u_{xx}}{u^2}-2(\lambda+2\mu)rR\frac{\theta u_x^2}{u^3}\\ &&\qquad\quad+\frac{2v_2v_{2x}}{r}-\frac{v_2^2 u}{r^3}, \end{eqnarray}$

which implies

$\begin{eqnarray} \|v_{1tx}\| \leq C_1(\|v_{1x}\|_{H^2}+\|u_{x}\|_{H^1} +\|\theta_{x}\|_{H^1}+\|v_{2x}\|), \end{eqnarray}$

$\begin{eqnarray} \|v_{1xxx}\| \leq C_1(\|v_{1x}\|_{H^1}+\|v_{1tx}\|+\|u_{x}\|_{H^1} +\|\theta_{x}\|_{H^1}+\|v_{2x}\|). \end{eqnarray}$

Differentiating (32) with respect to $x$ , using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to find

$\begin{eqnarray} &&v_{2tx} = (\mu+\mu_r)\frac{r_x (rv_2)_{xx}}{u}-(\mu+\mu_r)\frac{r_x (rv_2)_{x}u_x}{u^2} +(\mu+\mu_r)\frac{r (rv_2)_{xxx}}{u^3}\\ &&\qquad\quad-(\mu+\mu_r)\frac{r (rv_2)_{xx} u_x}{u^2}-(\mu+\mu_r)\frac{r (rv_2)_{xx} u_x}{u^2} -(\mu+\mu_r)\frac{r (rv_2)_{x} u_xx}{u^2}\\ &&\qquad\quad+2(\mu+\mu_r)\frac{r (rv_2)_{x} u_x^2}{u^3} -\frac{(v_{2x}v_1+v_2v_{1x})r-v_2v_1r_x}{r^2}\\ &&\qquad\quad-2\mu_r(r_xw_{3x}+rw_{3xx}), \end{eqnarray}$

thus

$\begin{eqnarray} \|v_{2tx}\| \leq C_1(\|v_{2x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{1x}\|+\|w_{3x}\|_{H^1}), \end{eqnarray}$

$\begin{eqnarray} \|v_{2xxx}\| \leq C_1(\|v_{2x}\|_{H^1}+\|v_{2tx}\|+\|u_{x}\|_{H^1} +\|w_{3x}\|_{H^1}+\|v_{1x}\|). \end{eqnarray}$

Similarly, differentiating (33) with respect to $x$ , and applying Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to deduce

$\begin{eqnarray} &&v_{3tx} = (\mu+\mu_r)\frac{r_x (rv_3)_{xx}}{u}-(\mu+\mu_r)\frac{r_x (rv_3)_{x}u_x}{u^2} +(\mu+\mu_r)\frac{r (rv_3)_{xxx}u-(rv_3)_{xx}u_x}{u^2}\\ &&-(\mu+\mu_r)r\frac{ \left[(rv_3)_{xx}u_x+(rv_3)_x u_{xx}\right]u^2-2(rv_3)_x u_x^2u}{u^4}+(\mu+\mu_r)\frac{u_xv_3+uv_{3x} }{r^2}\\ &&-(\mu+\mu_r)\frac{2uv_3r_x}{r^3}+2\mu_r(r_{xx}w_2+r_xw_{2x}+r_xw_{2x}+rw_{2xx}), \end{eqnarray}$

thus

$\begin{eqnarray} \|v_{3tx}\| \leq C_1(\|v_{3x}\|_{H^2}+\|u_{x}\|_{H^1} +\|w_{2x}\|_{H^1}), \end{eqnarray}$

$\begin{eqnarray} \|v_{3xxx}\| \leq C_1(\|v_{3x}\|_{H^1}+\|v_{3tx}\|+\|u_{x}\|_{H^1} +\|w_{2x}\|_{H^1}). \end{eqnarray}$

We can differentiate (34) with respect to $x$ , combine Lemma 2.1 and the Gagliaro-Nirenberg inequality to get

$\begin{eqnarray} &&j_I w_{1tx} = (c_0+2c_d)r_x\frac{(r w_1)_{xx}}{u}-(c_0+2c_d)r_x\frac{(r w_1)_{x}u_x}{u^2}\\ &&\qquad\qquad+(c_0+2c_d)r\frac{(r w_1){x x x}u-(r w_1)_{xx}u_x)_{xx}}{u^2}\\ &&\qquad\qquad-(c_0+2c_d)r\frac{\left[(r w_1)_{xx}u_x+(r w_1)_x u_{xx}\right]u^2-2(r w_1)_x u_x^2 u}{u^4}\\ &&\qquad\qquad+j_I\frac{w_{2x}v_2+w_2v_{2x}}{r}-j_I\frac{w_2 v_2 r_x}{r^2} -4\mu_r(u_x w_1+u w_{1x}). \end{eqnarray}$

Then

$\begin{eqnarray} \|w_{1tx}\| \leq C_1(\|w_{1x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|+\|w_{2x}\|), \end{eqnarray}$

$\begin{eqnarray} \|w_{1xxx}\| \leq C_1(\|w_{1x}\|_{H^1}+\|w_{1tx}\|+\|u_{x}\|_{H^1} \|w_{2x}\|+\|v_{2x}\|). \end{eqnarray}$

Differentiating (35) with respect to $x$ , using Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality to yield

$\begin{eqnarray} &&j_Iw_{2tx} = (c_a+c_d)r_x\frac{(r w_2)_{xx}}{u}-(c_a+c_d)r_x\frac{(rw_2)_{x}u_x}{u^2}\\ &&\qquad\qquad+(c_a+c_d)r\frac{(rw_2)_{xxx}u-(r w_2)_{xx}u_x}{u^2}-j_I\frac{w_{1x}v_2+w_1v_{2x}}{r}\\ &&\qquad\qquad-(c_a+c_d)r\frac{\left[(r w_2)_{xx}u_x+(rw_2)_x u_{xx}\right]u^2-2(rw_2)_x u_x^2u}{u^4}\\ &&\qquad\qquad+j_I\frac{w_1v_2r_x}{r^2}-2\mu_r(r_xw_{3xx}+rv_{3xx})-4\mu_r(u_xw_2+uw_{2x}), \end{eqnarray}$

thus

$\begin{eqnarray} &&\|w_{2tx}\| \leq C_1(\|w_{2x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|+\|w_{1x}\|\\ &&\quad+\|w_{3x}\|+\|v_{3x}\|_{H^1}), \end{eqnarray}$

$\begin{eqnarray} &&\|w_{2xxx}\| \leq C_1(\|w_{2x}\|_{H^1}+\|w_{2tx}\|+\|u_{x}\|_{H^1} +\|w_{1x}\|+\|v_{2x}\|\\ &&\quad+\|w_{3x}\|+\|v_{3x}\|_{H^1}). \end{eqnarray}$

Likewise, we differentiate (36) with respect to $x$ , apply Lemma 2.1 and the Gagliaro-Nirenberg inequality to arrive at

$\begin{eqnarray} &&j_Iw_{3xt} = (c_a+c_d)r_x\frac{(rw_3)_{xx}}{u}-(c_a+c_d)r_x\frac{(rw_3)_{x}u_x}{u^2}\\ &&\qquad\qquad+(c_a+c_d)r\frac{(rw_3)_{xxx}u-(rw_3)_{xx}u_x}{u^2} -(c_a+c_d)r\frac{(rw_3)_{xx}u_x+(rw_3)_x u_{xx}}{u^2}\\ &&\qquad\qquad+(c_a+c_d)r\frac{2(rw_3)_{x}u_x^2}{u^3} +(c_a+c_d)\frac{u_xw_{3}+u w_{3x}}{r^2} -2(c_a+c_d)\frac{u^2w_3}{r^4}\\ &&\qquad\qquad+2\mu_r(r_{xx}v_2+r_xv_{2x}+r_xv_{2x}+rv_{2xx}) -4\mu_r(u_xw_3+uw_{3x}). \end{eqnarray}$

Then we have

$\begin{eqnarray} \|w_{3tx}\| \leq C_1(\|w_{3x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|_{H^1}), \end{eqnarray}$

$\begin{eqnarray} \|w_{3xxx}\| \leq C_1(\|w_{3x}\|_{H^1}+\|w_{3tx}\|+\|u_{x}(t)\|_{H^1} +\|v_{2x}\|_{H^1}). \end{eqnarray}$

We can differentiate (37) with respect to $x$ , and easily deduce from Lemma 2.1, (50)-(52), (62)-(64), (74), (76) and the Gagliaro-Nirenberg inequality that

$\begin{eqnarray} &&\|\theta_{tx}\| \leq C_2(\|\theta_{x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{1x}\|_{H^1}+\|v_{2x}\|_{H^1}+\|v_{3x}\|_{H^1}\\ &&\quad+\|w_{1x}\|_{H^1}+\|w_{2x}\|_{H^1}+\|w_{3x}\|_{H^1}), \end{eqnarray}$

$\begin{eqnarray} &&\|\theta_{xxx}\| \leq C_2(\|\theta_{x}\|_{H^1}+\|\theta_{xt}\|+\|u_{x}\|_{H^1} +\|v_{1x}\|_{H^1}+\|v_{2x}\|_{H^1}\\ &&\quad+\|v_{3x}\|_{H^1}+\|w_{1x}\|_{H^1}+\|w_{2x}\|_{H^1}+\|w_{3x}\|_{H^1}). \end{eqnarray}$

According to the estimate above, we have

$\begin{eqnarray} &&\|v_{1tx}\| \leq C_2(\|v_{1x}\|_{H^2}+\|u_{x}\|_{H^1} +\|\theta_{x}\|_{H^1}+\|v_{2x}\|), \end{eqnarray}$

(81)

$\begin{eqnarray} &&\|v_{2tx}\| \leq C_2(\|v_{2x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{1x}\|+\|w_{3x}\|_{H^1}), \end{eqnarray}$

(82)

$\begin{eqnarray} &&\|v_{3tx}\| \leq C_2(\|v_{3x}\|_{H^2}+\|u_{x}\|_{H^1} +\|w_{2x}\|_{H^1}), \end{eqnarray}$

(83)

$\begin{eqnarray} &&\|w_{1tx}\| \leq C_2(\|w_{1x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|+\|w_{2x}\|), \end{eqnarray}$

(84)

$\begin{eqnarray} &&\|w_{2tx}\| \leq C_2(\|w_{2x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|+\|w_{1x}\|\\ &&\qquad\qquad+\|w_{3x}\|+\|v_{3x}\|_{H^1}), \end{eqnarray}$

(85)

$\begin{eqnarray} &&\|w_{3tx}\| \leq C_2(\|w_{3x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{2x}\|_{H^1}), \end{eqnarray}$

(86)

$\begin{eqnarray} &&\|\theta_{tx}\| \leq C_2(\|\theta_{x}\|_{H^2}+\|u_{x}\|_{H^1} +\|v_{1x}\|_{H^1}+\|v_{2x}\|_{H^1}\\ &&\qquad\qquad+\|v_{3x}\|_{H^1}+\|w_{1x}\|_{H^1}+\|w_{2x}\|_{H^1}+\|w_{3x}\|_{H^1}), \end{eqnarray}$

(87)

$\begin{eqnarray} &&\|v_{1xxx}\| \leq C_2(\|v_{1x}\|_{H^1}+\|v_{1tx}\|+\|u_{x}\|_{H^1} +\|\theta_{x}\|_{H^1}+\|v_{2x}\|), \end{eqnarray}$

(88)

$\begin{eqnarray} &&\|v_{2xxx}\| \leq C_2(\|v_{2x}\|_{H^1}+\|v_{2tx}\|+\|u_{x}\|_{H^1} +\|w_{3x}\|_{H^1}+\|v_{1x}\|), \end{eqnarray}$

(89)

$\begin{eqnarray} &&\|v_{3xxx}\| \leq C_2(\|v_{3x}\|_{H^1}+\|v_{3tx}\|+\|u_{x}\|_{H^1} +\|w_{2x}\|_{H^1}), \end{eqnarray}$

(90)

$\begin{eqnarray} &&\|w_{1xxx}\| \leq C_2(\|w_{1x}\|_{H^1}+\|w_{1tx}\|+\|u_{x}\|_{H^1} \|w_{2x}\|+\|v_{2x}\|), \end{eqnarray}$

(91)

$\begin{eqnarray} &&\|w_{2xxx}\| \leq C_2(\|w_{2x}\|_{H^1}+\|w_{2tx}\|+\|u_{x}\|_{H^1} +\|w_{1x}\|+\|v_{2x}\|\\ &&\qquad\qquad\quad+\|w_{3x}\|+\|v_{3x}\|_{H^1}), \end{eqnarray}$

(92)

$\begin{eqnarray} &&\|w_{3xxx}\| \leq C_2(\|w_{3x}\|_{H^1}+\|w_{3tx}\|+\|u_{x}\|_{H^1} +\|v_{2x}\|_{H^1}), \end{eqnarray}$

(93)

$\begin{eqnarray} &&\|\theta_{xxx}\| \leq C_2(\|\theta_{x}\|_{H^1}+\|\theta_{xt}\|+\|u_{x}\|_{H^1} +\|v_{1x}\|_{H^1}+\|v_{2x}\|_{H^1}\\ &&\qquad\quad\quad+\|v_{3x}\|_{H^1}+\|w_{1x}\|_{H^1}+\|w_{2x}\|_{H^1}+\|w_{3x}\|_{H^1}). \end{eqnarray}$

(94)

By (50)-(52), (62)-(64), (74), (76), (81)-(87) and Lemma 2.1, we get (77). The proof is complete.

Lemma 2.6. Under assumptions of Theorem 1.1, there exists a positive constant $\gamma_2 = \gamma_2(C_2)$ such that for any fixed $\gamma\in(0,\gamma_2]$ , the following estimate holds for any $t>0$

$\begin{eqnarray} && e^{\gamma t}(\|v_{1xx}\|^2+\|v_{2xx}\|^2+\|v_{3xx}\|^2+\|w_{1xx}\|^2 +\|w_{2xx}\|^2\\ &&\qquad+\|w_{3xx}\|^2+\|u_{xx}\|^2+\|\theta_{xx}\|^2+\|v_{1t}\|^2+\|v_{2t}\|^2\\ &&\qquad +\|v_{3t}\|^2+\|w_{1t}\|^2+\|w_{2t}\|^2+\|w_{3t}\|^2 +\|\theta_{t}\|^2)\\ &&\qquad+\int_0^t e^{\gamma s}(\|v_{1x}\|_{H^2}^2+\|v_{2x}\|_{H^2}^2 +\|v_{3x}\|_{H^2}^2+\|w_{1x}\|_{H^2}^2\\ &&\qquad+\|w_{2x}\|_{H^2}^2+\|w_{3x}\|_{H^2}^2+\|\theta_{x}\|_{H^2}^2+\|u_{x}\|_{H^1}^2 +\|v_{1tx}\|^2\\ &&\qquad+\|v_{2tx}\|^2+\|v_{3tx}\|^2+\|w_{1tx}\|^2+\|w_{2tx}\|^2+\|w_{3tx}\|^2\\ &&\qquad+\|\theta_{tx}\|^2)ds \leq C_1 . \end{eqnarray}$

(95)

Proof. We multiply (53) by $e^{\gamma t}$ , integrate the resulting over $[0,t]$ , and then apply Lemma 2.1 and Lemma 2.2 to deduce

$\begin{eqnarray} &&e^{\gamma t}\|v_{1t}\|^2+\int_0^t e^{\gamma s}\|v_{1tx}\|^2ds\\ &&\leq C_2+C_1\int_0^te^{\gamma s}(\|v_{1x}\|_{H^1}^2+\|v_{1t}\|^2 +\|v_{2t}\|^2+\|\theta_{t}\|^2+\|u_{x}\|^2+\|\theta_{x}\|^2)ds\\ && \leq C_2. \end{eqnarray}$

(96)

Multiplying (56), (59), (65), (68), (71) and (75) by $e^{\gamma t}$ , and adding them together, by virtue of Lemmas 2.1-2.2, we can conclude

$\begin{eqnarray} &&e^{\gamma t}(\|v_{2t}\|^2+\|v_{3t}\|^2+\|w_{1t}\|^2+\|w_{2t}\|^2 +\|w_{3t}\|^2+\|\theta_{t}\|^2)\\ &&\qquad+\int_0^te^{\gamma s}(\|v_{1tx}\|^2+\|v_{2tx}\|^2+\|v_{3tx}\|^2 +\|w_{1tx}\|^2+\|w_{2tx}\|^2\\ &&\qquad+\|w_{3tx}\|^2+\|\theta_{tx}\|^2)ds\leq C_2. \end{eqnarray}$

(97)

On the other hand, multiplying (79) by $e^{\gamma t}$ , integrating over $[0,t]$ to have

$\begin{eqnarray} &&e^{\gamma t}\|\frac{u_{xx}}{u}\|^2+\frac{1}{2C_1}\int_0^te^{\gamma s}\|\frac{u_{xx}}{u}\|^2ds\\ &&\qquad\leq C_2+\gamma\int_0^te^{\gamma s}\|\frac{u_{xx}}{u}\|^2+C_1\int_0^te^{\gamma s}(\|v_{1x}\|_{H^1}^2+\|u_{x}\|^2+\|v_{1t}\|^2\\ &&\qquad+\|\theta_{x}\|_{H^1}^2+\|v_{2x}\|^2)ds . \end{eqnarray}$

(98)

Picking $\gamma_2 = min\{\frac{1}{4C_1},\gamma_1\}$ such that for any fixed $\gamma\in(0,\gamma_2]$ , using Lemmas 2.1 - 2.2 and (96)-(97), we obtain

$\begin{eqnarray} e^{\gamma t}\|u_{xx}\|^2+\int_0^te^{\gamma s}\|u_{xx}\|^2ds \leq C_2. \end{eqnarray}$

(99)

By (31)-(37), (81)-(87), (96)-(97) and (99), we have

$\begin{eqnarray} && e^{\gamma t}(\|v_{1xx}\|^2+\|v_{2xx}\|^2+\|v_{3xx}\|^2 +\|v_{1xx}\|^2+\|w_{1xx}\|^2\\ &&\qquad+\|w_{2xx}\|^2+\|w_{3xx}\|^2+\|\theta_{xx}\|^2)+\int_0^te^{\gamma s}(\|v_{1xx}\|^2+\|v_{2xx}(t)\|^2\\ &&\qquad+\|v_{3xx}\|^2 +\|w_{1xx}\|^2+\|w_{2xx}\|^2+\|w_{3xx}\|^2+\|\theta_{xx}\|^2)\leq C_2, \end{eqnarray}$

which, combined with (96)-(97) and (99), yields (95). The proof is complete.

Proof of Theorem 1.1. According to Lemmas 2.2-2.6, Theorem 1.1 is complete.

Acknowledgments

The authors are grateful to the referees for their helpful suggestions which improved the presentation of this paper. This research was supported by the NSFC (No. 11871212 and No. 11501199) and Key Research Projects in Colleges of Henan Province (No. 20A110026).

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