In this section, we will give some useful a priori estimates of the solutions to complete the proofs of the theorems.
3.1 Global existence of \(H^{1}\)-solution
In this subsection, we shall complete the proof of Theorem 2.1. As in [1], we have the following mass conservation and energy-entropy inequality.
Lemma 3.1
Under the assumptions in Theorem
2.1, the following estimates hold, for any
\(t\in[0,T]\),
$$\begin{aligned}& \int_{0}^{M}\eta(x,t)\,dx= \int_{0}^{M}\eta_{0}(x)\,dx, \end{aligned}$$
(3.1)
$$\begin{aligned}& \int_{0}^{M} \biggl(\frac{1}{2}v^{2}+ \frac{A}{2(\beta-1)}\eta^{\beta-1}(\theta -\theta_{\Gamma})^{2} \biggr)\,dx \\& \quad{}+ \int_{0}^{t} \int_{0}^{M} \biggl(\frac{\kappa(\eta,\theta )r^{4}}{\eta\theta^{2}} \theta_{x}^{2}+\frac{\mu}{\eta\theta}\bigl(\bigl(r^{2}v \bigr)_{x}\bigr)^{2} \biggr)\,dx\,ds\leq C_{1}. \end{aligned}$$
(3.2)
Proof
See, e.g., Lemma 1 in [1]. □
Lemma 3.2
Under the assumptions in Theorem
2.1, the following estimates hold for all
\((x,t)\in\Omega\times[0,T]\):
$$\begin{aligned} 0< C_{1}^{-1}\leq\eta(x,t)\leq C_{1},\qquad 0< C_{1}^{-1}\leq\theta(x,t)\leq C_{1}. \end{aligned}$$
(3.3)
Proof
See, e.g., Propositions 2 and 5 in [1]. □
Lemma 3.3
Under the assumptions in Theorem
2.1, the following estimate holds for any
\(t\in[0,T]\):
$$\begin{aligned} \bigl\Vert \eta_{x}(t)\bigr\Vert ^{2}+\bigl\Vert v_{x}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{x}(t) \bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert \eta_{x}\Vert ^{2}+\Vert v_{xx}\Vert ^{2}+\Vert \theta_{t}\Vert ^{2}\bigr) (s)\,ds\leq C_{1}. \end{aligned}$$
(3.4)
Proof
See, e.g., Propositions 3-5 and Lemma 5 in [1]. □
Lemma 3.4
Under the assumptions in Theorem
2.1, the following estimate holds for any
\(t\in[0,T]\):
$$\begin{aligned} \int_{0}^{t}\bigl(\Vert \theta_{xx} \Vert ^{2}+\Vert v_{t}\Vert ^{2}\bigr) (s)\,ds \leq C_{1}(T). \end{aligned}$$
(3.5)
Proof
Multiplying (1.9) by \(v_{t}\) over \((0,M)\times (0,T)\), employing an integration by parts and using Lemmas 3.1-3.3 and the Young inequality, we have
$$\begin{aligned} &\bigl\Vert \bigl(r^{2}v\bigr)_{x}\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds \\ &\quad\leq C_{1}+C_{1} \int_{0}^{t} \int_{0}^{M} \bigl(\bigl\vert v_{t} \bigl(-r^{2}p_{x}+f\bigr)\bigr\vert +\bigl\vert \bigl(r^{2}v\bigr)_{x}\bigr\vert ^{3}+\vert vv_{x}\vert \bigr)\,dx\,ds \\ &\quad\leq C_{1}+\frac{1}{2} \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds +C_{1} \int_{0}^{t} \int_{0}^{M}\bigl(\theta_{x}^{2}+ \eta _{x}^{2}+f^{2}+v^{2}+v_{x}^{2}+ \bigl\vert \bigl(r^{2}v\bigr)_{x}\bigr\vert ^{3} \bigr)\,dx\,ds \\ &\quad\leq C_{1}(T)+\frac{1}{2} \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds +C_{1} \int_{0}^{t}\bigl\Vert \bigl(r^{2}v \bigr)_{x}\bigr\Vert _{L^{3}}^{3}\,ds \\ &\quad\leq C_{1}(T)+\frac{1}{2} \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds +C_{1} \int_{0}^{t}\bigl\Vert \bigl(r^{2}v \bigr)_{x}\bigr\Vert ^{2}\bigl\Vert \bigl(r^{2}v \bigr)_{xx}\bigr\Vert \,ds \\ &\quad\leq C_{1}(T)+\frac{1}{2} \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds +C_{1} \int_{0}^{t}\bigl\Vert \bigl(r^{2}v \bigr)_{xx}\bigr\Vert ^{2}\,ds \\ &\quad\leq C_{1}(T)+\frac{1}{2} \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds, \end{aligned}$$
which implies
$$\begin{aligned} \bigl\Vert \bigl(r^{2}v\bigr)_{x}\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{t}(s)\bigr\Vert ^{2}\,ds\leq C_{1}(T). \end{aligned}$$
(3.6)
Equation (1.10) can be rewritten as
$$ e_{\theta}\theta_{t}=Q_{x}-\theta p_{\theta}\bigl(r^{2}v\bigr)_{x}+\frac{\mu}{\eta}\bigl(r^{2}v \bigr)_{x}^{2}. $$
(3.7)
Multiplying (3.7) by \(e_{\theta}^{-1}\theta_{xx}\), then integrating the result with respect to x over \((0,M)\), using Hölder’s inequality, the Sobolev embedding theorem, and Lemmas 3.1-3.3, we have, for any \(\varepsilon>0\),
$$\begin{aligned} &\frac{d}{dt}\bigl\Vert \theta_{x}(t)\bigr\Vert ^{2}+2 \int_{0}^{M}\frac{r^{4}\kappa}{e_{\theta}\eta }\theta_{xx}^{2}\,dx \\ &\quad= \int_{0}^{M} \biggl( \biggl(\frac{r^{4}\kappa}{\eta} \biggr)_{x}\theta_{x}-\theta p_{\theta}\bigl(r^{2}v\bigr)_{x}+\frac{\mu}{\eta}\bigl(r^{2}v \bigr)_{x}^{2} \biggr)\frac{\theta _{xx}}{e_{\theta}}\,dx \\ &\quad\leq\varepsilon \Vert \theta_{xx}\Vert ^{2}+C_{1}( \varepsilon) \bigl(\Vert \theta_{x}\Vert ^{2}+\Vert \eta_{x}\theta_{x}\Vert ^{2}+\Vert \theta_{x}\Vert _{L^{4}}^{4}+\Vert v_{x} \Vert ^{2}+\Vert v\Vert _{L^{4}}^{4}+\Vert v_{x}\Vert _{L^{4}}^{4}\bigr) \\ &\quad\leq\varepsilon \Vert \theta_{xx}\Vert ^{2}+C_{1}( \varepsilon) \bigl(\Vert \theta_{x}\Vert ^{2}+\Vert \theta_{x}\Vert _{L^{\infty}}^{2}+\Vert \theta_{x}\Vert ^{3}\Vert \theta_{xx}\Vert + \Vert v_{x}\Vert ^{2}+\Vert v\Vert ^{3} \Vert v_{x}\Vert +\Vert v_{x}\Vert ^{3} \Vert v_{xx}\Vert \bigr) \\ &\quad\leq2\varepsilon \Vert \theta_{xx}\Vert ^{2}+C_{1}( \varepsilon) \bigl(\Vert \theta_{x}\Vert ^{2}+\Vert v \Vert ^{2}+\Vert v_{x}\Vert ^{2}+\Vert v_{xx}\Vert ^{2}\bigr). \end{aligned}$$
(3.8)
Integrating (3.8) with respect to t over \((0,t)\), taking \(\varepsilon >0\) small enough, and using Lemmas 3.1 and 3.3, we can obtain
$$\begin{aligned} \bigl\Vert \theta_{x}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \theta_{xx}(s) \bigr\Vert ^{2}\,ds\leq C_{1} , \end{aligned}$$
(3.9)
which, along with (3.6), leads to the estimate (3.5). □
Now combining Lemmas 3.1-3.4 and noting equation (1.8), we complete the proof of Theorem 2.1.
3.2 Global existence of \(H^{2}\)-solution
In this subsection, we shall deal with the \(H^{2}\)-regularity of the global solutions to problem (1.8)-(1.13).
Lemma 3.5
Under the assumptions in Theorem
2.2, the following estimate holds for any
\(t\in[0,T]\):
$$\begin{aligned} \bigl\Vert v_{xx}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{xx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{t}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{t}(t) \bigr\Vert ^{2}+ \int _{0}^{t}\bigl(\Vert v_{xt}\Vert ^{2}+\Vert \theta_{xt}\Vert ^{2}\bigr) (s)\,ds \leq C_{2}(T). \end{aligned}$$
(3.10)
Proof
See, e.g., Proposition 6 in [1]. □
Lemma 3.6
Under the assumptions in Theorem
2.2, the following estimate holds for any
\(t\in[0,T]\):
$$\begin{aligned} \bigl\Vert \eta_{xx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \eta_{xx}(s)\bigr\Vert ^{2}\,ds\leq C_{2}(T). \end{aligned}$$
(3.11)
Proof
Differentiating (1.9) with respect to x, we have
$$\begin{aligned} \mu\frac{d}{dt} \biggl(\frac{\eta_{xx}}{\eta} \biggr)-p_{\eta}\eta _{xx} =&\bigl(r^{-2}v_{t}\bigr)_{x}+p_{\theta}\theta_{xx}+p_{\eta\eta}\eta_{x}^{2}+p_{\theta \theta} \theta_{x}^{2} \\ &{} +2p_{\eta\theta}\eta_{x}\theta_{x}+2\mu \frac{\eta_{x}}{\eta} \biggl(\frac {(r^{2}v)_{x}}{\eta} \biggr)_{x}- \bigl(r^{-2}f\bigr)_{x} \\ =:&\mathcal{M}, \end{aligned}$$
(3.12)
where
$$\begin{aligned} \Vert \mathcal{M}\Vert \leq C_{1}(T) \bigl(\Vert \theta_{x}\Vert _{H^{1}}+\Vert v_{t}\Vert + \Vert v_{xt}\Vert +\Vert \eta _{x}\Vert _{L^{4}}^{2}+\Vert v_{x}\Vert _{H^{1}}+1 \bigr). \end{aligned}$$
By Theorem 2.1 and Lemma 3.5, using Young’s inequality, we get, for any \(\varepsilon>0\),
$$\begin{aligned} \int_{0}^{t}\Vert \mathcal{M}\Vert ^{2}\,ds\leq C_{2}(T)+\varepsilon \int_{0}^{t}\bigl\Vert \eta _{xx}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.13)
Multiplying (3.12) by \(\frac{\eta_{xx}}{\eta}\), then integrating the result over \([0,M]\times[0,t]\) and using Young’s inequality and (3.13), taking \(\varepsilon>0\) sufficiently small, we can obtain (3.11). Thus we complete the proof. □
Lemma 3.7
Under the assumptions in Theorem
2.2, the following estimate holds for any
\(t\in[0,T]\):
$$\begin{aligned} \int_{0}^{t}\bigl(\Vert v_{xxx}\Vert ^{2}+\Vert \theta_{xxx}\Vert ^{2}\bigr) (s)\,ds \leq C_{2}(T). \end{aligned}$$
(3.14)
Proof
Differentiating (1.9) and (1.10) with respect to x, respectively, and using the Cauchy inequality, we easily obtain
$$\begin{aligned} \bigl\Vert v_{xxx}(t)\bigr\Vert \leq C_{1}(T) \bigl( \bigl\Vert v_{xt}(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{1}} \bigr) \end{aligned}$$
(3.15)
and
$$\begin{aligned} \bigl\Vert \theta_{xxx}(t)\bigr\Vert \leq C_{1}(T) \bigl(\bigl\Vert \theta_{xt}(t)\bigr\Vert +\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t) \bigr\Vert _{H^{1}}\bigr). \end{aligned}$$
(3.16)
By virtue of Theorem 2.1 and Lemmas 3.5-3.6, we complete the proof. □
Now combining Lemmas 3.5-3.7, we have completed the proof of Theorem 2.2.
3.3 Global existence of \(H^{4}\)-solution
In this subsection, we shall complete the proof of Theorem 2.3, which can be divided into the following lemmas.
Lemma 3.8
Under the assumptions of Theorem
2.3, we see that for any
\(t\in[0,T]\)
and for
\(\varepsilon>0\)
small enough,
$$\begin{aligned} &\bigl\Vert v_{xt}(x,0)\bigr\Vert +\bigl\Vert \theta_{xt}(x,0)\bigr\Vert +\bigl\Vert v_{tt}(x,0)\bigr\Vert +\bigl\Vert \theta _{tt}(x,0)\bigr\Vert \\ &\quad{} +\bigl\Vert v_{txx}(x,0)\bigr\Vert +\bigl\Vert \theta_{txx}(x,0)\bigr\Vert \leq C_{4}(T), \end{aligned}$$
(3.17)
$$\begin{aligned} &\bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert v_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T)+C_{2}(T) \int _{0}^{t}\bigl(\Vert \theta_{txx} \Vert ^{2}+\Vert v_{txx}\Vert ^{2}\bigr) (s)\,ds, \end{aligned}$$
(3.18)
$$\begin{aligned} &\bigl\Vert \theta_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \theta_{ttx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}(T)+C_{2}(T) \varepsilon^{-1} \int_{0}^{t}\bigl\Vert \theta_{txx}(s) \bigr\Vert ^{2}\,ds \\ &\hphantom{\bigl\Vert \theta_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \theta_{ttx}(s) \bigr\Vert ^{2}\,ds\leq}{} +C_{1}\varepsilon \int_{0}^{t}\bigl(\Vert v_{ttx}\Vert ^{2}+\Vert v_{txx}\Vert ^{2}\bigr) (s)\,ds. \end{aligned}$$
(3.19)
Proof
Differentiating (1.9) and (1.10) with respect to x, respectively, using Theorems 2.1 and 2.2, we can get
$$\begin{aligned}& \bigl\Vert v_{xt}(t)\bigr\Vert \leq C_{2}(T) \bigl( \bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+1\bigr), \end{aligned}$$
(3.20)
$$\begin{aligned}& \bigl\Vert \theta_{xt}(t)\bigr\Vert \leq C_{2}(T) \bigl(\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \theta_{x}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.21)
Similarly, differentiating (1.9) and (1.10) with respect to x twice, respectively, we can infer from Theorems 2.1 and 2.2 that
$$\begin{aligned} \bigl\Vert v_{xxt}(t)\bigr\Vert \leq&C_{2}(T) \bigl( \bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert _{L^{\infty}}\bigl\Vert \eta_{xxx}(t) \bigr\Vert \\ &{} +\bigl\Vert \eta_{x}(t)\bigr\Vert _{L^{\infty}} \bigl\Vert v_{xxx}(t)\bigr\Vert +\bigl\Vert v_{xx}(t) \bigr\Vert _{L^{\infty}}\bigl\Vert \eta_{xx}(t)\bigr\Vert +\bigl\Vert \eta_{x}(t)\bigr\Vert \bigr) \\ \leq&C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+ \bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(3.22)
$$\begin{aligned} \bigl\Vert \theta_{xxt}(t)\bigr\Vert \leq& C_{2}(T) \bigl(\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(3.23)
or
$$\begin{aligned}& \bigl\Vert v_{xxxx}(t)\bigr\Vert \leq C_{2}(T) \bigl( \bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{txx}(t)\bigr\Vert \bigr), \end{aligned}$$
(3.24)
$$\begin{aligned}& \bigl\Vert \theta_{xxxx}(t)\bigr\Vert \leq C_{2}(T) \bigl(\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{txx}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.25)
It follows from (1.8) and (1.10) that
$$\begin{aligned}& \bigl\Vert \eta_{t}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert v(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert \bigr), \end{aligned}$$
(3.26)
$$\begin{aligned}& \bigl\Vert \theta_{t}(t)\bigr\Vert \leq C_{1}\bigl( \bigl\Vert \theta_{xx}(t)\bigr\Vert +\bigl\Vert \eta_{x}(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert + \bigl\Vert v_{xx}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.27)
Differentiating (1.9) and (1.10) with respect to t, respectively, using Theorems 2.1-2.2 and (3.20)-(3.27), we have
$$\begin{aligned} \bigl\Vert v_{tt}(t)\bigr\Vert \leq&C_{2}(T) \bigl( \bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert +\bigl\Vert \theta _{t}(t)\bigr\Vert \\ &{} +\bigl\Vert \theta_{xt}(t)\bigr\Vert +\bigl\Vert v_{tx}(t)\bigr\Vert +\bigl\Vert v_{txx}(t)\bigr\Vert + \bigl\Vert \eta _{t}(t)\bigr\Vert \bigr) \end{aligned}$$
(3.28)
$$\begin{aligned} \leq&C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+ \bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+1\bigr), \end{aligned}$$
(3.29)
$$\begin{aligned} \bigl\Vert \theta_{tt}(t)\bigr\Vert \leq&C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert +\bigl\Vert \theta_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{tx}(t)\bigr\Vert \bigr) \end{aligned}$$
(3.30)
$$\begin{aligned} \leq&C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+ \bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{3}}+1\bigr) . \end{aligned}$$
(3.31)
Thus the estimate (3.17) follows from (3.20)-(3.23), (3.29), and (3.31).
Differentiating (1.9) with respect to t twice, multiplying the resultant by \(v_{tt}\) and performing an integration by parts in \(L^{2}(0,M)\), and using Theorem 2.2, the embedding theorem, and the Young inequality, we can derive
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert v_{tt}\Vert ^{2} =&- \int_{0}^{M}\bigl(r^{2}v_{tt} \bigr)_{x} \biggl(\mu \frac{(r^{2}v)_{x}}{\eta}-p \biggr)_{tt}\,dx-2 \int_{0}^{M}\bigl(\bigl(r^{2} \bigr)_{t}v_{tt}\bigr)_{x} \biggl(\mu \frac{(r^{2}v)_{x}}{\eta}-p \biggr)_{t}\,dx \\ &{} - \int_{0}^{M}\bigl(\bigl(r^{2} \bigr)_{tt}v_{tt}\bigr)_{x} \biggl(\mu \frac{(r^{2}v)_{x}}{\eta}-p \biggr)\,dx \\ \leq&- \int_{0}^{M}\mu\frac{r^{4}_{x}}{\eta}v_{ttx}^{2}\,dx+C_{2}(T) \bigl(\Vert v_{tt}\Vert +\Vert v_{xt}v_{x} \Vert +\bigl\Vert v_{x}^{3}\bigr\Vert +\Vert \theta_{t}v_{x}\Vert \\ &{} +\Vert v_{xt}\Vert +\Vert \theta_{tt}\Vert +\bigl\Vert v_{x}^{2}\bigr\Vert \bigr)\Vert v_{ttx} \Vert \\ \leq&-C_{1}^{-1}\Vert v_{ttx}\Vert ^{2}+C_{2}(T) \bigl(\Vert v_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{t}\Vert ^{2}+\Vert v_{xt}\Vert ^{2}+\Vert \theta_{tt}\Vert ^{2}+\Vert v_{tt}\Vert ^{2}\bigr). \end{aligned}$$
(3.32)
Thus, by Theorem 2.2,
$$\begin{aligned} \bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T)+C_{2}(T) \int_{0}^{t}\bigl(\Vert v_{tt}\Vert ^{2}+\Vert \theta_{tt}\Vert ^{2}\bigr) (s)\,ds, \end{aligned}$$
which, together with (3.28) and (3.30), gives estimate (3.18).
Similarly, differentiating (1.10) with respect to t twice, multiplying the result by \(\theta_{tt}\) and performing an integration by parts over \(L^{2}(0,M)\), and using the embedding theorem and the Young inequality, we have
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int_{0}^{M}e_{\theta}\theta_{tt}^{2}\,dx \\ &\quad=- \int_{0}^{M} \biggl(\frac{r^{4}\kappa\theta_{x}}{\eta} \biggr)_{tt}\theta _{ttx}\,dx- \int_{0}^{M} \bigl(e_{\theta tt} \theta_{t}+e_{\eta tt}\bigl(r^{2}v\bigr)_{x} \bigr)\theta_{tt}\,dx-\frac{3}{2} \int_{0}^{M}e_{\theta t}\theta_{tt}^{2}\,dx \\ &\qquad{} - \int_{0}^{M} \biggl(e_{\eta}+p-\mu \frac{(r^{2}v)_{x}}{\eta} \biggr) \bigl(r^{2}v\bigr)_{xtt} \theta_{tt}\,dx+ \int_{0}^{M} \biggl(\mu\frac{(r^{2}v)_{x}}{v}-p \biggr)_{tt}\bigl(r^{2}v\bigr)_{x} \theta_{tt}\,dx \\ &\qquad{} -2 \int_{0}^{M} \biggl(e_{\eta t}+ \biggl(p-\mu \frac{(r^{2}v)_{x}}{\eta} \biggr)_{t} \biggr) \bigl(r^{2}v \bigr)_{xt}\theta_{tt}\,dx \\ &\quad=: \sum_{i=1}^{6}P_{i}. \end{aligned}$$
(3.33)
By virtue of Theorems 2.1-2.2 and the embedding theorem, we deduce that, for any \(\varepsilon\in(0,1)\),
$$\begin{aligned} P_{1} \leq&-C_{1}\Vert \theta_{ttx}\Vert ^{2}+C_{2}\bigl(\Vert \theta_{x}\Vert _{L^{\infty}} \Vert v_{xt}\Vert +\Vert v_{x}\Vert _{L^{\infty}} \Vert \theta_{xt}\Vert +\Vert v_{x} \Vert _{L^{\infty}}^{2}\Vert \theta_{x}\Vert \\ &{} +\Vert \theta_{x}\Vert _{L^{\infty}} \Vert \theta_{t}\Vert +\Vert \theta_{x}\Vert _{L^{\infty}} \Vert \theta_{tt}\Vert \bigr)\Vert \theta_{ttx}\Vert \\ \leq&-(2C_{1})^{-1}\Vert \theta_{ttx}\Vert ^{2}+C_{2}(T) \bigl(\Vert \theta_{xt}\Vert ^{2}+\Vert v_{xt}\Vert ^{2}+\Vert v_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{tt}\Vert ^{2}\bigr), \end{aligned}$$
(3.34)
$$\begin{aligned} P_{2} \leq&C_{1} \int_{0}^{M} \bigl(\bigl(\vert v_{x} \vert +\vert \theta_{t}\vert \bigr)^{2}+\vert v_{xt}\vert +\vert \theta_{tt}\vert \bigr) \bigl( \vert v_{x}\vert +\vert \theta_{t}\vert \bigr)\vert \theta_{tt}\vert \,dx \\ \leq&C_{1}\Vert \theta_{tt}\Vert _{L^{\infty}}\bigl( \Vert v_{x}\Vert +\Vert \theta_{t}\Vert \bigr) \bigl( \bigl(\Vert v_{x}\Vert _{L^{\infty}}+\Vert \theta_{t} \Vert _{L^{\infty}}\bigr) \bigl(\Vert v_{x}\Vert +\Vert \theta_{t}\Vert \bigr)+\Vert v_{xt}\Vert +\Vert \theta_{tt}\Vert \bigr) \\ \leq&C_{2}(T) \bigl(\Vert \theta_{tt}\Vert +\Vert \theta_{ttx}\Vert \bigr) \bigl(\Vert v_{x}\Vert _{H^{1}}+\Vert \theta _{t}\Vert +\Vert \theta_{xt}\Vert +\Vert v_{xt}\Vert +\Vert \theta_{tt}\Vert \bigr) \\ \leq&\varepsilon \Vert \theta_{ttx}\Vert ^{2}+C_{2}(T) \varepsilon^{-1}\bigl(\Vert v_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{t}\Vert ^{2}+ \Vert \theta_{xt}\Vert ^{2}+\Vert v_{xt} \Vert ^{2}+\Vert \theta_{tt}\Vert ^{2}\bigr), \end{aligned}$$
(3.35)
$$\begin{aligned} P_{3} \leq&C_{1} \int_{0}^{M}\bigl(\vert v_{x}\vert + \vert \theta_{t}\vert \bigr)\theta_{tt}^{2}\,dx \leq C_{1}\Vert \theta _{tt}\Vert _{L^{\infty}}\bigl( \Vert v_{x}\Vert +\Vert \theta_{t}\Vert \bigr)\Vert \theta_{tt}\Vert \\ \leq&C_{1}\bigl(\Vert \theta_{tt}\Vert +\Vert \theta_{ttx}\Vert \bigr) \bigl(\Vert v_{x}\Vert +\Vert \theta_{t}\Vert \bigr)\Vert \theta_{tt}\Vert \leq \varepsilon \Vert \theta_{ttx}\Vert ^{2}+C_{2}(T) \varepsilon^{-1}\Vert \theta_{tt}\Vert ^{2}, \end{aligned}$$
(3.36)
$$\begin{aligned} P_{4} \leq&\varepsilon \Vert v_{ttx}\Vert ^{2}+C_{2}(T)\varepsilon^{-1}\Vert \theta_{tt}\Vert ^{2}, \end{aligned}$$
(3.37)
$$\begin{aligned} P_{5} \leq&C_{2}(T)\Vert v_{x}\Vert _{L^{\infty}} \Vert \theta_{tt}\Vert \bigl(\bigl(\Vert v_{x}\Vert _{L^{\infty}}+\Vert \theta_{t}\Vert _{L^{\infty}}\bigr) \bigl(\Vert v_{x}\Vert +\Vert \theta_{t}\Vert \bigr)+\Vert v_{xt}\Vert \\ &{} +\Vert \theta_{tt}\Vert +\Vert v_{xtt}\Vert + \Vert v_{tt}\Vert +\Vert v_{x}\Vert \bigr) \\ \leq&C_{2}(T)\Vert \theta_{tt}\Vert \bigl(\Vert v_{x}\Vert _{H^{1}}+\Vert \theta_{t}\Vert + \Vert \theta_{xt}\Vert +\Vert v_{xt}\Vert +\Vert \theta_{tt}\Vert +\Vert v_{xtt}\Vert +\Vert v_{tt}\Vert \bigr) \\ \leq&\varepsilon \Vert v_{ttx}\Vert ^{2}+C_{2}(T) \varepsilon^{-1}\bigl(\Vert \theta_{tt}\Vert ^{2}+\Vert v_{x}\Vert _{H^{1}}^{2}+ \Vert \theta_{t}\Vert ^{2}+\Vert \theta_{xt} \Vert ^{2}+\Vert v_{xt}\Vert ^{2}\bigr), \end{aligned}$$
(3.38)
$$\begin{aligned} P_{6} \leq&C_{1} \int_{0}^{M}\bigl(\vert v_{x}\vert + \vert \theta _{t}\vert +\vert v_{xt}\vert +\vert v_{x}\vert ^{2}+\vert v_{t}\vert \bigr) \bigl(\vert v_{xt}\vert +\vert v_{t}\vert \bigr) \vert \theta_{tt}\vert \,dx \\ \leq&C_{2}(T)\Vert v_{tx}\Vert ^{\frac{1}{2}}\Vert v_{txx}\Vert ^{\frac{1}{2}}\bigl(\Vert v_{x}\Vert + \Vert \theta_{t}\Vert +\Vert v_{xt}\Vert \bigr)\Vert \theta_{tt}\Vert , \end{aligned}$$
(3.39)
which, by Hölder’s inequality, implies
$$\begin{aligned} \int_{0}^{t}P_{6}\,ds \leq&C_{2}(T) \sup_{0\leq s\leq t}\bigl\Vert \theta_{tt}(s)\bigr\Vert \biggl( \int_{0}^{t}\bigl\Vert v_{txx}(s)\bigr\Vert ^{2}\,ds \biggr)^{\frac{1}{4}} \biggl( \int_{0}^{t}\bigl\Vert v_{tx}(s)\bigr\Vert ^{2}\,ds \biggr)^{\frac{1}{4}} \\ &{} \times \biggl( \int_{0}^{t}\bigl(\Vert v_{x}\Vert ^{2}+\Vert \theta_{t}\Vert ^{2}+\Vert v_{tx}\Vert ^{2}\bigr) (s)\,ds \biggr)^{\frac{1}{2}} \\ \leq&\varepsilon \biggl(\sup_{0\leq s\leq t}\bigl\Vert \theta_{tt}(s)\bigr\Vert ^{2}+ \int _{0}^{t}\bigl\Vert v{txx}(s)\bigr\Vert ^{2}\,ds \biggr)+C_{2}(T)\varepsilon^{-3}. \end{aligned}$$
(3.40)
Thus it follows from (3.33)-(3.40) that, for any \(\varepsilon\in(0,1)\) small enough,
$$\begin{aligned} &\bigl\Vert \theta_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \theta_{ttx}(s) \bigr\Vert ^{2}\,ds \\ &\quad\leq C_{4}(T)\varepsilon^{-3}+C_{2}(T) \varepsilon^{-1} \int_{0}^{t}\bigl\Vert \theta_{tt}(s) \bigr\Vert ^{2}\,ds \\ &\qquad{} +C_{1}\varepsilon \biggl(\sup_{0\leq s\leq t}\bigl\Vert \theta_{tt}(s)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert v_{txx}\Vert ^{2}+\Vert v_{ttx}\Vert ^{2}\bigr) (s)\,ds \biggr). \end{aligned}$$
(3.41)
Therefore taking the supremum in t on the left-hand side of (3.41) and choosing \(\varepsilon\in(0,1)\) small enough, we can derive estimate (3.19) from (3.30). The proof is complete. □
Lemma 3.9
Under the assumptions of Theorem
2.3, the following estimates hold for any
\(t\in[0,T]\)
and for
\(\varepsilon>0\)
small enough:
$$\begin{aligned}& \bigl\Vert v_{xt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert v_{xxt}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T)+C_{2}(T) \varepsilon^{2} \int_{0}^{t} \bigl(\Vert v_{xtt}\Vert ^{2}+\Vert \theta_{xxt}\Vert ^{2}\bigr) (s)\,ds, \end{aligned}$$
(3.42)
$$\begin{aligned}& \bigl\Vert \theta_{xt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \theta_{xxt}(s) \bigr\Vert ^{2}\,ds\leq C_{4}(T)+C_{2}(T) \varepsilon^{2} \int_{0}^{t} \bigl(\Vert v_{xxt}\Vert ^{2}+\Vert \theta_{xtt}\Vert ^{2}\bigr) (s)\,ds. \end{aligned}$$
(3.43)
Proof
Differentiating (1.9) with respect to x and t, multiplying the result by \(v_{xt}\) and integrating by parts in \(L^{2}(0,M)\), we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert v_{xt}\Vert ^{2}=N_{0}(t)+N_{1}(t) \end{aligned}$$
(3.44)
with
$$\begin{aligned} N_{0}(t)= \biggl(r^{2} \biggl(\mu \frac{(r^{2}v)_{x}}{v}-p \biggr)_{x} \biggr)_{t}v_{xt}\bigg|_{x=0}^{x=L},\qquad N_{1}(t)=- \int_{0}^{M} \biggl(r^{2} \biggl(\mu \frac{(r^{2}v)_{x}}{v}-P \biggr)_{x} \biggr)_{t}v_{xxt}\,dx. \end{aligned}$$
Using Theorem 2.2 and Lemma 3.8, the interpolation inequality, and Poincaré’s inequality, we can get
$$\begin{aligned} N_{0}(t) \leq&C_{1} \bigl(\bigl(\Vert v_{x} \Vert _{L^{\infty}}+\Vert \theta_{x}\Vert _{L^{\infty}}\bigr) \bigl(\Vert v_{x}\Vert _{L^{\infty}}+\Vert \theta_{x}\Vert _{L^{\infty}}+\Vert \eta_{x}\Vert _{L^{\infty}}\bigr)+\Vert v_{xx}\Vert _{L^{\infty}}+\Vert \theta_{xt}\Vert _{L^{\infty}} \\ &{} +\Vert v_{xxt}\Vert _{L^{\infty}}+\Vert \eta_{x} \Vert _{L^{\infty}} \Vert v_{xt}\Vert _{L^{\infty}}+\Vert v_{x}\Vert _{L^{\infty}} \Vert v_{xx}\Vert _{L^{\infty}}+\bigl\Vert v_{x}^{2}\bigr\Vert _{L^{\infty}}+\Vert \eta_{xt}\Vert _{L^{\infty}} \\ &{} +\Vert \eta_{x}\Vert _{L^{\infty}} \Vert \theta_{t}\Vert _{L^{\infty}}+\Vert v_{x}\Vert _{L^{\infty}} \Vert \theta_{x}\Vert _{L^{\infty}} \\ &{}+\Vert \theta_{x}\Vert _{L^{\infty}} \Vert \theta _{t}\Vert _{L^{\infty}}+\Vert v_{x}\Vert _{L^{\infty}} \Vert \eta_{x}\Vert _{L^{\infty}} \bigr)\Vert v_{xt}\Vert _{L^{\infty}} \\ \leq&C_{2}(T) (N_{01}+N_{02})\Vert v_{xt}\Vert ^{\frac{1}{2}}\Vert v_{xxt}\Vert ^{\frac{1}{2}}, \end{aligned}$$
(3.45)
where
$$N_{01}=\Vert v_{x}\Vert _{H^{2}}+\Vert \theta_{t}\Vert +\Vert \theta_{xt}\Vert $$
and
$$N_{02}=\Vert \theta_{xt}\Vert ^{\frac{1}{2}}\Vert \theta_{xxt}\Vert ^{\frac{1}{2}}+\Vert v_{xxt}\Vert ^{\frac{1}{2}}\Vert v_{xxxt}\Vert ^{\frac{1}{2}}+\Vert v_{xxt}\Vert +\Vert v_{xt}\Vert ^{\frac{1}{2}}\Vert v_{xxt}\Vert ^{\frac{1}{2}}. $$
Applying Young’s inequality several times, we have, for any \(\varepsilon \in(0,1)\),
$$\begin{aligned} C_{2}(T)N_{01}\Vert v_{xt}\Vert ^{\frac{1}{2}}\Vert v_{xxt}\Vert ^{\frac{1}{2}} \leq& \frac {\varepsilon^{2}}{2}\Vert v_{xxt}\Vert w^{2} \\ &{}+C_{2}(T) \varepsilon^{-1}\bigl(\Vert v_{x}\Vert _{H^{2}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2}+\Vert v_{xt}\Vert ^{2} \bigr) \end{aligned}$$
(3.46)
and
$$\begin{aligned} C_{2}(T)N_{02}\Vert v_{xt}\Vert ^{\frac{1}{2}}\Vert v_{xxt}\Vert ^{\frac{1}{2}} \leq& \frac {\varepsilon^{2}}{2}\Vert v_{xxt}\Vert ^{2}+ \varepsilon^{2}\bigl(\Vert \theta_{txx}\Vert ^{2}+\Vert v_{xxxt}\Vert ^{2} \bigr) \\ &{}+C_{2}(T)\varepsilon^{-6}\bigl(\Vert \theta_{tx}\Vert ^{2}+\Vert v_{xt}\Vert ^{2}\bigr). \end{aligned}$$
(3.47)
Thus it follows from (3.45)-(3.47) and Theorem 2.1 and Lemma 3.8 that
$$\begin{aligned} N_{0}(t) \leq&\varepsilon^{2}\bigl(\Vert v_{xxt} \Vert ^{2}+\Vert \theta_{txx}\Vert ^{2}+\Vert v_{xxxt}\Vert ^{2}\bigr) \\ &{}+C_{2}(T) \varepsilon^{-6}\bigl(\Vert \theta_{x}\Vert ^{2}+\Vert v_{x}\Vert _{H^{2}}^{2}+ \Vert \theta _{tx}\Vert ^{2}+\Vert v_{xt} \Vert ^{2}\bigr) , \end{aligned}$$
(3.48)
which, along with Theorem 2.2, further yields
$$\begin{aligned} \int_{0}^{t}N_{0}(s)\,ds\leq \varepsilon^{2} \int_{0}^{t}\bigl(\Vert v_{xxt}\Vert ^{2}+\Vert \theta_{txx}\Vert ^{2}+\Vert v_{xxxt}\Vert ^{2}\bigr) (s)\,ds+C_{2}(T) \varepsilon^{-6}. \end{aligned}$$
(3.49)
Analogously, from Lemma 3.8, Theorem 2.1, and the embedding theorem, we can also derive that, for any \(\varepsilon\in(0,1)\),
$$\begin{aligned} N_{1}(t) \leq&- \int_{0}^{M}\mu\frac{r^{4}}{\eta}v_{txx}^{2}\,dx \\ &{}+C_{1} \bigl(\bigl(\Vert v_{x}\Vert +\Vert \theta_{t}\Vert +\Vert \eta_{x}\Vert \bigr) \bigl(\Vert \theta_{x} \Vert _{L^{\infty}}+\Vert \eta_{x}\Vert _{L^{\infty}}\bigr)+ \Vert v_{xx}\Vert +\Vert \theta_{xt}\Vert \\ &{} +\Vert \eta_{x}\Vert _{L^{\infty}} \Vert v_{xt} \Vert +\Vert v_{x}\Vert _{L^{\infty}} \Vert v_{xx} \Vert +\Vert v_{x}\Vert _{L^{\infty}}^{2}\Vert \eta_{x}\Vert +\Vert v_{x}\Vert +\Vert \theta_{x}\Vert +\Vert v_{x}\Vert ^{2} \bigr)\Vert v_{xxt}\Vert \\ \leq&-(2C_{1})^{-1}\Vert v_{xxt}\Vert ^{2}+C_{2}(T) \bigl(\Vert v_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2}+\Vert v_{xt}\Vert ^{2}+ \Vert \eta_{x}\Vert ^{2}\bigr) , \end{aligned}$$
(3.50)
which, combined with (3.44), (3.49), and Theorem 2.2, shows that, for any \(\varepsilon\in(0,1)\) small enough,
$$\begin{aligned} &\bigl\Vert v_{xt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{xxt}(s)\bigr\Vert ^{2}\,ds \\ &\quad\leq C_{2}(T)\varepsilon ^{-6}+C_{1} \varepsilon^{2} \int_{0}^{t}\bigl(\Vert \theta_{txx} \Vert ^{2}+\Vert v_{xxxt}\Vert ^{2}\bigr) (s)\,ds. \end{aligned}$$
(3.51)
On the other hand, differentiating (1.9) with respect to x and t, we can derive from Theorem 2.2 and Lemma 3.8 that
$$\begin{aligned} \bigl\Vert v_{xxxt}(t)\bigr\Vert \leq& C_{1}\bigl\Vert v_{xtt}(t)\bigr\Vert +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta _{x}(t)\bigr\Vert _{H^{1}} \\ &{}+\bigl\Vert \eta_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert \theta_{t}(t)\bigr\Vert _{H^{2}}\bigr). \end{aligned}$$
(3.52)
Thus inserting (3.52) into (3.51) leads to (3.42).
Similarly, by (1.10), we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{0}^{L}e_{\theta}\theta_{tx}^{2}\,dx=: \sum_{i=0}^{3}L_{i}(t), \end{aligned}$$
(3.53)
where
$$\begin{aligned}& L_{0}(t)= \biggl(\frac{r^{4}\kappa\theta_{x}}{\eta} \biggr)_{xt} \theta _{xt}\bigg|_{x=0}^{x=M},\qquad L_{1}(t)=- \int_{0}^{M} \biggl(\frac{r^{4}\kappa\theta_{x}}{\eta} \biggr)_{tx}\theta _{txx}\,dx,\\& L_{2}(t)=- \int_{0}^{M} \biggl( \biggl(e_{\eta}+p-\mu \frac{(r^{2}v)_{x}}{\eta} \biggr) \bigl(r^{2}v\bigr)_{x} \biggr)_{xt}\theta_{tx}\,dx,\\& L_{3}(t)=- \int_{0}^{M} \biggl(e_{\theta tx} \theta_{t}+\frac{1}{2}e_{\theta t}\theta_{tx}+e_{\theta x} \theta_{tt} \biggr)\theta_{tx}\,dx. \end{aligned}$$
By virtue of the embedding theorem and the Young inequality, we derive from Lemmas 3.1, 3.8, and (3.42) that, for any \(\varepsilon\in(0,1)\),
$$\begin{aligned} L_{0}(t) \leq&C_{2}(T) \bigl(\Vert v_{x}\Vert _{H^{2}}+\Vert \theta_{x}\Vert _{H^{2}}+\Vert \theta_{t}\Vert _{H^{2}}+\Vert \theta_{xt}\Vert ^{\frac{1}{2}}\Vert \theta_{xxt}\Vert ^{\frac {1}{2}} \\ &{} +\Vert \theta_{xxt}\Vert ^{\frac{1}{2}}\Vert \theta_{xxxt}\Vert ^{\frac{1}{2}}\bigr)\Vert \theta_{xt} \Vert ^{\frac{1}{2}}\Vert \theta_{xxt}\Vert ^{\frac{1}{2}} \\ \leq&\varepsilon^{2}\bigl(\Vert \theta_{txx}\Vert ^{2}+\Vert \theta_{txxx}\Vert ^{2} \bigr)+C_{2}(T)\varepsilon^{-6}\bigl(\Vert v_{x} \Vert _{H^{2}}^{2}+\Vert \theta_{x}\Vert _{H^{2}}^{2}+\Vert \theta _{xt}\Vert ^{2}\bigr), \end{aligned}$$
(3.54)
$$\begin{aligned} L_{1}(t) \leq&-(2C_{1})^{-1}\Vert \theta_{txx}\Vert ^{2}+C_{2}(T) \bigl(\Vert v_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{x}\Vert _{H^{1}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2}\bigr), \end{aligned}$$
(3.55)
$$\begin{aligned} L_{2}(t) \leq&\varepsilon^{2}\Vert v_{txx} \Vert ^{2}+C_{2}(T)\varepsilon^{-2}\bigl(\Vert v_{x}\Vert _{H^{2}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2}+\Vert v_{xt}\Vert ^{2}+\Vert \eta_{x}\Vert _{H^{1}}^{2}\bigr), \end{aligned}$$
(3.56)
$$\begin{aligned} L_{3}(t) \leq&\varepsilon^{2}\Vert \theta_{txx} \Vert ^{2}+C_{2}(T)\varepsilon^{-2}\bigl(\Vert v_{x}\Vert w_{H^{1}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2}+\Vert \theta_{x}\Vert _{H^{2}}^{2}+\Vert v_{xt}\Vert ^{2}+\Vert \eta_{x}\Vert ^{2}\bigr). \end{aligned}$$
(3.57)
Differentiating (1.10) with respect to x and t, we can derive from Theorems 2.1-2.2 and Lemma 3.8 that
$$\begin{aligned} \bigl\Vert \theta_{txxx}(t)\bigr\Vert \leq& C_{1}\bigl( \bigl\Vert \theta_{ttx}(t)\bigr\Vert +\bigl\Vert v_{xxt}(t)\bigr\Vert \bigr) \\ &{} +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+ \bigl\Vert \theta _{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{xt}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.58)
Inserting (3.54)-(3.58) into (3.53) yields (3.43). □
Lemma 3.10
Under the assumptions of Theorem
2.3, we have, for any
\(t\in[0,T]\),
$$\begin{aligned} &\bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xt}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{tt}(t) \bigr\Vert ^{2}+\bigl\Vert \theta _{xt}(t)\bigr\Vert ^{2} \\ &\quad{}+ \int_{0}^{t} \bigl(\Vert v_{ttx}\Vert ^{2}+\Vert v_{xxt}\Vert ^{2}+\Vert \theta_{ttx}\Vert ^{2}+\Vert \theta_{xxt}\Vert ^{2}\bigr) (s)\,ds\leq C_{4}(T), \end{aligned}$$
(3.59)
$$\begin{aligned} &\bigl\Vert \eta_{xxx}(t)\bigr\Vert _{H^{1}}^{2}+ \bigl\Vert v_{xxx}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert \theta_{xxx}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert v_{txx}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{txx}(t)\bigr\Vert ^{2} \\ &\quad{} + \int_{0}^{t} \bigl(\Vert v_{tt}\Vert ^{2}+\Vert v_{xxt}\Vert _{H^{1}}^{2}+ \Vert \theta_{tt}\Vert ^{2}+\Vert \theta_{xxt} \Vert _{H^{1}}^{2}\bigr) (s)\,ds\leq C_{4}(T), \end{aligned}$$
(3.60)
$$\begin{aligned} &\int_{0}^{t} \bigl(\Vert \eta_{xxx} \Vert _{H^{1}}^{2}+\Vert v_{xxxx}\Vert _{H^{1}}^{2}+\Vert \theta _{xxxx}\Vert _{H^{1}}^{2}\bigr) (s)\,ds\leq C_{4}(T). \end{aligned}$$
(3.61)
Proof
Adding (3.42)-(3.43) and choosing \(\varepsilon>0\) small enough, we get
$$\begin{aligned} &\bigl\Vert v_{xt}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{xt}(t)\bigr\Vert ^{2}+ \int_{0}^{t} \bigl(\Vert v_{xxt}\Vert ^{2}+\Vert \theta _{xxt}\Vert ^{2}\bigr) (s)\,ds \\ &\quad \leq C_{4}(T)+C_{2}(T)\varepsilon^{2} \int_{0}^{t} \bigl(\Vert v_{xtt}\Vert ^{2}+\Vert \theta_{xtt}\Vert ^{2}\bigr) (s)\,ds. \end{aligned}$$
(3.62)
Now multiplying (3.18) and (3.19) by ε and \(\varepsilon ^{\frac{3}{2}}\), respectively, then adding the results to (3.62) and taking ε sufficiently small, we obtain (3.59).
Differentiating (1.9) with respect to x and noting that \(\eta _{xxt}=(r^{2}v)_{xxx}\), we get
$$\begin{aligned} \mu\frac{\partial}{\partial t} \biggl(\frac{\eta_{xx}}{\eta} \biggr)-p_{\eta}\eta_{xx}=r^{-2}v_{tx}+K(x,t)-\bigl(r^{-2}f \bigr)_{x}-2r^{-5}\eta v_{t}, \end{aligned}$$
(3.63)
where
$$\begin{aligned} K(x,t) =&p_{\eta\eta}\eta_{x}^{2}+2p_{\eta\theta} \theta_{x}\eta_{x}+p_{\theta \theta}\theta_{x}^{2}+p_{\theta}\theta_{xx}-2\mu\frac{\eta_{x}^{2}}{\eta ^{3}}\bigl(r^{2}v \bigr)_{x}+2\mu\frac{\eta_{x}}{\eta^{2}}\bigl(r^{2}v \bigr)_{xx} \\ =&\frac{A(\beta-2)(\beta-3)}{2}\theta^{2}\eta^{\beta-4} \eta_{x}^{2}+2A(\beta -2)\theta\eta^{\beta-3} \theta_{x} \eta_{x}+A\eta^{\beta-2}\theta _{x}^{2} \\ &{} +A\theta\eta^{\beta-2}\theta_{xx}+2\mu \biggl( \frac{\eta_{x}}{\eta ^{2}}\bigl(r^{2}v\bigr)_{xx}-\frac{\eta_{x}^{2}}{\eta^{3}} \bigl(r^{2}v\bigr)_{x} \biggr). \end{aligned}$$
Differentiating (3.63) with respect to x, we have
$$\begin{aligned} \mu\frac{\partial}{\partial t} \biggl(\frac{\eta_{xxx}}{\eta} \biggr)-p_{\eta}\eta_{xxx}=K_{1}(x,t), \end{aligned}$$
(3.64)
where
$$\begin{aligned} K_{1}(x,t) =&K_{x}(x,t)+p_{\eta x} \eta_{xx}+\mu \biggl(\frac{\eta_{xx}\eta _{x}}{\eta^{2}} \biggr)_{t}+r^{-2}v_{txx}-4r^{-5} \eta v_{tx}\\ &{} +10r^{-8}\eta^{2}v_{t}-2r^{-5} \eta_{x}v_{t}-\bigl(r^{-2}f\bigr)_{xx}. \end{aligned}$$
Obviously, it follows from Theorem 2.1 and Lemmas 3.8-3.9 that
$$\begin{aligned} \bigl\Vert K_{1}(t)\bigr\Vert \leq C_{2}(T) \bigl(\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta _{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{txx}(t) \bigr\Vert \bigr) \end{aligned}$$
(3.65)
and
$$\begin{aligned} \int_{0}^{t}\bigl\Vert K_{1}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.66)
Multiplying (3.64) by \(\frac{\eta_{xxx}}{\eta}\) over \(L^{2}(0,M)\), we can obtain
$$\begin{aligned} \frac{d}{dt}\biggl\Vert \frac{\eta_{xxx}}{\eta}\biggr\Vert ^{2}+C_{1}^{-1}\biggl\Vert \frac {\eta_{xxx}}{\eta} \biggr\Vert ^{2}\leq C_{1}\bigl\Vert K_{1}(t) \bigr\Vert ^{2} , \end{aligned}$$
(3.67)
which, along with (3.66), gives
$$\begin{aligned} \bigl\Vert \eta_{xxx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \eta_{xxx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.68)
It follows from (1.8)-(1.10) that
$$\begin{aligned}& \bigl\Vert v_{xxx}(t)\bigr\Vert \leq C_{2}(T) \bigl( \bigl\Vert v(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert \theta _{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{xt}(t)\bigr\Vert \bigr), \end{aligned}$$
(3.69)
$$\begin{aligned}& \bigl\Vert \theta_{xxx}(t)\bigr\Vert \leq C_{2}(T) \bigl(\bigl\Vert \theta(t)\bigr\Vert _{H^{2}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert \theta_{xt}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.70)
Using the embedding theorem, Theorems 2.1-2.2 and Lemmas 3.8-3.9, we can derive from (3.24)-(3.25), (3.59), and (3.68)-(3.70) that, for any \(t\in[0,T]\),
$$\begin{aligned} &\bigl\Vert v_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xx}(t)\bigr\Vert _{L^{\infty}}^{2}+\bigl\Vert \theta_{xx}(t)\bigr\Vert _{L^{\infty}}^{2} \\ &\quad{} + \int_{0}^{t}\bigl(\Vert v_{xxx}\Vert _{H^{1}}^{2}+\Vert \theta_{xxx}\Vert _{H^{1}}^{2}+\Vert v_{xx}\Vert _{W^{1,\infty}}^{2}+\Vert \theta_{xx}\Vert _{W^{1,\infty}}^{2}\bigr) (s)\,ds\leq C_{4}(T). \end{aligned}$$
(3.71)
Differentiating (1.9)-(1.10) with respect to t and using Theorems 2.1-2.2 and Lemmas 3.8-3.9, we can deduce from (3.59), (3.68)-(3.71) that
$$\begin{aligned} \bigl\Vert v_{txx}(t)\bigr\Vert \leq& C_{1}\bigl\Vert v_{tt}(t)\bigr\Vert +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta _{x}(t)\bigr\Vert +\bigl\Vert \theta_{x}(t)\bigr\Vert \\ &{} +\bigl\Vert \theta_{t}(t)\bigr\Vert +\bigl\Vert \theta_{xt}(t)\bigr\Vert +\bigl\Vert v_{xt}(t)\bigr\Vert \bigr)\leq C_{4}(T), \end{aligned}$$
(3.72)
$$\begin{aligned} \bigl\Vert \theta_{txx}(t)\bigr\Vert \leq& C_{1}\bigl\Vert \theta_{tt}(t)\bigr\Vert +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \eta_{x}(t)\bigr\Vert +\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{2}} \\ &{} +\bigl\Vert \theta_{t}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{xt}(t)\bigr\Vert \bigr)\leq C_{4}(T), \end{aligned}$$
(3.73)
which, combined with (3.24)-(3.25) and (3.72), implies
$$ \begin{aligned}[b] &\bigl\Vert v_{xxxx}(t)\bigr\Vert ^{2}+\bigl\Vert \theta_{xxxx}(t)\bigr\Vert ^{2}\\ &\quad{}+ \int_{0}^{t}\bigl(\Vert v_{txx}\Vert ^{2}+\Vert \theta _{txx}\Vert ^{2}+\Vert v_{xxxx}\Vert ^{2}+\Vert \theta_{xxxx}\Vert ^{2}\bigr) (s)\,ds\leq C_{4}(T). \end{aligned} $$
(3.74)
Therefore it follows from (3.71), (3.74), and the embedding theorem that
$$\begin{aligned} \bigl\Vert v_{xxx}(t)\bigr\Vert _{L^{\infty}}^{2}+\bigl\Vert \theta_{xxx}(t)\bigr\Vert _{L^{\infty}}^{2}+ \int_{0}^{t}\bigl(\Vert v_{xxx}\Vert _{L^{\infty}}^{2}+\Vert \theta_{xxx}\Vert _{L^{\infty}}^{2}\bigr) (s)\,ds\leq C_{4}(T). \end{aligned}$$
(3.75)
Now differentiating (3.64) with respect to x, we find
$$\begin{aligned} \epsilon\frac{\partial}{\partial t} \biggl(\frac{\eta_{xxxx}}{\eta} \biggr)-p_{\eta}\eta_{xxxx}=K_{2}(x,t), \end{aligned}$$
(3.76)
where
$$\begin{aligned} K_{2}(x,t)=K_{1x}(x,t)+p_{\eta x}\eta_{xxx}+ \mu \biggl(\frac{\eta_{xxx}\eta _{x}}{\eta^{2}} \biggr)_{t}. \end{aligned}$$
From the embedding theorem and Lemmas 3.8-3.9 and (3.68)-(3.75), we can derive
$$\begin{aligned} \bigl\Vert K_{xx}(t)\bigr\Vert \leq& C_{4}(T) \bigl( \bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta _{x}(t)\bigr\Vert _{H^{2}}\bigr),\\ \bigl\Vert K_{1x}(t)\bigr\Vert \leq& C_{1} \biggl( \bigl\Vert K_{xx}(t)\bigr\Vert +\Vert v_{xxxt}\Vert + \Vert v_{xxt}\Vert +\bigl\Vert (p_{\eta x} \eta_{xx})_{x}\bigr\Vert +\Vert \eta_{x}v_{xt} \Vert \\ &{} +\Vert \eta_{xx}\Vert +\Vert \eta_{x}v_{t} \Vert +\Vert \eta_{xx}v_{t}\Vert +\biggl\Vert \biggl( \frac{\eta _{x}\eta_{xx}}{\eta^{2}}\biggr)_{xt}\biggr\Vert \biggr)\\ \leq&C_{1}\bigl\Vert v_{xxxt}(t)\bigr\Vert +C_{4}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{3}}+ \bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
whence
$$\begin{aligned} \bigl\Vert K_{2}(t)\bigr\Vert \leq C_{1}\bigl\Vert v_{xxxt}(t)\bigr\Vert +C_{4}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \theta _{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t) \bigr\Vert _{H^{2}}\bigr). \end{aligned}$$
(3.77)
It follows from (3.28)-(3.31) that
$$\begin{aligned} \int_{0}^{t}\bigl(\Vert v_{tt}\Vert ^{2}+\Vert \theta_{tt}\Vert ^{2}\bigr) (s)\,ds \leq C_{4}(T) , \end{aligned}$$
(3.78)
which, along with (3.52) and (3.59), gives
$$\begin{aligned} \int_{0}^{t}\bigl\Vert v_{xxxt}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.79)
Thus from (3.68), (3.74), (3.77), and (3.79), it follows that
$$\begin{aligned} \int_{0}^{t}\bigl\Vert K_{2}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.80)
Multiplying (3.76) by \(\frac{\eta_{xxxx}}{\eta}\) in \(L^{2}(0,M)\), we can get
$$\begin{aligned} \frac{d}{dt} \biggl\Vert \frac{\eta_{xxxx}}{\eta} \biggr\Vert ^{2}+C_{1}^{-1} \biggl\Vert \frac {\eta_{xxxx}}{\eta} \biggr\Vert ^{2}\leq C_{1}\bigl\Vert K_{2}(t) \bigr\Vert ^{2} , \end{aligned}$$
(3.81)
whence, by (3.80),
$$\begin{aligned} \bigl\Vert \eta_{xxxx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \eta_{xxxx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.82)
Differentiating (1.10) with respect to x and t, we can derive from Theorems 2.1-2.2 and Lemmas 3.8-3.9 that
$$\begin{aligned} \bigl\Vert \theta_{txxx}(t)\bigr\Vert \leq& C_{1}\bigl\Vert \theta_{ttx}(t)\bigr\Vert +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \theta_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \theta_{xt}(t)\bigr\Vert \bigr). \end{aligned}$$
(3.83)
Thus,
$$\begin{aligned} \int_{0}^{t}\bigl\Vert \theta_{txxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.84)
Differentiating (1.9) with respect to x three times, applying Lemmas 3.8-3.9, Theorems 2.1-2.2, and Poincaré’s inequality, we have
$$\begin{aligned} \bigl\Vert v_{xxxxx}(t)\bigr\Vert \leq C_{1}\bigl\Vert v_{txxx}(t)\bigr\Vert +C_{2}(T) \bigl(\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \eta _{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \theta_{x}(t) \bigr\Vert _{H^{3}}\bigr). \end{aligned}$$
(3.85)
Thus it follows from (3.74), (3.79), and (3.82) that
$$\begin{aligned} \int_{0}^{t}\bigl\Vert v_{xxxxx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.86)
Similarly, we can differentiate (1.10) with respect to x three times and use Lemmas 3.8-3.9, Theorems 2.1-2.2, Poincaré’s inequality, (3.74), (3.82), and (3.84) to find
$$\begin{aligned} \int_{0}^{t}\bigl\Vert \theta_{xxxxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}(T). \end{aligned}$$
(3.87)
Hence, (3.60)-(3.61) follow from (3.74), (3.82), (3.86), and (3.87). □
Finally, combining Lemmas 3.8-3.10, we complete the proof of Theorem 2.3.