In this section, we will prove the local existence of strong solutions. Because equation (1.1)2 always possesses a singularity, we overcome this difficulty by a regularized process, then taking the limiting process back to the original problem. First of all, we consider the following system:
$$\begin{aligned}& \rho_{t}+(\rho u)_{x}=0, \end{aligned}$$
(2.1)
$$\begin{aligned}& (\rho u )_{t}+ \bigl(\rho u^{2} \bigr)_{x}- \biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{x}+(P+\eta)_{x}= -(\eta+\beta\rho) \Phi_{x}, \end{aligned}$$
(2.2)
$$\begin{aligned}& \eta_{t}+ \bigl(\eta(u-\Phi_{x}) \bigr)_{x}= \eta_{xx}, \end{aligned}$$
(2.3)
with the initial and boundary conditions,
$$\begin{aligned}& (\rho,u,\eta)|_{t=0}=(\rho_{0},u_{0}, \eta_{0}), \quad x\in\Omega, \end{aligned}$$
(2.4)
$$\begin{aligned}& u|_{\partial\Omega}=(\eta_{x}+\eta\Phi_{x})|_{\partial\Omega}=0, \quad t\in [0,T], \end{aligned}$$
(2.5)
and \(u_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\) is the smooth solution of the boundary value problem
$$ \textstyle\begin{cases} [ (\frac{\varepsilon u_{0x}^{2}+1}{u_{0x}^{2}+\varepsilon} )^{\frac{2-p}{2}}u_{0x} ]_{x}- (P(\rho_{0})+\eta_{0} )_{x}-\eta_{0}\Phi _{x}=\rho_{0}(g+\beta\Phi_{x}), \\ u_{0}(0)=u_{0}(1)=0. \end{cases} $$
(2.6)
By using the iterative method step by step, the nonlinear coupled system admits a smooth solution (see Section 3). Provided that \((\rho ,u,\eta)\) is a smooth solution of (2.1)-(2.5) and \(\rho_{0}\geq\delta\), where \(0<\delta\ll 1\) is a positive number. We denote \(M_{0}=1+ \mu_{0}+ \mu^{-1}_{0}+|\rho_{0}|_{H^{1}}+|g|_{L^{2}}\).
First we obtain the estimate of \(|u_{0xx}|_{L^{2}}\). From (2.6), we have
$$u_{0xx}= \biggl(\frac{\varepsilon u_{0x}^{2}+1}{u_{0x}^{2}+\varepsilon} \biggr)^{p\over 2} \frac{(u_{0x}^{2}+\varepsilon)^{2}[(P(\rho_{0})+\eta_{0})_{x}+\eta_{0}\Phi_{x}+\rho _{0}(g+\beta\Phi_{x})]}{(\varepsilon u_{0x}^{2}+1)(u_{0x}^{2}+\varepsilon)-(2-p)(1-\varepsilon^{2})u_{0x}^{2}}. $$
Then
$$\begin{aligned} \vert u_{0xx}\vert _{L^{2}} \leq&\frac{1}{p-1} \biggl\vert \biggl(\frac {u_{0x}^{2}+\varepsilon}{\varepsilon u_{0x}^{2}+1} \biggr)^{1-\frac {p}{2}} \biggr\vert _{L^{\infty}} \bigl\vert \bigl(P(\rho_{0})+\eta_{0} \bigr)_{x}+\eta_{0}\Phi_{x}+\rho _{0}(g+ \beta\Phi_{x}) \bigr\vert _{L^{2}} \\ \leq&\frac{1}{p-1} \bigl(\vert u_{0x}\vert _{L^{\infty}}^{2}+1 \bigr)^{1-\frac{p}{2}} \bigl( \bigl\vert \bigl(P( \rho _{0})+\eta_{0} \bigr)_{x}+ \eta_{0} \Phi_{x}+\rho_{0}(g+\beta \Phi_{x}) \bigr\vert _{L^{2}} \bigr) \\ \leq&\frac{1}{p-1} \bigl(\vert u_{0xx}\vert _{L^{2}}^{2}+1 \bigr)^{1-\frac{p}{2}} \bigl(a\gamma \vert \rho _{0}\vert _{L^{\infty}}^{\gamma-1}\vert \rho_{0x} \vert _{L^{2}}+\vert \eta_{0x}\vert _{L^{2}}+\vert \eta _{0}\vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}} \\ &{} +\vert \rho_{0}\vert _{L^{\infty}} \vert g\vert _{L^{2}}+\beta \vert \rho_{0}\vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}} \bigr). \end{aligned}$$
Using Young’s inequality, we have
$$|u_{0xx}|_{L^{2}}\leq C, $$
where C is a positive constant, depending only on \(M_{0}\).
Next, we introduce an auxiliary function,
$$\Psi(t)=\sup_{0\leq s\leq t} \bigl(1+ \bigl\vert \rho (s) \bigr\vert _{H^{1}}+ \bigl\vert u(s) \bigr\vert _{W_{0}^{1,p}}+ \bigl\vert \sqrt {\rho}u_{t}(s) \bigr\vert _{L^{2}}+ \bigl\vert \eta _{t}(s) \bigr\vert _{L^{2}}+ \bigl\vert \eta(s) \bigr\vert _{H^{1}} \bigr). $$
Then we estimate each term of \(\Psi(t)\) in terms of some integrals of \(\Psi(t)\), apply arguments of Gronwall-type, and thus prove that \(\Psi(t)\) is locally bounded.
Estimate for \(|\rho|_{H^{1}}\)
First we need the following estimates for u and η. By virtue of (2.2)
$$\biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{x}=\rho u_{t}+\rho u u_{x}+(P+ \eta)_{x}+(\eta+\beta\rho)\Phi_{x}. $$
Then, we have
$$ |u_{xx}|\leq\frac{1}{p-1} \bigl(u_{x}^{2}+ \varepsilon \bigr)^{1-\frac{p}{2}} \bigl\vert \rho u_{t}+\rho u u_{x}+(P+\eta)_{x}+(\eta+\beta\rho)\Phi_{x} \bigr\vert . $$
Taking the \(L^{2}\) norm and using Young’s inequality, we get
$$\begin{aligned} \vert u_{xx}\vert _{L^{2}}^{p-1} \leq& C \bigl(1+ \vert \rho u_{t}\vert _{L^{2}}+\vert \rho u u_{x}\vert _{L^{2}}+ \bigl\vert (P+\eta)_{x} \bigr\vert _{L^{2}}+ \bigl\vert (\eta+\beta\rho)\Phi_{x} \bigr\vert _{L^{2}} \bigr) \\ \leq& C \bigl(1+\vert \rho \vert _{L^{\infty}}^{1\over 2}\vert \sqrt{ \rho}u_{t}\vert _{L^{2}}+\vert \rho \vert _{L^{\infty}} \vert u\vert _{L^{\infty}} \vert u_{x}\vert _{L^{p}}^{\frac{p}{2}}\vert u_{x}\vert _{L^{\infty}}^{1-\frac{p}{2}} \\ &{} +\vert \rho \vert _{L^{\infty}}^{\gamma-1}\vert \rho_{x}\vert _{L^{2}}+\vert \eta _{x}\vert _{L^{2}}+\vert \eta \vert _{L^{\infty}} \vert \Phi_{x} \vert _{L^{2}}+\vert \rho \vert _{L^{\infty}} \vert \Phi _{x}\vert _{L^{2}} \bigr) \\ \leq& C \bigl[1+\vert \rho \vert _{L^{\infty}}^{1\over 2}\vert \sqrt{ \rho}u_{t}\vert _{L^{2}}+ \bigl(\vert \rho \vert _{L^{\infty}} \vert u\vert _{L^{\infty}} \vert u_{x}\vert _{L^{p}}^{\frac{p}{2}} \bigr)^{\frac {2(p-1)}{3p-4}}+\vert \rho \vert _{L^{\infty}}^{\gamma-1}\vert \rho_{x}\vert _{L^{2}} \\ &{} +\vert \eta_{x}\vert _{L^{2}}+\vert \eta \vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}}+\vert \rho \vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}} \bigr]+ \frac{1}{2}\vert u_{xx}\vert _{L^{2}}^{p-1} \\ \leq& C \Psi^{\frac{(4+p)(p-1)}{3p-4}\gamma}(t)+\frac{1}{2}\vert u_{xx} \vert _{L^{2}}^{p-1}. \end{aligned}$$
(2.7)
Hence, we deduce that
$$ |u_{xx}|_{L^{2}}\leq C\Psi^{\frac{6\gamma}{3p-4}}(t). $$
(2.8)
From (2.3), taking the \(L^{2}\) norm, we get
$$\begin{aligned} \vert \eta_{xx}\vert _{L^{2}}&\leq \bigl\vert \eta_{t}+ \bigl(\eta(u-\Phi_{x}) \bigr)_{x} \bigr\vert _{L^{2}} \\ &\leq \vert \eta_{t}\vert _{L^{2}}+\vert \eta_{x}\vert _{L^{2}}\vert u\vert _{L^{\infty}}+\vert \eta_{x}\vert _{L^{2}}\vert \Phi _{x}\vert _{L^{\infty}}+\vert \eta \vert _{L^{2}}\vert u_{xx} \vert _{L^{2}}+\vert \eta \vert _{L^{\infty}} \vert \Phi _{xx}\vert _{L^{2}} \\ &\leq C\Psi^{\frac{8\gamma}{3p-4}}(t). \end{aligned}$$
(2.9)
Multiplying (2.1) by ρ, integrating over Ω, we have
$$ {1\over 2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}|\rho|^{2}\, {\mathrm{d}} s+ \int_{\Omega}(\rho u)_{x}\rho\, {\mathrm{d}} x=0. $$
Integrating by parts, using the Sobolev inequality, we deduce that
$$ \frac{\mathrm{d}}{\mathrm{d}t}\bigl|\rho(t)\bigr|_{L^{2}}^{2}\leq \int_{\Omega}|u_{x}||\rho|^{2}\, { \mathrm{d}}x \leq|u_{xx}|_{L^{2}}|\rho|_{L^{2}}^{2}. $$
(2.10)
Differentiating (1.1)1 with respect to x, and multiplying it by \(\rho_{x}\), integrating over Ω, and using the Sobolev inequality, we have
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}|\rho_{x}|^{2}{\mathrm{d}} x=&- \int_{\Omega}\biggl[{3\over 2} u_{x}( \rho_{x})^{2}+\rho\rho_{x} u_{xx} \biggr](t){\mathrm{d}} x \\ &\leq C \bigl[|u_{x}|_{L^{\infty}}|\rho_{x}|_{L^{2}}^{2}+| \rho|_{L^{\infty}}|\rho _{x}|_{L^{2}}|u_{xx}|_{L^{2}} \bigr] \\ &\leq C|\rho|_{H^{1}}^{2}|u_{xx}|_{L^{2}}. \end{aligned}$$
(2.11)
From (2.10)and (2.11), by Gronwall’s inequality, it follows that
$$ \sup_{0\leq t\leq T}\bigl|\rho(t)\bigr|_{H^{1}}^{2} \leq|\rho_{0}|_{H^{1}}^{2}\exp \biggl\{ C \int _{0}^{t}|u_{xx}|_{L^{2}}\, { \mathrm{d}} s \biggr\} \leq C\exp \biggl(C \int_{0}^{t}\Psi^{\frac{6}{3p-4}\gamma}(s)\, {\mathrm{d}} s \biggr), $$
(2.12)
we can also get the following estimates. Using (1.1)1 we obtain
$$ \bigl\vert \rho_{t}(t) \bigr\vert _{L^{2}} \leq \bigl\vert \rho_{x}(t) \bigr\vert _{L^{2}} \bigl\vert u(t) \bigr\vert _{L^{\infty}}+ \bigl\vert \rho (t) \bigr\vert _{L^{\infty}} \bigl\vert u_{x}(t) \bigr\vert _{L^{2}}\leq C \Psi^{2}(t), $$
(2.13)
where C is a positive constant, depending only on \(M_{0}\).
Estimate for \(|\eta_{t}|_{L^{2}}\) and \(|\eta|_{H^{1}}\)
Multiplying (1.1)3 by η, integrating the resulting equation over \(\Omega_{T}\), using the boundary conditions (1.3) and Young’s inequality, we have
$$\begin{aligned}& \int_{0}^{t} \bigl\vert \eta_{x}(s) \bigr\vert _{L^{2}}^{2}\, {\mathrm{d}} s +\frac{1}{2} \bigl\vert \eta(t) \bigr\vert _{L^{2}}^{2} \\& \quad \leq \iint_{\Omega_{T}} \bigl(\vert \eta u\eta_{x}\vert + \vert \eta\Phi_{x}\eta_{x}\vert \bigr)\, {\mathrm{d}} x \, {\mathrm{d}} s \\& \quad \leq\frac{1}{4} \int_{0}^{t} \bigl\vert \eta_{x}(s) \bigr\vert _{L^{2}}^{2}\, {\mathrm{d}} s+C \int _{0}^{t}\vert u_{x}\vert _{L^{p}}^{2}\vert \eta \vert _{H^{1}}^{2}\, {\mathrm{d}} s+C \int_{0}^{t}\vert \eta \vert _{H^{1}}^{2}+C \\& \quad \leq\frac{1}{4} \int_{0}^{t} \bigl\vert \eta_{x}(s) \bigr\vert _{L^{2}}^{2}\, {\mathrm{d}} s+C \biggl(1+ \int_{0}^{t}\Psi ^{4}(t)\, {\mathrm{d}} s \biggr). \end{aligned}$$
(2.14)
Multiplying (1.1)3 by \(\eta_{t}\), integrating (by parts) over \(\Omega_{T}\), using the boundary conditions (1.3) and Young’s inequality, we have
$$\begin{aligned}& \int_{0}^{t} \bigl\vert \eta_{t}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s +\frac{1}{2} \bigl\vert \eta _{x}(t) \bigr\vert _{L^{2}}^{2} \\& \quad \leq \iint_{\Omega_{T}} \bigl\vert \eta(u-\Phi_{x}) \eta_{xt} \bigr\vert \,{\mathrm{d}} x\,{\mathrm{d}} s \\& \quad \leq\frac{1}{4} \int_{0}^{t} \bigl\vert \eta_{xt}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s+C \int_{0}^{t}\vert \eta \vert _{H^{1}}^{2} \vert u_{x}\vert _{L^{p}}^{2}\,{\mathrm{d}} s+C \int_{0}^{t}\vert \eta \vert _{H^{1}}^{2} \,{\mathrm{d}} s+C \\& \quad \leq\frac{1}{4} \int_{0}^{t} \bigl\vert \eta_{xt}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s+C \biggl(1+ \int _{0}^{t}\Psi^{4}(t)\,{\mathrm{d}} s \biggr). \end{aligned}$$
(2.15)
Differentiating (1.1)3 with respect to t, multiplying the resulting equation by \(\eta_{t}\), integrating (by parts) over \(\Omega_{T}\), we get
$$\begin{aligned}& \int_{0}^{t} \bigl\vert \eta_{xt}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s +\frac{1}{2} \bigl\vert \eta _{t}(t) \bigr\vert _{L^{2}}^{2} \\& \quad = \iint_{\Omega_{T}} \bigl(\eta(u-\Phi_{x}) \bigr)_{t} \eta_{xt}\,{\mathrm{d}} x\,{\mathrm{d}} s \\& \quad \leq C+ \iint_{\Omega_{T}} \bigl(\vert \eta_{t} u \eta_{xt} \vert +\vert \eta_{t}\Phi_{x}\eta _{xt}\vert +\vert \eta_{x} u_{t} \eta_{t}\vert +\vert \eta u_{xt}\eta_{t}\vert \bigr)\,{\mathrm{d}} x\,{ \mathrm{d}} s \\& \quad \leq C \biggl(1+ \int_{0}^{t} \bigl(\vert \eta_{t}\vert _{L^{2}}^{2}\vert u_{x}\vert _{L^{p}}^{2}+ \vert \eta _{t}\vert _{L^{2}}^{2}+\vert \eta_{x}\vert _{L^{2}}^{2}\vert \eta_{t} \vert _{L^{2}}^{2}+\vert \eta \vert _{H^{1}}^{2} \vert \eta _{t}\vert _{L^{2}}^{2} \bigr)\,{ \mathrm{d}} x \biggr) \\& \qquad {} +\frac{1}{2} \int_{0}^{t}\vert \eta_{xt}\vert _{L^{2}}^{2}+\frac{1}{2} \int _{0}^{t}\vert u_{xt}\vert _{L^{2}}^{2} \\& \quad \leq C \biggl(1+ \int_{0}^{t} \Psi^{4}(s)\,{\mathrm{d}} s \biggr). \end{aligned}$$
(2.16)
Combining (2.14)-(2.16) and (2.28), we get
$$ |\eta|_{H^{1}}^{2}+|\eta_{t}|_{L^{2}}^{2}+ \int_{0}^{t} \bigl(|\eta_{x}|_{L^{2}}^{2}+| \eta _{t}|_{L^{2}}^{2}+|\eta_{xt}|_{L^{2}}^{2} \bigr) (s)\,{\mathrm{d}} s\leq C \biggl(1+ \int_{0}^{t} \Psi ^{4}(s)\,{\mathrm{d}} s \biggr). $$
(2.17)
Estimate for \(|u|_{W_{0}^{1,p}}\)
Using (2.1), we rewrite (2.1) as
$$ \rho u_{t}+\rho u u_{x}- \biggl[ \biggl( \frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{x}+(P+ \eta )_{x}=-(\eta+\beta\rho)\Phi_{x}. $$
(2.18)
Multiplying (2.18) by \(u_{t}\), integrating (by parts) over \(\Omega _{T}\), we have
$$\begin{aligned}& \iint_{\Omega_{T}}\rho|u_{t}|^{2}\,{\mathrm{d}} x \,{\mathrm{d}} s + \iint_{\Omega_{T}} \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} u_{xt}\,{\mathrm{d}} x\,{\mathrm{d}} s \\& \quad =- \iint_{\Omega_{T}} \bigl(\rho uu_{x}+P_{x}+ \eta_{x}+(\eta+\beta\rho)\Phi_{x} \bigr)u_{t}\,{ \mathrm{d}} x\,{\mathrm{d}} s. \end{aligned}$$
(2.19)
We deal with each term as follows:
$$\begin{aligned}& \int_{\Omega}\biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} u_{xt}\,{\mathrm{d}}x =\frac{1}{2} \int_{\Omega}\biggl(\frac {\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac {2-p}{2}} \bigl(u_{x}^{2} \bigr)_{t}\,{\mathrm{d}}x \\& \hphantom{\int_{\Omega}\biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} u_{xt}\,{\mathrm{d}}x} =\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\biggl( \int_{0}^{u_{x}^{2}} \biggl(\frac{\varepsilon s+1}{s+\varepsilon} \biggr)^{\frac{2-p}{2}}\,{\mathrm{d}} s \biggr)\,{\mathrm{d}} x, \\& \int_{0}^{u_{x}^{2}} \biggl(\frac{\varepsilon s+1}{s+\varepsilon} \biggr)^{\frac{2-p}{2}}\,{\mathrm{d}} s\geq \int_{0}^{u_{x}^{2}}(s+1)^{\frac{2-p}{2}}\,{\mathrm{d}} s= \frac{2}{p} \bigl[ \bigl(u_{x}^{2}+1 \bigr)^{\frac{p}{2}}-1 \bigr], \\& - \iint_{\Omega_{T}}P_{x}u_{t}\,{\mathrm{d}} x\,{ \mathrm{d}} s= \iint_{\Omega_{T}}P u_{xt}\,{\mathrm{d}} x\,{\mathrm{d}} s= \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Omega_{T}}P u_{x}\,{\mathrm{d}} x\,{\mathrm{d}} s- \iint_{\Omega_{T}}P_{t} u_{x}\,{\mathrm{d}} x\,{ \mathrm{d}} s. \end{aligned}$$
From (1.1)1 we have
$$\begin{aligned}& P_{t}=-\gamma P u_{x}-P_{x} u, \\& - \iint_{\Omega_{T}}\eta\Phi_{x} u_{t}\,{\mathrm{d}} x\,{\mathrm{d}} s=-\frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Omega_{T}}\eta\Phi_{x} u\,{\mathrm{d}} x\,{ \mathrm{d}} s+ \iint_{\Omega_{T}}\eta_{t} \Phi_{x} u\,{ \mathrm{d}} x\,{\mathrm{d}} s. \end{aligned}$$
(2.20)
Substituting the above into (2.19), using the Sobolev inequality and Young’s inequality, we obtain
$$\begin{aligned}& \int_{0}^{t} \bigl\vert \sqrt{\rho}u_{t}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s+ \bigl\vert u_{x}(t) \bigr\vert _{L^{p}}^{p} \\& \quad \leq \iint_{\Omega_{T}} \bigl(\vert \rho uu_{x} u_{t} \vert + \bigl\vert \gamma P u_{x}^{2} \bigr\vert + \vert P_{x} u u_{x}\vert +\vert \eta_{t} u_{x}\vert +\vert \eta_{t}\Phi_{x} u\vert + \vert \beta\rho\Phi_{x} u_{t}\vert \bigr)\,{\mathrm{d}} x \,{\mathrm{d}} s \\& \qquad {} + \int_{\Omega}\bigl(\vert P u_{x}\vert +\vert \eta u_{x}\vert +\vert \eta\Phi_{x} u\vert \bigr)\,{ \mathrm{d}} x +C \\& \quad \leq C+ \int_{0}^{t} \bigl(\vert \rho \vert _{L^{2}} \vert u\vert _{L^{\infty}} \vert u_{x}\vert _{L^{p}}^{\frac {p}{2}}\vert u_{x}\vert _{L^{\infty}}^{1-\frac{p}{2}}\vert u_{t}\vert _{L^{\infty}}+ \gamma \vert P\vert _{L^{2}}\vert u_{x}\vert _{L^{p}}^{\frac{p}{2}}\vert u_{x}\vert _{L^{\infty}}^{1-\frac {p}{2}}\vert u_{xx}\vert _{L^{2}} \\& \qquad {} +a\gamma \vert \rho \vert _{L^{\infty}}^{\gamma-1}\vert \rho _{x}\vert _{L^{2}}\vert u_{x}\vert _{L^{p}}\vert u_{xx}\vert _{L^{2}}+\vert \eta_{t}\vert _{L^{2}}\vert u_{x}\vert _{L^{p}}^{\frac {p}{2}}\vert u_{x}\vert _{L^{\infty}}^{1-\frac{p}{2}}+\vert \eta_{t}\vert _{L^{2}} \vert \Phi _{x}\vert _{L^{2}}\vert u\vert _{L^{\infty}} \\& \qquad {} +\beta \vert \rho \vert _{L^{\infty}}^{\frac{1}{2}}\vert \Phi_{x}\vert _{L^{2}}\vert \sqrt{\rho}u_{t}\vert _{L^{2}} \bigr)\,{\mathrm{d}} s+\vert P\vert _{L^{\frac{p}{p-1}}}\vert u_{x}\vert _{L^{p}}+\vert \eta \vert _{L^{\frac {p}{p-1}}} \vert u_{x}\vert _{L^{p}}+\vert \eta \vert _{L^{\frac{p}{p-1}}}\vert \Phi _{x}\vert _{L^{p}}\vert u \vert _{L^{\infty}} \\& \quad \leq C \biggl(1+ \int_{0}^{t} \bigl(\vert \rho \vert _{L^{2}}^{2}\vert u_{x}\vert _{L^{p}}^{2+p} \vert u_{xx}\vert _{L^{2}}^{2-p}+\vert P\vert _{L^{\infty}} \vert u_{x}\vert _{L^{p}}^{\frac{p}{2}} \vert u_{xx}\vert _{L^{2}}^{2-\frac{p}{2}} \\& \qquad {} +\vert \rho \vert _{L^{\infty}}^{\gamma-1}\vert \rho _{x}\vert _{L^{2}}\vert u_{x}\vert _{L^{p}}\vert u_{xx}\vert _{L^{2}}+\vert \eta_{t}\vert _{L^{2}}\vert u_{x}\vert _{L^{p}}^{\frac {p}{2}}\vert u_{xx}\vert _{L^{2}}^{1-\frac{p}{2}}+\vert \eta_{t}\vert _{L^{2}} \vert u_{x}\vert _{L^{p}}+\vert \rho \vert _{L^{\infty}} \bigr)\,{\mathrm{d}} s \biggr) \\& \qquad {} +\vert P\vert _{L^{\frac{p}{p-1}}}^{\frac{p}{p-1}}+\vert \eta \vert _{L^{\frac {p}{p-1}}}^{\frac{p}{p-1}}+\frac{1}{2} \int_{0}^{t} \bigl\vert \sqrt{\rho }u_{t}(s) \bigr\vert _{L^{2}}^{2}\,{\mathrm{d}} s+ \frac{1}{2} \bigl\vert u_{x}(t) \bigr\vert _{L^{p}}^{p}. \end{aligned}$$
(2.21)
To estimate (2.21), combining (2.20) we have the following estimates:
$$\begin{aligned} \int_{\Omega}\bigl\vert P(t) \bigr\vert ^{\frac{p}{p-1}}\,{ \mathrm{d}} x&= \int_{\Omega}\bigl\vert P(0) \bigr\vert ^{\frac {p}{p-1}}\,{ \mathrm{d}} x+ \int_{0}^{t}\frac{\partial}{\partial s} \biggl( \int_{\Omega}P(s)^{\frac{p}{p-1}}\,{\mathrm{d}} x \biggr)\,{ \mathrm{d}} s \\ &\leq \int_{\Omega}\bigl\vert P(0) \bigr\vert ^{\frac{p}{p-1}}\,{ \mathrm{d}} x+{\frac{p}{p-1}} \int _{0}^{t} \int_{\Omega}a\gamma\rho^{\gamma-1} P(s)^{\frac{p}{p-1}}(- \rho_{x} u-\rho u_{x})\,{\mathrm{d}} x\,{\mathrm{d}} s \\ &\leq C+C \int_{0}^{t}\vert \rho \vert _{L^{\infty}}^{\gamma-1} \vert P\vert _{L^{\infty}}^{\frac {p}{p-1}}\vert \rho \vert _{H^{1}} \vert u_{x}\vert _{L^{p}}\,{\mathrm{d}} s \\ &\leq C \biggl(1+ \int_{0}^{t}\Psi^{{\frac{1}{p-1}}+\gamma+1}(s)\,{\mathrm{d}} s \biggr), \end{aligned}$$
(2.22)
following the same method, we get
$$ \int_{\Omega}\bigl\vert \eta(t) \bigr\vert ^{\frac{p}{p-1}}\,{ \mathrm{d}} x\leq C \biggl(1+ \int_{0}^{t}\Psi ^{\frac{1}{p-1}+1}(s)\,{\mathrm{d}} s \biggr). $$
(2.23)
Combining (2.21)-(2.23) yields
$$ \int_{0}^{t} \bigl\vert \sqrt{\rho}u_{t}(s) \bigr\vert _{L^{2}}^{2}(s)\,{\mathrm{d}} s+ \bigl\vert u_{x}(t) \bigr\vert _{L^{p}}^{p}\leq C \biggl(1+ \int_{0}^{t}\Psi^{\frac{18\gamma}{3p-4}}(s)\,{\mathrm{d}} s \biggr), $$
(2.24)
where C is a positive constant, depending only on \(M_{0}\).
Estimate for \(|\sqrt{\rho}u_{t}|_{L^{2}}\)
Differentiating (1.1)2 with respect to t, we get
$$\begin{aligned}& \rho u_{tt}+\rho uu_{xt}+\rho_{t} u_{t}+ \rho_{t} u u_{x}+\rho u_{t} u_{x}+(P+ \eta)_{xt} \\& \quad = \biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{xt}-(\eta _{t}+\beta\rho_{t}) \Phi_{x}. \end{aligned}$$
Multiplying the result equation by \(u_{t}\), integrating over Ω, we derive
$$\begin{aligned}& {1\over 2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\rho|u_{t}|^{2}\,{\mathrm{d}} x + \int_{\Omega}\biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac {2-p}{2}}u_{x} \biggr]_{t} u_{xt}\,{ \mathrm{d}} x \\& \quad = \int_{\Omega}\bigl[(\rho u)_{x} \bigl(u_{t}^{2}+u u_{x} u_{t} \bigr)-(P+\eta)_{t} u_{xt}-( \eta _{t}+\beta\rho_{t})\Phi_{x} u_{t} \bigr]\,{\mathrm{d}} x. \end{aligned}$$
(2.25)
Note that
$$\begin{aligned}& \int_{\Omega}\biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{t} u_{xt}\,{ \mathrm{d}}x \\& \quad = \int_{\Omega}\biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon } \biggr)^{-\frac{p}{2}}u_{x} \biggr]\frac{(\varepsilon u_{x}^{2}+1)(u_{x}^{2}+\varepsilon)-(2-p)(1-\varepsilon ^{2})u_{x}^{2}}{(u_{x}^{2}+\varepsilon)^{2}}u_{xt}^{2} \,{\mathrm{d}}x \\& \quad \geq(p-1) \int_{\Omega}\bigl(u_{x}^{2}+1 \bigr)^{\frac{p-2}{2}}|u_{xt}|^{2} \,{\mathrm{d}} x. \end{aligned}$$
Let
$$\zeta= \bigl(u_{x}^{2}+1 \bigr)^{\frac{p-2}{4}}, $$
from (2.8), it follows that
$$\bigl\vert \zeta^{-1} \bigr\vert _{L^{\infty}}= \bigl\vert \bigl(u_{x}^{2}+1 \bigr)^{\frac{2-p}{4}} \bigr\vert _{L^{\infty}}\leq C \bigl(\vert u_{xx}\vert _{L^{2}}^{\frac{2-p}{2}}+1 \bigr)\leq C\Psi^{\frac{3\gamma}{3p-4}}(t). $$
Combining (2.20), (2.25) can be rewritten as
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\rho|u_{t}|^{2}\,{\mathrm{d}} x+ \int_{\Omega}|u_{xt}|^{2}\,{\mathrm{d}} x \\& \quad \leq2 \int_{\Omega}\rho|u||u_{t}||u_{xt}|\,{ \mathrm{d}} x+ \int_{\Omega}|\rho _{x}||u|^{2}|u_{x}||u_{t}| \,{\mathrm{d}} x+ \int_{\Omega}\rho|u||u_{x}|^{2}|u_{t}| \,{\mathrm{d}} x \\& \qquad {} + \int_{\Omega}\gamma P|u_{x}||u_{xt}|\,{ \mathrm{d}} x+ \int_{\Omega}|P_{x}||u||u_{xt}|\,{ \mathrm{d}} x+ \int_{\Omega}|\eta_{t}||\Phi_{x}||u_{t}| \,{\mathrm{d}} x \\& \qquad {} + \int_{\Omega}|\beta\rho_{x}||u||\Phi_{x}||u_{t}| \,{\mathrm{d}} x+ \int_{\Omega}|\beta\rho||u_{x}||\Phi_{x}||u_{t}| \,{\mathrm{d}} x=\sum_{j=1}^{8}I_{j}. \end{aligned}$$
(2.26)
Using the Sobolev inequality, Young’s inequality, (1.1)2, (2.8), and (2.9), we obtain
$$\begin{aligned}& I_{1}\leq2\vert \rho \vert _{L^{\infty}}^{1\over 2}\vert u \vert _{L^{\infty}} \vert \sqrt{\rho}u_{t}\vert _{L^{2}} \vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta^{-1} \bigr\vert _{L^{\infty}}\leq C\Psi^{\frac {8\gamma}{3p-4}}(t)+ \frac{1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{2}\leq \vert \rho_{x}\vert _{L^{2}}\vert u \vert _{L^{\infty}}^{2}\vert u_{x}\vert _{L^{p}}\vert u_{x}\vert _{L^{\infty}}^{1-\frac{p}{2}} \vert u_{t}\vert _{L^{\infty}}\leq \vert \rho _{x} \vert _{L^{2}}\vert u_{x}\vert _{L^{p}}^{3} \vert u_{xx}\vert _{L^{2}}^{1-\frac{p}{2}}\vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta^{-1} \bigr\vert _{L^{\infty}} \\& \hphantom{I_{2}}\leq C\Psi^{\frac{12\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{3}\leq \vert \rho \vert _{L^{\frac{p}{p-1}}}\vert u\vert _{L^{\infty}} \vert u_{x}\vert _{L^{p}}\vert u_{x}\vert _{L^{\infty}} \vert u_{t}\vert _{L^{\infty}}\leq \vert \rho \vert _{L^{\infty}} \vert u_{x} \vert _{L^{p}}^{2}\vert u_{xx}\vert _{L^{2}}\vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta^{-1} \bigr\vert _{L^{\infty}} \\& \hphantom{I_{3}}\leq C\Psi^{\frac{15\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{4}\leq C\vert P\vert _{L^{2}}\vert u_{x} \vert _{L^{\infty}} \vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta ^{-1} \bigr\vert _{L^{\infty}}\leq C \Psi^{\frac{7\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{5}\leq \vert P_{x}\vert _{L^{2}}\vert u \vert _{L^{\infty}} \vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta ^{-1} \bigr\vert _{L^{\infty}}\leq C \Psi^{\frac{7\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{6}\leq \vert \eta_{t}\vert _{L^{2}}\vert \Phi_{x}\vert _{L^{2}}\vert u_{t}\vert _{L^{\infty}}\leq C\vert \eta _{t}\vert _{L^{2}}\vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta^{-1} \bigr\vert _{L^{\infty}}\leq C\Psi^{\frac {11\gamma}{3p-4}}(t)+\frac{1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{7}\leq\beta \vert \rho_{x}\vert _{L^{2}} \vert u\vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}} \vert u_{t}\vert _{L^{\infty}}\leq C\vert \rho_{x} \vert _{L^{2}}\vert u_{x}\vert _{L^{p}}\vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta ^{-1} \bigr\vert _{L^{\infty}} \\& \hphantom{I_{7}}\leq C\Psi^{\frac{9\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}, \\& I_{8}\leq\beta \vert \rho \vert _{L^{2}}\vert u_{x}\vert _{L^{\infty}} \vert \Phi_{x}\vert _{L^{2}}\vert u_{t}\vert _{L^{\infty}}\leq C\vert \rho \vert _{L^{2}}\vert u_{xx}\vert _{L^{2}}\vert \zeta u_{xt}\vert _{L^{2}} \bigl\vert \zeta ^{-1} \bigr\vert _{L^{\infty}} \\& \hphantom{I_{8}}\leq C\Psi^{\frac{11\gamma}{3p-4}}(t)+\frac {1}{8}\vert u_{xt}\vert _{L^{2}}^{2}. \end{aligned}$$
Substituting \(I_{j}\) (\(j=1,2,\ldots,8\)) into (2.26), and integrating over \((\tau,t)\subset(0,T)\) over the time variable, we have
$$ \bigl\vert \sqrt{\rho}u_{t}(t) \bigr\vert _{L^{2}}^{2}+ \int_{0}^{t}\vert u_{xt}\vert _{L^{2}}^{2}(s)\,{\mathrm{d}} s\leq \bigl\vert \sqrt{ \rho}u_{t}(\tau) \bigr\vert _{L^{2}}^{2}+ \int_{0}^{t}\Psi^{\frac{15\gamma }{3p-4}}(s)\,{\mathrm{d}} s. $$
(2.27)
To obtain the estimate of \(|\sqrt{\rho}u_{t}(t)|_{L^{2}}^{2}\), we need to estimate \(\lim_{\tau\rightarrow 0}|\sqrt{\rho}u_{t}(\tau)|_{L^{2}}^{2}\). Multiplying (2.18) by \(u_{t}\) and integrating over Ω, we get
$$\begin{aligned} \begin{aligned} &\int_{\Omega}\rho \vert u_{t}\vert ^{2}\,{ \mathrm{d}} x \\ &\quad \leq 2 \int_{\Omega}\biggl(\rho \vert u\vert ^{2}\vert u_{x}\vert ^{2}+\beta^{2}\rho \vert \Phi_{x}\vert ^{2}+\rho^{-1} \biggl\vert \biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon} \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{x}+(P+\eta)_{x}+\eta\Phi _{x} \biggr\vert ^{2} \biggr)\,{\mathrm{d}}x. \end{aligned} \end{aligned}$$
According to the smoothness of \((\rho,u,\eta)\), we obtain
$$\begin{aligned}& \lim_{\tau\rightarrow0} \int_{\Omega}\biggl(\rho \vert u\vert ^{2}\vert u_{x}\vert ^{2}+\beta^{2}\rho \vert \Phi_{x}\vert ^{2}+\rho^{-1} \biggl\vert \biggl[ \biggl(\frac{\varepsilon u_{x}^{2}+1}{u_{x}^{2}+\varepsilon } \biggr)^{\frac{2-p}{2}}u_{x} \biggr]_{x}+(P+\eta)_{x}+\eta\Phi_{x} \biggr\vert ^{2} \biggr)\,{\mathrm{d}}x \\& \quad = \int_{\Omega}\biggl(\rho \vert u_{0}\vert ^{2}\vert u_{0x}\vert ^{2}+\beta^{2} \rho_{0}\vert \Phi_{x}\vert ^{2} \\& \qquad {}+\rho _{0}^{-1} \biggl\vert \biggl[ \biggl(\frac{\varepsilon u_{0x}^{2}+1}{u_{0x}^{2}+\varepsilon } \biggr)^{\frac{2-p}{2}}u_{0x} \biggr]_{x}+(P_{0}+ \eta_{0})_{x}+\eta_{0}\Phi_{x} \biggr\vert ^{2} \biggr)\,{\mathrm{d}}x \\& \quad \leq \vert \rho_{0}\vert _{L^{\infty}} \vert u_{0}\vert _{L^{\infty}}^{2}\vert u_{0x} \vert _{L^{2}}^{2}+\beta^{2}\vert \rho _{0}\vert _{L^{\infty}} \vert \Phi_{x}\vert ^{2}+\vert g\vert _{L^{2}}^{2}+\beta \vert \Phi_{x}\vert _{L^{2}}^{2}\leq C. \end{aligned}$$
Therefore, taking the limit on τ in (2.27), as \(\tau\rightarrow0\), we conclude that
$$ \bigl\vert \sqrt{\rho}u_{t}(t) \bigr\vert _{L^{2}}^{2}+ \int_{0}^{t}\vert u_{xt}\vert _{L^{2}}^{2}(s)\,{\mathrm{d}} s\leq C \biggl(1+ \int_{0}^{t}\Psi^{\frac{15\gamma}{3p-4}}(s)\,{\mathrm{d}} s \biggr), $$
(2.28)
where C is a positive constant, depending only on \(M_{0}\).
Combining the estimates of (2.8), (2.9), (2.12), (2.13), (2.17), (2.24), (2.28), and the definition of \(\Psi(t)\), we conclude that
$$ \Psi(t)\leq C\exp \biggl(\tilde{C} \int_{0}^{t}\Psi^{\frac{18\gamma}{3p-4}}(s) \,{\mathrm{d}} s \biggr), $$
(2.29)
where C, C̃ are positive constant, depending only on \(M_{0}\). This means that there exist a time \(T_{1}>0\) and a constant C, such that
$$\begin{aligned}& \operatorname{ess}\sup_{0\leq t\leq T_{1}} \bigl(\vert \rho \vert _{H^{1}}+|u|_{W_{0}^{1,p} \cap H^{2}}+|\eta|_{H^{2}}+| \eta_{t}|_{L^{2}}+|\sqrt{\rho}u_{t}|_{L^{2}}+| \rho _{t}|_{L^{2}} \bigr) \\& \quad {}+ \int_{0}^{T_{1}} \bigl(|\sqrt{\rho}u_{t}|_{L^{2}}^{2}+|u_{xt}|_{L^{2}}^{2}+| \eta _{x}|_{L^{2}}^{2}+|\eta_{t}|_{L^{2}}^{2}+| \eta_{xt}|_{L^{2}}^{2} \bigr) \,{\mathrm{d}} s\leq C, \end{aligned}$$
(2.30)
where C is a positive constant, depending only on \(M_{0}\).