In this section, we consider system (1.2) with degenerate diffusion (i.e., \(D(u)\geq0\) for all \(u\geq0\)). We first consider the following regularized system with nondegenerate diffusion for \(\varepsilon\in(0,1)\), which satisfies all the formal arguments:
$$ \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{\varepsilon t}=\nabla\cdot(D_{\varepsilon}(u_{\varepsilon})\nabla u_{\varepsilon})-\nabla\cdot(\chi u_{\varepsilon}\nabla v_{\varepsilon})+\nabla\cdot(\xi u_{\varepsilon}\nabla w_{\varepsilon}),& x\in\Omega, t>0,\\ 0=\Delta v_{\varepsilon}+\alpha u_{\varepsilon}-\beta v_{\varepsilon},& x\in \Omega, t>0,\\ 0=\Delta w_{\varepsilon}+\gamma u_{\varepsilon}-\delta w_{\varepsilon},& x\in \Omega, t>0,\\ \frac{\partial u_{\varepsilon}}{\partial\nu}=\frac{\partial v_{\varepsilon}}{\partial\nu}=\frac{\partial w_{\varepsilon}}{\partial\nu }=0,& x\in\partial\Omega, t>0,\\ u_{\varepsilon}(x,0)=u_{0}(x), &x\in\Omega, \end{array}\displaystyle \right . $$
(4.1)
where \(D_{\varepsilon}\) is defined by
$$D_{\varepsilon}(s):=D(s+\varepsilon) \quad\mbox{for all }s\geq0. $$
Thus, \(D_{\varepsilon}\) satisfies (1.3), (1.4), and (1.5). The following proposition is a direct consequence of Theorem 1.1.
Proposition 4.1
Let
\(\varepsilon\in(0,1)\), and let
\(u_{0}\in W^{1,\infty}(\Omega)\)
be a nonnegative function. Suppose that
\(\xi\gamma-\chi\alpha>0\)
or
\(\xi\gamma-\chi\alpha\le0\)
and
\(m>2-\frac{2}{n}\). Then system (4.1) admits a unique global bounded classical solution
\((u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon})\).
Next, we go to find some estimates to \((u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon})\), which are independent of ε and used to obtain some convergence properties. By taking \(\varepsilon\to0\) we will establish the existence of global bounded weak solutions. The following two lemmas based on the ideas in [18] are used to prove the existence of the limit function of \(\nabla\int^{u_{\varepsilon}+\varepsilon}_{0} D(z)\,dz\).
Lemma 4.1
Let
\(T>0\), and let the assumptions in Proposition
4.1
hold. Let
\((u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})\)
be a solution to system (4.1) on
\((0,T)\). Then
$$ \bigl\| D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)\nabla u_{\varepsilon}\bigr\| _{L^{2} (0,T;L^{2}(\Omega) )}^{2}\leq\frac {1}{2} \|u_{0}\|^{2}_{L^{2}(\Omega)}+C_{1} T, $$
(4.2)
where
\(C_{1}\)
is a positive constant independent of
ε.
Proof
Taking \(u_{\varepsilon}\) as a test function on the first equation in (4.1) and integrating it over \(\Omega\times(0,T)\), we derive
$$\begin{aligned} &\frac{1}{2} \bigl(\bigl\| u_{\varepsilon}(T)\bigr\| _{L^{2}(\Omega )}^{2}- \|u_{0}\|_{L^{2}(\Omega)}^{2} \bigr) \\ &\quad= - \int_{0}^{T} \int_{\Omega} D(u_{\varepsilon}+\varepsilon)|\nabla u_{\varepsilon}|^{2}\,dx \,dt + \int_{0}^{T} \int_{\Omega} \chi u_{\varepsilon }\nabla v_{\varepsilon}\cdot \nabla u_{\varepsilon}\,dx \,dt\\ &\qquad{} - \int _{0}^{T} \int_{\Omega} \xi u_{\varepsilon}\nabla w_{\varepsilon}\cdot \nabla u_{\varepsilon}\,dx \,dt \\ &\quad= - \int_{0}^{T} \int_{\Omega} D(u_{\varepsilon}+\varepsilon)|\nabla u_{\varepsilon}|^{2}\,dx \,dt +\frac{\chi}{2} \int_{0}^{T} \int_{\Omega} \nabla v_{\varepsilon}\cdot\nabla u_{\varepsilon}^{2}\,dx \,dt \\ &\qquad{}-\frac{\xi }{2} \int_{0}^{T} \int_{\Omega} \nabla w_{\varepsilon}\cdot\nabla u_{\varepsilon}^{2}\,dx \,dt \\ &\quad= - \int_{0}^{T} \int_{\Omega} D(u_{\varepsilon}+\varepsilon)|\nabla u_{\varepsilon}|^{2}\,dx \,dt -\frac{\chi}{2} \int_{0}^{T} \int_{\Omega} u_{\varepsilon}^{2} \Delta v_{\varepsilon}\,dx \,dt +\frac{\xi}{2} \int _{0}^{T} \int_{\Omega} u_{\varepsilon}^{2} \Delta w_{\varepsilon}\,dx \,dt. \end{aligned}$$
It then follows from the second and third equations in (4.1) that
$$\begin{aligned} &\frac{1}{2} \bigl(\bigl\| u_{\varepsilon}(T)\bigr\| _{L^{2}(\Omega )}^{2}- \|u_{0}\|_{L^{2}(\Omega)}^{2} \bigr) \\ &\quad= - \int_{0}^{T} \int_{\Omega} D(u_{\varepsilon}+\varepsilon)|\nabla u_{\varepsilon}|^{2}\,dx \,dt -\frac{\chi}{2} \int_{0}^{T} \int_{\Omega} (\beta v_{\varepsilon}-\alpha u_{\varepsilon}) u_{\varepsilon}^{2}\,dx \,dt \\ &\qquad{}+\frac{\xi}{2} \int_{0}^{T} \int_{\Omega} (\delta w_{\varepsilon}-\gamma u_{\varepsilon}) u_{\varepsilon}^{2}\,dx \,dt. \end{aligned}$$
From Proposition 4.1 we obtain that there exist some positive constants \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), and \(c_{5}\) independent of ε such that
$$ \begin{aligned} &\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{1},\qquad \|v_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{2},\qquad \|w_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{3},\\ & \|\nabla v_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{4},\quad \mbox{and}\quad \|\nabla w_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{5}. \end{aligned} $$
(4.3)
Therefore, we have
$$\begin{aligned} &\frac{1}{2} \bigl(\bigl\| u_{\varepsilon}(T)\bigr\| _{L^{2}(\Omega )}^{2}- \|u_{0}\|_{L^{2}(\Omega)}^{2} \bigr) \\ &\quad\leq - \int_{0}^{T} \int_{\Omega} D(u_{\varepsilon}+\varepsilon )|\nabla u_{\varepsilon}|^{2}\,dx \,dt+\frac{1}{2}c_{1}^{2} \bigl(\chi\beta c_{2} +(\xi\gamma+\chi\alpha)c_{1}+\xi\delta c_{3} \bigr)|\Omega|T, \end{aligned}$$
which yields the desired estimate
$$\bigl\| D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)\nabla u_{\varepsilon}\bigr\| _{L^{2} (0,T;L^{2}(\Omega) )}\leq\frac {1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}+C_{1} T, $$
where \(C_{1}:=\frac{1}{2}c_{1}^{2} (\chi\beta c_{2}+(\xi\gamma+\chi\alpha )c_{1}+\xi\delta c_{3} )|\Omega|\). □
Lemma 4.2
Let
\(T>0\), and let the assumptions in Proposition
4.1
hold. Let
\((u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})\)
be a solution to system (4.1) on
\((0,T)\). Then
$$\begin{aligned} &\biggl\Vert \sqrt{t}\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T;L^{2}(\Omega) )}^{2} +\sup_{t\in(0,T)} t\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon }D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} \\ &\quad\leq C_{2}+C_{2} T +C_{2} T^{2}, \end{aligned}$$
(4.4)
where
\(C_{2}\)
is a positive constant independent of
ε.
Proof
We multiply the first equation in (4.1) by \(\frac{d}{dt}\int _{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\) and then integrate it over Ω to obtain
$$\begin{aligned} & \int_{\Omega} \biggl(\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon }D^{\frac{1}{2}}(z)\,dz \biggr)^{2}\,dx \\ & \quad= \int_{\Omega}\nabla\cdot\bigl(D(u_{\varepsilon}+\varepsilon) \nabla u_{\varepsilon}\bigr)\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz \,dx - \int_{\Omega}\nabla\cdot(\chi u_{\varepsilon}\nabla v_{\varepsilon})\frac {d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz \,dx \\ &\qquad{}+ \int_{\Omega}\nabla\cdot(\xi u_{\varepsilon}\nabla w_{\varepsilon})\frac {d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz \,dx \\ &\quad= - \int_{\Omega}D(u_{\varepsilon}+\varepsilon)\nabla u_{\varepsilon}\cdot\frac{d}{dt} \bigl(D(u_{\varepsilon}+\varepsilon) \nabla u_{\varepsilon}\bigr)\,dx \\ &\qquad{}-\chi \int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla v_{\varepsilon}+ u_{\varepsilon}\triangle v_{\varepsilon}) D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon) \biggl(\frac{d}{dt} \int _{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz \biggr)\,dx \\ &\qquad{} + \xi \int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla w_{\varepsilon}+u_{\varepsilon}\triangle w_{\varepsilon}) D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon) \biggl(\frac{d}{dt} \int _{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz \biggr)\,dx, \end{aligned}$$
where we used \(\frac{d}{dt}\int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz =D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)\frac{d}{dt}\int _{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\). By Young’s inequality we obtain
$$\begin{aligned} & \int_{\Omega} \biggl(\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon }D^{\frac{1}{2}}(z)\,dz \biggr)^{2}\,dx \\ &\quad\leq -\frac{1}{2}\frac{d}{dt}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} +\frac{1}{4} \int_{\Omega} \biggl(\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz \biggr)^{2}\,dx \\ &\qquad{}+\frac{1}{4} \int_{\Omega} \biggl(\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz \biggr)^{2}\,dx \\ &\qquad{} +\chi^{2} \int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla v_{\varepsilon}+ u_{\varepsilon}\Delta v_{\varepsilon})^{2} D(u_{\varepsilon}+\varepsilon)\,dx \\ &\qquad{} +\xi^{2} \int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla w_{\varepsilon}+ u_{\varepsilon}\Delta w_{\varepsilon})^{2} D(u_{\varepsilon}+\varepsilon)\,dx. \end{aligned}$$
(4.5)
Since \(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{1}\) and \(D\in C^{2} ([0,\infty) )\), we have \(\|D(u_{\varepsilon}+\varepsilon)\|_{L^{\infty}(\Omega)}< c_{\infty}\) with some constant \(c_{\infty}>0\). Thus, by Young’s inequality and (4.3) we derive
$$\begin{aligned} &\int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla v_{\varepsilon}+ u_{\varepsilon}\triangle v_{\varepsilon})^{2} D(u_{\varepsilon}+\varepsilon )\,dx \\ &\quad\leq2 \int_{\Omega} \bigl(|\nabla u_{\varepsilon}|^{2} | \nabla v_{\varepsilon}|^{2} +u_{\varepsilon}^{2} | \triangle v_{\varepsilon}|^{2} \bigr)D(u_{\varepsilon}+ \varepsilon)\,dx \\ &\quad \leq2 \| \nabla v_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2} \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} +2c_{1} ^{2} c_{\infty}\int_{\Omega}|\triangle v_{\varepsilon}|^{2}\,dx \\ &\quad\leq2 \| \nabla v_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2} \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2}+4c_{1} ^{2} c_{\infty}\int_{\Omega}\bigl(\alpha^{2} u_{\varepsilon}^{2}+ \beta^{2} v_{\varepsilon}^{2} \bigr)\,dx \\ &\quad\leq2c_{4} ^{2}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2}+4c_{1} ^{2} c_{\infty}\bigl(\alpha^{2} c_{1} ^{2} +\beta^{2} c_{2} ^{2} \bigr)|\Omega|. \end{aligned}$$
Similarly,
$$\begin{aligned} &\int_{\Omega} (\nabla u_{\varepsilon}\cdot\nabla w_{\varepsilon}+ u_{\varepsilon}\triangle w_{\varepsilon})^{2} D(u_{\varepsilon}+\varepsilon)\,dx \\ &\quad\leq2c_{5} ^{2}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2}+4c_{1} ^{2} c_{\infty}\bigl(\gamma^{2} c_{1} ^{2} +\delta^{2} c_{3} ^{2} \bigr)|\Omega|. \end{aligned}$$
Substituting the last two inequalities into (4.5), we obtain
$$\begin{aligned} & \int_{\Omega} \biggl(\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon }D^{\frac{1}{2}}(z)\,dz \biggr)^{2}\,dx +\frac{d}{dt}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon }D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} \\ &\quad\leq 4 \bigl(\chi^{2} c_{4} ^{2} + \xi^{2} c_{5} ^{2} \bigr)\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2}\\ &\qquad{}+8c_{1} ^{2} c_{\infty}\chi^{2} \bigl(\alpha^{2} c_{1} ^{2} + \beta ^{2} c_{2} ^{2} \bigr)|\Omega|+8c_{1} ^{2} c_{\infty}\xi^{2} \bigl(\gamma^{2} c_{1} ^{2} +\delta^{2} c_{3} ^{2} \bigr)|\Omega|. \end{aligned}$$
Setting \(C_{\max}:=4\max \{ (\chi^{2} c_{4} ^{2} + \xi^{2} c_{5} ^{2} ), [2c_{1} ^{2} c_{\infty}\chi^{2}(\alpha^{2} c_{1} ^{2} +\beta^{2} c_{2} ^{2})|\Omega |+2c_{1} ^{2} c_{\infty}\xi^{2}(\gamma^{2} c_{1} ^{2} +\delta c_{3} ^{2})|\Omega| ] \}\) yields
$$\begin{aligned} &\biggl\Vert \frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} +\frac{d}{dt}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon }D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} \\ &\quad\leq C_{\max} \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} +C_{\max}. \end{aligned}$$
(4.6)
Multiplying (4.6) by t and integrating it over \((0, T)\), we obtain
$$ \begin{aligned}[b] &\biggl\Vert \sqrt{t}\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2}+ t\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2}\\ &\quad\leq \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2} +C_{\max} T \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2}\\ &\qquad{} + C_{\max} T. \end{aligned} $$
(4.7)
By (4.2) the integrals on the right-hand side of (4.7) can be estimated as
$$\begin{aligned} \biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2} &=\biggl\Vert D^{\frac{1}{2}}(u_{\varepsilon}+ \varepsilon)\nabla \int _{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2} \\ &\leq c_{\infty}\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2} \\ &\leq c_{\infty}\biggl(\frac{1}{2}\|u_{0} \|^{2}_{L^{2}(\Omega)}+C_{1} T \biggr) \end{aligned}$$
(4.8)
Then substituting (4.8) into (4.7) and using (4.2) again, we have
$$\begin{aligned} &\biggl\Vert \sqrt{t}\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2} (0,T; L^{2}(\Omega) )}^{2}+ t\biggl\Vert \nabla \int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\biggr\Vert _{L^{2}(\Omega)}^{2} \\ &\quad\leq c_{\infty}\biggl(\frac{1}{2}\|u_{0} \|^{2}_{L^{2}(\Omega)}+C_{1} T \biggr)+C_{\max} T \biggl(\frac{1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}+C_{1} T \biggr)+ C_{\max} T \\ &\quad\leq C_{2}+C_{2} T +C_{2} T^{2}, \end{aligned}$$
where \(C_{2}:=\max \{\frac{1}{2}c_{\infty}\|u_{0}\|^{2}_{L^{2}(\Omega)}, C_{\max}C_{1}, [\frac{1}{2}C_{\max}\|u_{0}\|^{2}_{L^{2}(\Omega)}+C_{\max }+c_{\infty}C_{1} ] \} \). By taking the supremum with respect to t on \((0, T)\) we complete the proof of (4.4). □
We now prove Theorem 1.2. Our method is also partially inspired by [18].
Proof of Theorem 1.2
For any given \(T>0\), we have \(\| u_{\varepsilon}\|_{L^{\infty} (0,T; L^{p}(\Omega) )}< C\) (\(p\in[1,\infty]\)), where C is a positive constant independent of T and ε. Then there exist a subsequence \(\{u_{\varepsilon_{j}} \}_{j\in\mathbb{N}}\) and a function \(u\in L^{\infty} (0,T; L^{p}(\Omega) )\) such that
$$ u_{\varepsilon_{j}}\rightharpoonup u \quad\mbox{weakly}^{*}\mbox{ in } L^{\infty} \bigl(0,T; L^{p}(\Omega) \bigr) $$
(4.9)
for any \(p\in[1,\infty]\), where \(\varepsilon_{j}\rightarrow0\) as \(j\rightarrow\infty\). By using \(D\in C^{2} ([0,\infty) )\) and \(\| u_{\varepsilon}(t)\|_{L^{\infty}(\Omega)}< c_{1}\) again, from Lemma 4.1 we deduce that \(\int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\) is bounded in \(L^{2} (0,T; H^{1}(\Omega) ) \). Hence, there exist a subsequence (still denoted by \(\{u_{\varepsilon_{j}}\} _{j\in\mathbb{N}}\)) and a function \(\vartheta\in L^{2} (0,T; H^{1}(\Omega) )\) such that
$$ \begin{aligned} & \int_{0}^{u_{\varepsilon_{j}}+\varepsilon_{j}} D^{\frac {1}{2}}(z)\,dz\rightharpoonup \vartheta \quad\mbox{weakly in } L^{2} \bigl(0,T; L^{2}( \Omega) \bigr), \\ &\nabla \int_{0}^{u_{\varepsilon_{j}}+\varepsilon_{j} }D^{\frac {1}{2}}(z)\,dz\rightharpoonup \nabla\vartheta \quad\mbox{weakly in } L^{2} \bigl(0,T; L^{2}( \Omega) \bigr). \end{aligned} $$
(4.10)
On the other hand, by letting \(\tau>0\), from Lemma 4.2 we have
$$\begin{aligned} \tau\biggl\Vert \frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\biggr\Vert _{L^{2} (\tau,T; L^{2}(\Omega) )}^{2} &\leq\biggl\Vert \sqrt{t}\frac{d}{dt} \int_{0}^{u_{\varepsilon}+\varepsilon }D^{\frac{1}{2}}(z)\,dz\biggr\Vert _{L^{2} (\tau,T; L^{2}(\Omega) )}^{2} \\ &\leq C_{2}+C_{2} T +C_{2} T^{2}, \end{aligned}$$
which implies that \(\int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\) is bounded in \(H^{1} (\tau,T; L^{2}(\Omega) )\) (in particular, it is bounded in \(H^{1} (\tau,T; H^{-1}(\Omega) )\). Thus, by the Aubin-Lions lemma there exists a subsequence (still denoted by \(\{u_{\varepsilon_{j}}\}_{j\in\mathbb{N}}\)) such that
$$\int_{0}^{u_{\varepsilon_{j}}+\varepsilon_{j}} D^{\frac {1}{2}}(z)\,dz\rightarrow \vartheta\quad \mbox{strongly in } L^{2} \bigl(\tau,T; L^{2}( \Omega) \bigr) \mbox{ and a.e. on } \Omega\times(\tau,T). $$
Set \(f(r):=\int_{0} ^{r} D^{\frac{1}{2}}(z)\,dz\). We see that \(f(r)\) is a strictly increasing and continuous function. Thus, the inverse function \(f^{-1}(r)\) of f exists and is continuous. Moreover, we can obtain that
$$ u_{\varepsilon_{j}}\rightarrow u=f^{-1}(\vartheta) \quad \mbox{strongly in } L^{2} \bigl(\tau,T; L^{2}(\Omega) \bigr) \mbox{ and a.e. on } \Omega\times(\tau,T). $$
(4.11)
Since \(\tau>0\) is arbitrary, we deduce from (4.10) and (4.11) that
$$ \vartheta= \int_{0} ^{u} D^{\frac{1}{2}}(z)\,dz\in L^{2} \bigl(0,T; H^{1}(\Omega) \bigr). $$
(4.12)
Since \(\|v_{\varepsilon}(t)\|_{W^{1,\infty}(\Omega)}< c_{2}+c_{4}\), there exist a subsequence \(\{v_{\varepsilon_{j}}\}_{j \in\mathbb{N}}\) (hereafter, we still denote the subscript of the subsequence by \(\{ v_{\varepsilon_{j}}\}_{j\in\mathbb{N}}\) for simplicity) and functions v such that
$$ \begin{aligned} & v_{\varepsilon_{j}}\rightharpoonup v \quad\mbox{weakly}^{*}\mbox{ in } L^{\infty} \bigl(0,T; L^{\infty}( \Omega) \bigr), \\ & \nabla v_{\varepsilon_{j}}\rightharpoonup\nabla v \quad\mbox{weakly}^{*}\mbox{ in } L^{\infty} \bigl(0,T; L^{\infty}(\Omega) \bigr). \end{aligned} $$
(4.13)
Similarly, there exist subsequence \(\{w_{\varepsilon_{j}}\}_{n\in\mathbb {N}}\) and functions w such that
$$ \begin{aligned} & w_{\varepsilon_{j}}\rightharpoonup w \quad\mbox{weakly}^{*}\mbox{ in } L^{\infty} \bigl(0,T; L^{\infty}( \Omega) \bigr), \\ & \nabla w_{\varepsilon_{j}}\rightharpoonup\nabla w \quad\mbox{weakly}^{*}\mbox{ in } L^{\infty} \bigl(0,T; L^{\infty}(\Omega) \bigr) \end{aligned} $$
(4.14)
due to \(\|w_{\varepsilon}(t)\|_{W^{1,\infty}(\Omega)}< c_{3}+c_{5}\).
For any given \(T\in(0,\infty)\), we take \(\varphi\in C_{0} ^{\infty} (\Omega\times[0,T) )\). Then multiplying the first, second, and third equations in (4.1) by φ and integrating those on \(\Omega \times(0,T)\) we see that
$$ \left \{ \textstyle\begin{array}{@{}l} \int_{0} ^{T}\int_{\Omega} (D(u_{\varepsilon_{j}}+\varepsilon_{j})\nabla u_{\varepsilon_{j}}\cdot\nabla\varphi-\chi u_{\varepsilon_{j}} \nabla v_{\varepsilon_{j}}\cdot\nabla\varphi+\xi u_{\varepsilon_{j}} \nabla w_{\varepsilon_{j}}\cdot\nabla\varphi-u_{\varepsilon_{j}}\varphi_{t} )\,dx\,dt\\ \quad=\int_{\Omega}u_{0}(x)\varphi(x,0)\,dx,\\ \int_{0} ^{T}\int_{\Omega} (\nabla v_{\varepsilon_{j}}\cdot\nabla\varphi +\beta v_{\varepsilon_{j}} \varphi )\,dx\,dt=\int_{0} ^{T}\int_{\Omega} \alpha u_{\varepsilon_{j}} \varphi \,dx\,dt,\\ \int_{0} ^{T}\int_{\Omega} (\nabla w_{\varepsilon_{j}}\cdot\nabla\varphi +\delta w_{\varepsilon_{j}} \varphi )\,dx\,dt=\int_{0} ^{T}\int_{\Omega} \gamma u_{\varepsilon_{j}} \varphi \,dx\,dt. \end{array}\displaystyle \right . $$
(4.15)
Noting that \(D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)|\nabla\varphi |\leq c^{\frac{1}{2}}_{\infty}\|\nabla\varphi\|_{L^{\infty}}\) and thus \(D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)|\nabla\varphi|\in L^{2} (0,T; L^{2}(\Omega) )\), we see from (4.11) that
$$D^{\frac{1}{2}}(u_{\varepsilon_{j}}+\varepsilon_{j})\nabla\varphi \rightarrow D^{\frac{1}{2}}(u)\nabla\varphi \quad\mbox{strongly in } L^{2} \bigl(0,T; L^{2}(\Omega) \bigr), $$
which, together with (4.10) and (4.12), yields that
$$\begin{aligned} &\int_{0} ^{T} \int_{\Omega} \bigl(D(u_{\varepsilon_{j}}+\varepsilon_{j}) \nabla u_{\varepsilon_{j}}\cdot\nabla\varphi \bigr)\,dx\,dt \\ &\quad= \int_{0} ^{T} \int_{\Omega} \biggl(D^{\frac{1}{2}}(u_{\varepsilon _{j}}+ \varepsilon_{j})\nabla\varphi\cdot\nabla \int_{0}^{u_{\varepsilon _{j}}+\varepsilon_{j}}D^{\frac{1}{2}}(z)\,dz \biggr)\,dx\,dt \\ &\quad\rightarrow \int_{0} ^{T} \int_{\Omega} \biggl(D^{\frac{1}{2}}(u)\nabla \varphi\cdot\nabla \int_{0}^{u}D^{\frac{1}{2}}(z)\,dz \biggr)\,dx\,dt \\ &\quad= \int_{0} ^{T} \int_{\Omega} \bigl(D(u)\nabla u\cdot\nabla\varphi \bigr)\,dx\,dt \end{aligned}$$
(4.16)
as \(j\to\infty\). Similarly, since
$$u_{\varepsilon_{j}} \nabla\varphi\rightarrow u\nabla\varphi \quad\mbox{strongly in } L^{2} \bigl(0,T; L^{2}(\Omega) \bigr) $$
by (4.11), from (4.13) and (4.14) we see that
$$\begin{aligned} &\int_{0} ^{T} \int_{\Omega} (-\chi u_{\varepsilon_{j}} \nabla v_{\varepsilon _{j}} \cdot\nabla\varphi+\xi u_{\varepsilon_{j}} \nabla w_{\varepsilon _{j}}\cdot\nabla \varphi )\,dx\,dt \\ &\quad\rightarrow \int_{0} ^{T} \int_{\Omega} (-\chi u\nabla v\cdot\nabla\varphi+\xi u\nabla w\cdot \nabla\varphi )\,dx\,dt \end{aligned}$$
(4.17)
as \(j\to\infty\). Summarily, by collecting (4.9), (4.13), (4.14), (4.16), and (4.17), from (4.15) we obtain that
$$ \left \{ \textstyle\begin{array}{@{}l} \int_{0} ^{T}\int_{\Omega} (D(u)\nabla u \cdot\nabla\varphi-\chi u \nabla v\cdot\nabla\varphi+\xi u \nabla w\cdot\nabla\varphi-u \varphi _{t} )\,dx\,dt\\ \quad=\int_{\Omega}u_{0}(x)\varphi(x,0)\,dx,\\ \int_{0} ^{T}\int_{\Omega} (\nabla v\cdot\nabla\varphi+\beta v \varphi )\,dx\,dt=\int_{0} ^{T}\int_{\Omega} \alpha u \varphi \,dx\,dt,\\ \int_{0} ^{T}\int_{\Omega} (\nabla w\cdot\nabla\varphi+\delta w \varphi )\,dx\,dt=\int_{0} ^{T}\int_{\Omega} \gamma u \varphi \,dx\,dt \end{array}\displaystyle \right . $$
(4.18)
upon letting \(j\rightarrow\infty\). Hence, \((u,v,w)\) is a global weak solution to system (1.2). Moreover, we deduce from (4.9), (4.13), (4.14), and Theorem 1.1 that
$$\begin{aligned}& \| u\|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq\liminf_{j\rightarrow\infty}\| u_{\varepsilon_{j}} \|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq c_{1}, \\& \| v\|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq\liminf_{j\rightarrow\infty}\| v_{\varepsilon_{j}} \|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq c_{2}, \\& \|w\|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq\liminf_{j\rightarrow\infty}\| w_{\varepsilon_{j}} \|_{L^{\infty} (0,T; L^{\infty}(\Omega) )}\leq c_{3}, \end{aligned}$$
which implies the uniform boundedness of \((u,v,w)\). Thus, we complete the proof of Theorem 1.2. □