A Appendix
The proof of Lemma 3.1 is based on energy estimates in a regularized system. More precisely, in order to solve the equation of u, we replace the nonlinear slow diffusion term with a non-degenerate one, that is to say, we work on the approximate problems given by
$$\begin{aligned} \left\{ \begin{aligned}&{u_{\varepsilon t}} = \Delta {(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{u_\varepsilon } - \chi \nabla \cdot ({u_\varepsilon }\nabla {v_\varepsilon }) - \xi \nabla \cdot \left( {{u_\varepsilon }\nabla {w_\varepsilon }} \right) - \mu {u_\varepsilon }(1 - {u_\varepsilon }) + \beta {u_\varepsilon }{v_\varepsilon },\quad in\;Q,\\&\quad {v_{\varepsilon t}} = D\Delta {v_\varepsilon } + {u_\varepsilon } - {u_\varepsilon }{v_\varepsilon },\quad in\;Q,\\&\quad {w_{\varepsilon t}} = - \delta {v_\varepsilon }{w_\varepsilon } + \eta {w_\varepsilon }\left( {1 - {w_\varepsilon }} \right) ,\quad in\;Q,\\&\quad {\left. {{{\left. {\frac{{\partial u_\varepsilon ^{}}}{{\partial n}}} \right| }_{\partial n}} = {{\left. {\frac{{\partial {v_\varepsilon }}}{{\partial n}}} \right| }_{\partial n}} = \frac{{\partial {w_\varepsilon }}}{{\partial n}}} \right| _{\partial n}} = 0,\\&\quad {u_\varepsilon }(x,0) = {u_{\varepsilon 0}}(x),\quad {v_\varepsilon }(x,0) = {v_{\varepsilon 0}}(x),\quad {w_\varepsilon }(x,0) = {w_{\varepsilon 0}}(x),\quad x \in \Omega , \end{aligned}\right. \end{aligned}$$
(A.1)
where \({u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}}\) satisfy
$$\begin{aligned} \left\{ \begin{aligned}&{u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}} \in {C^{2 + \alpha }}(\bar{\Omega }),\;\alpha \in (0,1), \\&\quad {u_{\varepsilon 0}},{v_{\varepsilon 0}},{w_{\varepsilon 0}} \ge 0,\;{\left\| {{v_{\varepsilon 0}}} \right\| _{{L^\infty }}} \le {\left\| {{v_0}} \right\| _{{L^\infty }}},\;{\left\| {{w_{\varepsilon 0}}} \right\| _{{L^\infty }}} \le {\left\| {{w_0}} \right\| _{{L^\infty }}}, \\&\quad {\left. {\frac{{\partial {u_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = {\left. {\frac{{\partial {v_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = {\left. {\frac{{\partial {w_{\varepsilon 0}}}}{{\partial n}}} \right| _{\partial \Omega }} = 0, \\&\quad {u_{\varepsilon 0}} \rightarrow {u_0}\;in\;{L^p}\;for\;any\;p \in (1,\infty ),\;{v_{\varepsilon 0}} \rightarrow {v_0},\;{w_{\varepsilon 0}} \rightarrow {w_0},\quad uniformly. \\ \end{aligned} \right. , \end{aligned}$$
(A.2)
It is not difficult to obtain
$$\begin{aligned} 0 \le \left\| {{v_\varepsilon }} \right\| \le \max \{ 1,{\left\| {{v_{\varepsilon 0}}} \right\| _{{L^\infty }}}\} \le \max \{ 1,{\left\| {{v_0}} \right\| _{{L^\infty }}}\} , \end{aligned}$$
(A.3)
by comparison principle. By the third equation of (A.1), it is easy to see that
$$\begin{aligned} 0 \le \left\| {{w_\varepsilon }} \right\| \le \max \{ 1,{\left\| {{w_{\varepsilon 0}}} \right\| _{{L^\infty }}}\} \le \max \{ 1,{\left\| {{w_0}} \right\| _{{L^\infty }}}\} . \end{aligned}$$
(A.4)
We must point out that, \({v_\varepsilon }\) and \({w_\varepsilon }\) are bounded uniformly.
Using fixed point theory, or similar to the study of the chemotaxis model [15, 24], it is not difficult to obtain the following local existence result of classical solution to the problem (A.1) for any \(\varepsilon > 0\).
Lemma A.1
Assume that \({u_{\varepsilon 0}},\;{v_{\varepsilon 0}},\;{w_{\varepsilon 0}}\) satisfy (A.2). Then for any \(\varepsilon >0\), there exists \({T_{\max }} \in (0, + \infty ]\) such that the problem (A.1) admits a unique classical solution \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon }) \in {C^{2 + \alpha ,1 + \frac{\alpha }{2}}}(\Omega \times [0,{T_{\max }}))\) with
$$\begin{aligned} {u_\varepsilon } \ge 0,\;{v_\varepsilon } \ge 0,\;{w_\varepsilon } \ge 0\quad for\;all\;(x,t) \in \Omega \times (0,{T_{\max }}) \end{aligned}$$
such that either \({T_{\max }} = \infty \), or
$$\begin{aligned} \mathop {\lim }\limits _{t \nearrow {T_{\max }}} ({\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} + {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + {\left\| {{w_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}}) = + \infty . \end{aligned}$$
(A.5)
In order to verify that the problem (A.1) admits a unique global solution, we may assume \({T_{\max }} < \infty \), our purpose is to show the boundedness of \({\left\| {u( \cdot ,t)} \right\| _{{L^\infty }}} + {\left\| {v( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + {\left\| {w( \cdot ,t)} \right\| _{{L^\infty }}}\) for any \(t \in [0,{T_{\max }}]\).
It is not difficult to obtain the following lemma.
Lemma A.2
Assume (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have
$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^1}}} + \int \limits _0^T {\int \limits _\Omega {u_\varepsilon ^2} } {\mathrm{d}}x{\mathrm{d}}t \le {C_T}, \end{aligned}$$
(A.6)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{H^1}}} + \int \limits _0^T {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{H^2}}^2} {\mathrm{d}}t \le {C_T}, \end{aligned}$$
(A.7)
$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{w_{\varepsilon t}}} \right\| _{{L^\infty }}} \le C, \end{aligned}$$
(A.8)
where the constants \({C_T}\) at most depend on \(\chi ,\xi ,\mu ,\beta ,D,{u_0},{v_0},{w_0},T\) and \(\left| \Omega \right| \), C at most depend on \(\chi ,\xi ,\mu ,\beta ,D,{u_0},{v_0},{w_0}, \left| \Omega \right| \), all of them are independent of \(\varepsilon \).
Proof
By a direct integration for the first equation of (A.1), noticing that \({{v_\varepsilon }}\) is bounded, and applying Young’s inequality, we have
$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x}&= \mu \int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x} - \mu \int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} + \beta \int \limits _\Omega {{u_\varepsilon }{v_\varepsilon }{\hbox {d}}x}\\&\le - \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} + {C_1} , \end{aligned}$$
it means
$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} \le {C_1}, \end{aligned}$$
(A.9)
then (A.6) is obtained by a direct integration.
Multiplying the second equation of (A.1) by \({v_\varepsilon }\) and \(\Delta {v_\varepsilon }\), respectively, we have
$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x} + D\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x}&= \int \limits _\Omega {{u_\varepsilon }(1 - {v_\varepsilon }){v_\varepsilon }{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x}\\&\le {C_2}\int \limits _\Omega {u_\varepsilon ^{}{\hbox {d}}x} + \frac{1}{2}\int \limits _\Omega {v_\varepsilon ^2{\hbox {d}}x} \end{aligned}$$
and
$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + D\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + \int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} {\text { }}&= - \int \limits _\Omega {({u_\varepsilon }(1 - {v_\varepsilon }) + {v_\varepsilon })\Delta {v_\varepsilon }{\hbox {d}}x}\\&\le \frac{D}{2}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_3}\left( {1 + \int \limits _\Omega {u_\varepsilon ^2{\hbox {d}}x} } \right) , \end{aligned}$$
adding the above two formulas, since \({v_\varepsilon }\) is bounded, and using (A.6), we obtain (A.7) by a direct integration.
By the third equation of (A.1), since \({{v_\varepsilon }},{{w_\varepsilon }}\) are bounded, we easily see that
$$\begin{aligned} {\left\| {{w_{\varepsilon t}}} \right\| _{{L^\infty }}} \le \delta {\left\| {{v_\varepsilon }{w_\varepsilon }} \right\| _{{L^\infty }}} + \eta {\left\| {{w_\varepsilon }(1 - {w_\varepsilon })} \right\| _{{L^\infty }}} \le {C_4}, \end{aligned}$$
then (A.8) is achieved. \(\square \)
Lemma A.3
Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have
$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} {\mathrm{d}}x \le C\mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^p}}}{{{w_\varepsilon }}}{\mathrm{d}}x} \le Cp\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^p}{\mathrm{d}}x} {\mathrm{d}}s + {C_T}} ,\;for\;any\;p \ge 2, \end{aligned}$$
(A.10)
$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} {\mathrm{d}}x \le C\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^2}\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^2}}}{{{w_\varepsilon }}}{\mathrm{d}}x{\mathrm{d}}t} } + \int \limits _0^T {\int \limits _\Omega {\frac{{{{\left| {\nabla {w_\varepsilon }} \right| }^4}}}{{{w_\varepsilon }}}{\mathrm{d}}x} } {\mathrm{d}}t} \right) + {C_T}, \end{aligned}$$
(A.11)
where C at most depend on \(\delta ,\eta ,{v_0},{w_0}\), \({C_T}\) also depend on T, all of them are independent of \(\varepsilon \).
Proof
The proof is similar to Lemma 3.5, so we omit it here. \(\square \)
Lemma A.4
Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\), \(m>1\). Then, for any \(q=1,2,3, \cdots \), we have
$$\begin{aligned}&\sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^q} - 1}}}^{2{m^q} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^q} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\mathrm{d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^q}}} {\mathrm{d}}x} \right) } {\mathrm{d}}s \le {C_T}, \end{aligned}$$
(A.12)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} + \int \limits _0^T {\left( {\left\| {{v_\varepsilon }} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} + \left\| {{v_{\varepsilon t}}} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} \right) } {\mathrm{d}}s \le {C_T},\quad for\;any\;q > 1. \end{aligned}$$
(A.13)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in [0,T]} \left( {\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} {\mathrm{d}}x + \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} } \right) {\mathrm{d}}x \le {C_T},\quad for\;any\;p \ge 2 \end{aligned}$$
(A.14)
where \({C_T}\) depends on q and T and is independent of \(\varepsilon \).
Proof
Multiplying the first equation of (A.1) by \(u_\varepsilon ^r\) for any \(r>0\), and integrating it over \(\Omega \). Since \({v_\varepsilon }\) is bounded, we infer that
$$\begin{aligned}&\frac{1}{{r + 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + r\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \mu \int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x}\\&\quad \le r\chi \int \limits _\Omega {u_\varepsilon ^r\nabla {u_\varepsilon }\nabla {v_\varepsilon }{\hbox {d}}x} + r\xi \int \limits _\Omega {u_\varepsilon ^r\nabla {u_\varepsilon }\nabla {w_\varepsilon }{\hbox {d}}x} + \mu \int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \beta \int \limits _\Omega {u_\varepsilon ^{r + 1}{v_\varepsilon }{\hbox {d}}x}\\&\quad \le \frac{r}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{13}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{14}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x} +{C_{15}}, \end{aligned}$$
which means
$$\begin{aligned}&\frac{1}{{r + 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \frac{r}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{r + 2}{\hbox {d}}x} \nonumber \\&\quad \le {C_{13}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{14}}\int \limits _\Omega {u_\varepsilon ^{r + 2 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{15}}. \end{aligned}$$
(A.15)
By Lemma 2.1, for any \(q=1,2, \cdots \), we have
$$\begin{aligned} \int \limits _0^T {\left\| {\nabla {v_\varepsilon }( \cdot ,t)} \right\| _{{L^{4{m^q}}}}^{4{m^q}}{\hbox {d}}s} {\text { }}&\le C\mathop {\sup }\limits _{t \in (0,T)} \left\| {{v_\varepsilon }} \right\| _{{L^\infty }}^{2{m^q}}\int \limits _0^T {\left\| {\Delta {v_\varepsilon }( \cdot ,s)} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s + C'\int \limits _0^T {\left\| {{v_\varepsilon }} \right\| _{{L^\infty }}^{4{m^q}}} {\hbox {d}}s \nonumber \\&\le {{C_T}} + {C''}\int \limits _0^T {\left\| {\Delta {v_\varepsilon }( \cdot ,s)} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s, \end{aligned}$$
(A.16)
where \( {{C_T}}\) depends on \(m,q,\left| \Omega \right| ,{\left\| {{v_0}} \right\| _{{L^\infty }}},T\). Taking \(r=2(m-1)\) in (A.15), then by (A.7), (A.10) and (A.16) with \(q=0\), we infer that
$$\begin{aligned}&\frac{1}{{2m - 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{2m - 1}{\hbox {d}}x} + (m - 1)\int \limits _\Omega {u_\varepsilon ^{3m - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} \nonumber \\&\quad \le (m - 1)\int \limits _\Omega {u_\varepsilon ^m{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {u_\varepsilon ^m{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_{17}}\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + \left( {{C_{19}}\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} } {\hbox {d}}t + {C_T}} \right) + {C_{17}} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_T}\int \limits _0^T {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}{\hbox {d}}x} } {\hbox {d}}t + {C_T} \nonumber \\&\quad \le \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} + {C_{16}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^4}{\hbox {d}}x} + {C_T}, \end{aligned}$$
(A.17)
then by a direct integration, and using (A.16) with \(q=0\), we have
$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2m - 1}}}^{2m - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{3m - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^{2m}{\hbox {d}}x} } \right) {\hbox {d}}s} \le {C_T}. \end{aligned}$$
(A.18)
Next, we use recursive method to prove (A.12). Assume that
$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^q} - 1}}}^{2{m^q} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^q} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^q}}} {\hbox {d}}x} \right) } {\hbox {d}}s \le {C_T}. \end{aligned}$$
(A.19)
By the second equation of (A.1), we see that
$$\begin{aligned} {v_{\varepsilon t}} - D\Delta {v_\varepsilon } + {v_\varepsilon } = {u_\varepsilon }(1 - {v_\varepsilon }) + {v_\varepsilon }, \end{aligned}$$
using the \(L^p\) theory of linear parabolic equations, we obtain
$$\begin{aligned} \int \limits _0^T {\left( {\left\| {{v_\varepsilon }} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} + \left\| {{v_{\varepsilon t}}} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} \right) {\hbox {d}}s} \le C\int \limits _0^T {\left\| {{u_\varepsilon }} \right\| _{{L^{2{m^q}}}}^{2{m^q}}} {\hbox {d}}s + {C_T}\left\| {{v_{\varepsilon 0}}} \right\| _{{W^{2,2{m^q}}}}^{2{m^q}} \le {C_T} . \end{aligned}$$
(A.20)
Taking \(r = 2{m^{q + 1}} - 2\) in (A.15), and using (A.10) and (A.16), we see that
$$\begin{aligned}&\frac{1}{{2{m^{q + 1}} - 1}}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - 1}{\hbox {d}}x} + \frac{{2{m^{q + 1}} - 2}}{2}\int \limits _\Omega {u_\varepsilon ^{m + 2{m^{q + 1}} - 4}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} {\hbox {d}}x + \frac{\mu }{2}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x}\\&\quad \le {C_{20}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - m}{{\left| {\nabla {v_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{21}}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}} - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_{24}}\int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_{25}}\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} } + {C_T}} \right) + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + {C_T}\left( {\int \limits _0^T {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^{2{m^q}}}{\hbox {d}}x} } + 1} \right) + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_{22}}\\&\quad \le {C_{23}}\int \limits _\Omega {{{\left| {\nabla {v_\varepsilon }} \right| }^{4{m^q}}}{\hbox {d}}x} + \frac{\mu }{4}\int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}{\hbox {d}}x} + {C_T}, \end{aligned}$$
then by a direct integration and using (A.16) and (A.20), we have
$$\begin{aligned} \sup _{t \in (0,T)} \left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^{q + 1}} - 1}}}^{2{m^{q + 1}} - 1} + \int \limits _0^T {\left( {\int \limits _\Omega {u_\varepsilon ^{m + 2{m^{q + 1}} - 4}} {{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^{2{m^{q + 1}}}} {\hbox {d}}x} \right) } {\hbox {d}}s \le {C_T}. \end{aligned}$$
Then, (A.19) is valid for any \(q=1,2,3, \cdots \).
By (A.19) and (A.20), we see that there exists a large enough \(q'\) such that \(2{m^{q'}} \ge 10\). Then, by t-anisotropic embedding theorem, we have
$$\begin{aligned} {\left\| {{v_\varepsilon }} \right\| _{{C^{\frac{3}{2},\frac{3}{4}}}({Q_T})}} \le C{\left\| {{v_\varepsilon }} \right\| _{W_{10}^{2,1}({Q_T})}} \le {C_T}, \end{aligned}$$
it means
$$\begin{aligned} \sup _{t \in (0,T )} {\left\| {{v_\varepsilon }( \cdot ,t)} \right\| _{{W^{1,\infty }}}} \le {C_T}. \end{aligned}$$
Combining this with (A.20), we obtain (A.13).
Using (A.10) and (A.13), we obtain
$$\begin{aligned} \mathop {\sup }\limits _{t \in [0,T]} \int \limits _\Omega {{{\left| {\nabla {w_\varepsilon }} \right| }^p}} \mathrm{d}x\mathrm{d}x \le {C_T},\quad for\;any\;p \ge 2 \end{aligned}$$
Combining this with (A.11) and (A.13), we obtain (A.14). The lemma is proved. \(\square \)
Next, we estimate the \({\left\| {{u_\varepsilon }} \right\| _{{L^\infty }}}\) by Moser’s iterative technique.
Lemma A.5
Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have
$$\begin{aligned}&\sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} \le {C_T}, \end{aligned}$$
(A.21)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {{u_\varepsilon }\ln {u_\varepsilon }} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {u_\varepsilon ^{m - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} } \mathrm{d}x\mathrm{d}s \le {C_T}, \end{aligned}$$
(A.22)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} {\left\| v \right\| _{W_{p'}^{2,1}({Q_T})}} \le {C_T}\quad for\;any\;p' > 1, \end{aligned}$$
(A.23)
where \({C_T}\) depends on \(m,\chi ,\xi ,\mu ,\beta ,\left| \Omega \right| ,{w_\varepsilon },T \), are independent of \(\varepsilon \).
Proof
In this section, all the constants \({C_i}(i = 25,26,27,28 )\) are also independent of r. We just consider it in dimension 2, the case in dimension 3 is almost the same, so we omit it. Multiplying the first equation by \(ru_\varepsilon ^{r - 1}\) with \(r \ge 2m\), and integrating it over \(\Omega \), by (A.3), (A.13) and (A.14), we see that
$$\begin{aligned}&\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^rdx} + r(r - 1)\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + \mu r\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^rdx} \nonumber \\&\quad \le \chi r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r - 1}\nabla {u_\varepsilon }\nabla {v_\varepsilon }{\hbox {d}}x} + \xi r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r - 1}\nabla {u_\varepsilon }\nabla {w_\varepsilon }{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^r(\mu r + \beta r{v_\varepsilon } + 1){\hbox {d}}x} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{26}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{\hbox {d}}x}+ {C_{27}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{{\left| {\nabla {w_\varepsilon }} \right| }^2}{\hbox {d}}x} \nonumber \\&\qquad + {C_{28}}r\int \limits _\Omega {u_\varepsilon ^rdx} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_{29}}r(r - 1)\int \limits _\Omega {u_\varepsilon ^{r + 1 - m}{\hbox {d}}x}+ {C_T}r(r - 1){\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}} \nonumber \\&\qquad + \frac{{\mu r}}{2}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x} \nonumber \\&\quad \le \frac{{r(r - 1)}}{2}\int \limits _\Omega {u_\varepsilon ^{m + r - 3}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}} + \frac{{\mu r}}{2}\int \limits _\Omega {u_\varepsilon ^{r + 1}{\hbox {d}}x}, \end{aligned}$$
(A.24)
it means
$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} + \int \limits _\Omega {{{\left| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {u_\varepsilon ^rdx} \le {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}} {\hbox {d}}x} \right) ^{\frac{1}{2}}}. \end{aligned}$$
(A.25)
It is not difficult to see that by Lemma 2.1
$$\begin{aligned} {C_T}{r^2}{\left( {\int \limits _\Omega {u_\varepsilon ^{2(r + 1 - m)}{\hbox {d}}x} } \right) ^{\frac{1}{2}}}&= {C_T}{r^2}\left\| {u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^{\frac{{4(r + 1 - m)}}{{r + m - 1}}}}}^{\frac{{2(r + 1 - m)}}{{r + m - 1}}} \\&\le {C_T}{r^2}\left\| {u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^{\frac{r}{{r + m - 1}}}}}^{\frac{r}{{2(r + m - 1)}}}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^{\frac{{3r + 4(1 - m)}}{{2(r + m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{\frac{r}{2}}^{r + 1 - m} \\&\le \frac{1}{2}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^2 + {C_T}{r^{\frac{{8r + 8(m - 1)}}{{r + 8(m - 1)}}}}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m} \\&\le \frac{1}{2}\left\| {\nabla u_\varepsilon ^{\frac{{r + m - 1}}{2}}} \right\| _{{L^2}}^2 + {C_T}{r^8}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m}, \end{aligned}$$
then substituting the above inequality into (A.25), we have
$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} + \int \limits _\Omega {u_\varepsilon ^r{\hbox {d}}x} \le {C_T}{r^8}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r \cdot \frac{{r + m - 1}}{{r + 8(m - 1)}}} + {C_T}{r^2}\left\| {{u_\varepsilon }} \right\| _{{L^{\frac{r}{2}}}}^{r + 1 - m}. \end{aligned}$$
By (A.12), there exists \({q'}>0\) such that \(2{m^{q'}} - 1 \ge 2m\) and
$$\begin{aligned} \mathop {\sup }\limits _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{2{m^{q'}} - 1}}}} \le C. \end{aligned}$$
Taking \({r_j} = 2{r_{j - 1}}\) with \({r_0} = 2{m^{q'}} - 1\), \({M_j} = \max \{ \mathop {\sup }\limits _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^{{r_j}}}}},{\left\| {{u_{\varepsilon 0}}} \right\| _{{L^\infty }}},1\} \), then by a direct calculation, we obtain
$$\begin{aligned} {M_j} \le C_T^{\frac{1}{{{r_j}}}}r_j^{\frac{8}{{{r_j}}}}{M_{j - 1}} = C_T^{\frac{1}{{{2^j}{r_0}}}}r_j^{\frac{8}{{{2^j}{r_0}}}}{M_{j - 1}} = C_T^{\frac{1}{{{2^j}{r_0}}}} \cdot {r_0}^{\frac{8}{{{2^j}{r_0}}}} \cdot {2^{\frac{{8j}}{{{2^j}{r_0}}}}}{M_{j - 1}}, \end{aligned}$$
then through the iterative process, we have
$$\begin{aligned} {M_n} \le C_T^{\frac{1}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{1}{{{2^j}}}} }{r_0}^{\frac{8}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{1}{{{2^j}}}} }{2^{\frac{8}{{{r_0}}}\sum \limits _{j = 1}^n {\frac{j}{{{2^j}}}} }}{M_0}. \end{aligned}$$
Let \(n \rightarrow \infty \), it is not difficult to see that \({\sum \limits _{j = 1}^n {\frac{1}{{{2^{j }}}}} }\) and \({\sum \limits _{j = 1}^n {\frac{{j}}{{{2^{j }}}}} }\) are convergent, so we have
$$\begin{aligned} \sup _{t \in (0,T)} {\left\| {{u_\varepsilon }( \cdot ,t)} \right\| _{{L^\infty }}} \le {C_T}. \end{aligned}$$
Multiplying \(1+\ln {u_\varepsilon }\) to the first equation of (A.1), using (A.13), (A.14), (A.21), we have
$$\begin{aligned}&\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{u_\varepsilon }\ln {u_\varepsilon }} {\hbox {d}}x + m\int \limits _\Omega {u_\varepsilon ^{m - 2}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} {\hbox {d}}x + \mu \int \limits _\Omega {u_\varepsilon ^2(1 + \ln {u_\varepsilon })} {\hbox {d}}x\\&\quad \le - \chi \int \limits _\Omega {{u_\varepsilon }\Delta {v_\varepsilon }} {\hbox {d}}x - \xi \int \limits _\Omega {{u_\varepsilon }\Delta {w_\varepsilon }} {\hbox {d}}x + \int \limits _\Omega {\left( {\beta {v_\varepsilon } + \mu } \right) u_\varepsilon ^{}} \left( {1 + \ln {u_\varepsilon }} \right) {\hbox {d}}x \\&\quad \le \int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_{T}}\int \limits _\Omega {u_\varepsilon ^2} {\hbox {d}}x, \\&\quad \le \int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_T}, \end{aligned}$$
then by a direct integration, and by (A.7), we obtain (A.22).
By (A.21) and the \(L^p\) theory of linear parabolic equations, (A.23) is valid obviously. The proof is completed. \(\square \)
Lemma A.6
Assume \(N=2, 3\), \(m>1\), and (A.2) holds. Then, for any \(\varepsilon >0\), the problem (A.1) admits a unique global classical solution \((u_\varepsilon , v_\varepsilon , w_\varepsilon )\in C^{2+\alpha , 1+\alpha /2}(\Omega \times (0, +\infty ))\).
Proof
By Lemma A.1, suppose that \(T_{\max }<+\infty \). Then, take \(T=T_{\max }\) in Lemma A.5, recalling (A.4), (A.13), (A.21). It is a contradiction. We complete the proof. \(\square \)
In order to derive the global weak solution of (1.4), we also need some more estimates, that is
Lemma A.7
Assume that (A.2) holds, let \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon })\) be the solution of (A.1) in \(\Omega \times (0,T)\) for any \(T>0\). Then, we have
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {u_\varepsilon ^2} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \mathrm{d}x\mathrm{d}s \le {C_T},} \end{aligned}$$
(A.26)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,\infty )} {\left\| {\nabla {w_\varepsilon }} \right\| _{{L^\infty }}} \le {C_T}, \end{aligned}$$
(A.27)
$$\begin{aligned}&\mathop {\sup }\limits _{t \in (0,T)} \int \limits _\Omega {{{\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| }^2}} \mathrm{d}x + \int \limits _0^T {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} \mathrm{d}x} \mathrm{d}t \le {{C_T}} , \end{aligned}$$
(A.28)
where \(C_T\) at most depends on \(\mu ,\chi ,\xi ,\beta ,{u_0},{v_0},{w_0},T\).
Proof
Multiplying the first equation of (A.1) by \({u_\varepsilon }\), and integrating it over \(\Omega \), since \({u_\varepsilon },{v_\varepsilon }\) are bounded, then by (A.14) and (A.21), we obtain
$$\begin{aligned} \frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {u_\varepsilon ^2} {\hbox {d}}x + \int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}{\hbox {d}}x} {\text { }}&\le - \frac{1}{2}\chi \int \limits _\Omega {u_\varepsilon ^2} \Delta {v_\varepsilon }{\hbox {d}}x - \frac{1}{2}\xi \int \limits _\Omega {u_\varepsilon ^2} \Delta {w_\varepsilon }{\hbox {d}}x + {C_T}\\&\le {C_1}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_2}\int \limits _\Omega {{{\left| {\Delta {w_\varepsilon }} \right| }^2}{\hbox {d}}x} + {C_T}\\&\le {C_1}\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2}} {\hbox {d}}x + {C_T}, \end{aligned}$$
then by a direct integration and (A.7), we obtain (A.26).
Add \(\nabla \) to the third equation of (A.1), by a direct calculation, we have
$$\begin{aligned} \nabla {w_\varepsilon }( \cdot ,t) = {e^{ - \int \limits _0^t {(2\eta {w_\varepsilon } + \delta {v_\varepsilon } - \eta ){\hbox {d}}s} }}\left( {w_{\varepsilon 0}} - \int \limits _0^t {\delta \nabla {v_\varepsilon }{w_\varepsilon } \cdot {e^{\int \limits _0^t {(2\eta {w_\varepsilon } + \delta {v_\varepsilon } - \eta ){\hbox {d}}s} }}{\hbox {d}}s}\right) \end{aligned}$$
since \({w_\varepsilon }\) and \({\left\| {\nabla {v_\varepsilon }} \right\| _{{L^\infty }}}\) are bounded, we obtain (A.27).
Multiplying the first equation of (A.1) by \(\frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}\), and integrating it over \(\Omega \), since \({u_\varepsilon },{v_\varepsilon }\) are bounded, then by (A.13) and (A.27), we have
$$\begin{aligned}&\frac{1}{2}\frac{{\hbox {d}}}{{{\hbox {d}}t}}{\int \limits _\Omega {\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| } ^2}{\hbox {d}}x + {\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| } ^2}{\hbox {d}}x\\&\quad \le - \chi \int \limits _\Omega {\nabla ({u_\varepsilon }\nabla {v_\varepsilon })} \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x - \xi \int \limits _\Omega {\nabla ({u_\varepsilon }\nabla {w_\varepsilon })} \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x\\&\qquad + \mu \int \limits _\Omega {{u_\varepsilon }\left( {1 + \frac{\beta }{\mu }{v_\varepsilon } - {u_\varepsilon }} \right) } \frac{{\partial ({{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon })}}{{\partial t}}{\hbox {d}}x\\&\quad \le {C_3}\int \limits _\Omega {{{\left| {\nabla ({u_\varepsilon }\nabla {v_\varepsilon })} \right| }^2}{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{\hbox {d}}x} + {C_4}\int \limits _\Omega {{{\left| {\nabla ({u_\varepsilon }\nabla {w_\varepsilon })} \right| }^2}{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{\hbox {d}}x}\\&\qquad + \frac{1}{2}\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x + {C_5}{\int \limits _\Omega {u_\varepsilon ^2\left( {1 + \frac{\beta }{\mu }{v_\varepsilon } - {u_\varepsilon }} \right) } ^2}{(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{\hbox {d}}x\\&\quad \le {C_T}\left( {1 + \left( {\int \limits _\Omega {{{\left| {\Delta {v_\varepsilon }} \right| }^2} + } {{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \right) {\hbox {d}}x} \right) + \frac{1}{2}\int \limits _\Omega {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x, \end{aligned}$$
it means
$$\begin{aligned} \frac{{\hbox {d}}}{{{\hbox {d}}t}}\int \limits _\Omega {{{\left| {\nabla \left( {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }} \right) } \right| }^2}} {\hbox {d}}x + \int \limits _\Omega {{{\left( {u_\varepsilon ^2 + \varepsilon } \right) }^{\frac{{m - 1}}{2}}}{{\left| {\frac{{\partial {u_\varepsilon }}}{{\partial t}}} \right| }^2}} {\hbox {d}}x \le {C_T}\left( {1 + \int \limits _\Omega {\left( {{{\left| {\Delta {v_\varepsilon }} \right| }^2} + {{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{{\left| {\nabla {u_\varepsilon }} \right| }^2}} \right) } {\hbox {d}}x} \right) , \end{aligned}$$
then by a direct integration and (A.7), (A.26), we obtain (A.28). \(\square \)
Proof of Lemma 3.1.
We take \(T={T_{\max }}\) and then the problem (A.1) admits a unique global classical solution \(({u_\varepsilon },{v_\varepsilon },{w_\varepsilon }) \in {C^{2 + \alpha ,1 + \alpha /2}}(\Omega \times (0,\infty ))\). Next, we will prove the existence of weak solutions. In what follows, we let ‘\(\rightarrow \)’ denote the strong convergence, and ‘\(\rightharpoonup \)’ denote the weak convergence.
By Lemma A.7, noticing that \(m>1\), we see that
$$\begin{aligned}&u_\varepsilon ^{\frac{{m + 1}}{2}} \rightharpoonup {u^{\frac{{m + 1}}{2}}}\quad in\;W_2^{1,1}({Q_T}),\\&{u_\varepsilon }\mathop \rightharpoonup \limits ^* u,\quad in\;{L^\infty }({Q_T}), \end{aligned}$$
then by Aubin-lions lemma [25], we have
$$\begin{aligned} u_\varepsilon ^{} \rightarrow {u^{}}\quad in\;{L^p}({Q_T})\;,\quad for\;any\;p \in (1,\infty ). \end{aligned}$$
Noticing that \({(a + b)^\alpha } \le {a^\alpha } + {b^\alpha }\) for any \(a,b>0,\alpha \in (0,1)\). When \(m\le 3\),
$$\begin{aligned} \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - {u^m}} \right| \le \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - u_\varepsilon ^m} \right| + \left| {u_\varepsilon ^m - {u^m}} \right| \le {\varepsilon ^{\frac{{m - 1}}{2}}}{u_\varepsilon } + \left| {u_\varepsilon ^m - {u^m}} \right| , \end{aligned}$$
when \(m>3\),
$$\begin{aligned} \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - {u^m}} \right|&\le \left| {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon } - u_\varepsilon ^m} \right| + \left| {u_\varepsilon ^m - {u^m}} \right| \\&\le \frac{m-1}{2}\varepsilon (u_\varepsilon ^2 + \varepsilon )^{\frac{m-1}{2}-1} u_\varepsilon + \left| {u_\varepsilon ^m - {u^m}} \right| \\&\le c\varepsilon u_\varepsilon ^{m - 2} + c{\varepsilon ^{\frac{{m - 1}}{2}}}{u_\varepsilon } + \left| {u_\varepsilon ^m - {u^m}} \right| , \end{aligned}$$
which implies that
$$\begin{aligned} {(u_\varepsilon ^2 + \varepsilon )^{\frac{{m - 1}}{2}}}{u_\varepsilon } \rightarrow {u^m}\;in\;{L^p}({Q_T}),\;for\;any\;p \in (1, + \infty ). \end{aligned}$$
Noting that \(W_{p'}^{2,1}({Q_T}) \hookrightarrow {C^{\alpha ,\frac{\alpha }{2}}}({Q_T})\) for any \(0 < \alpha \le 2 - \frac{5}{{p'}} ({p'} > \frac{5}{2})\), so by (A.23), we have
$$\begin{aligned} {v_\varepsilon } \rightarrow v\quad uniformly. \end{aligned}$$
By (A.14), we have
$$\begin{aligned} \nabla {w_\varepsilon } \rightarrow \nabla w\quad in\;{L^p}({Q_T})\;, for\;any\;p \in (1, + \infty ), \end{aligned}$$
since \({w_\varepsilon }\) is bounded, and using (A.8) and Aubin-lions lemma, we also have
$$\begin{aligned} {w_\varepsilon } \rightarrow w\quad in\;{L^p}({Q_T})\;, for\;any\;p \in (1, + \infty ). \end{aligned}$$
Then, we have
$$\begin{aligned}&{u_\varepsilon }\nabla {v_\varepsilon } \rightharpoonup u\nabla v\quad in\;{L^q}({Q_T})\;for\;any\;1< q<\infty ,\\&\quad {u_\varepsilon } {v_\varepsilon } \rightarrow u v\quad in\;{L^q}({Q_T})\;for\;any\;1< q<\infty ,\\&\quad {u_\varepsilon }\nabla {w_\varepsilon } \rightarrow u\nabla w\quad in\;{L^q}({Q_T})\;, for\;any\;1< q< \infty ,\\&\quad {v_\varepsilon } {w_\varepsilon } \rightarrow v w\quad in\;{L^q}({Q_T})\;for\;any\;1< q< \infty ,\\&\quad w_\varepsilon ^2 \rightarrow {w^2}\quad in\;{L^q}({Q_T})\;for\;any\;1< q < \infty , \end{aligned}$$
recalling that for any \(\varphi ,\phi ,\psi \in {C^\infty }(\overline{{Q_T}} )\) with \({\left. {\frac{{\partial \varphi }}{{\partial n}}} \right| _{\partial \Omega }} = 0\),
$$\begin{aligned}&\int \limits _\Omega {{u_\varepsilon }(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {{u_{\varepsilon 0}}\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {{u_\varepsilon }{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {{{(u_\varepsilon ^2 + \varepsilon )}^{\frac{{m - 1}}{2}}}{u_\varepsilon }\Delta \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {{u_\varepsilon }\nabla {v_\varepsilon }\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {{u_\varepsilon }\nabla {w_\varepsilon }\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {{u_\varepsilon }(1 - {u_\varepsilon })\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {{u_\varepsilon }{v_\varepsilon }\varphi }{\hbox {d}}x{\hbox {d}}t,\\&\quad \iint \limits _{{Q_T}} {{v_{\varepsilon t}}\phi {\hbox {d}}x{\hbox {d}}t + D\iint \limits _{{Q_T}} {\nabla {v_\varepsilon }\nabla \phi {\hbox {d}}x{\hbox {d}}t} + \iint \limits _{{Q_T}} {{u_\varepsilon }{v_\varepsilon }\phi {\hbox {d}}x{\hbox {d}}t} - \iint \limits _{{Q_T}} {{u_\varepsilon }\phi }{\hbox {d}}x{\hbox {d}}t = 0},\\&\quad \iint \limits _{{Q_T}} {{w_{\varepsilon t}}\psi {\hbox {d}}x{\hbox {d}}t + \delta \iint \limits _{{Q_T}} {{v_\varepsilon }{w_\varepsilon }\psi {\hbox {d}}x{\hbox {d}}t} - \eta \iint \limits _{{Q_T}} {\psi {w_\varepsilon }(1 - {w_\varepsilon }){\hbox {d}}x{\hbox {d}}t} = 0}. \end{aligned}$$
Let \(\varepsilon \rightarrow 0,\) we conclude that
$$\begin{aligned}&\int \limits _\Omega {u(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {u(x,0)\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {u{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {{u^m}\Delta \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {u\nabla v\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {u\nabla w\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {u(1 - u)\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {uv\varphi }{\hbox {d}}x{\hbox {d}}t,\\&\quad \iint \limits _{{Q_T}} {{v_t}\phi {\hbox {d}}x{\hbox {d}}t + D\iint \limits _{{Q_T}} {\nabla v\nabla \phi {\hbox {d}}x{\hbox {d}}t} + \iint \limits _{{Q_T}} {uv\phi {\hbox {d}}x{\hbox {d}}t} - \iint \limits _{{Q_T}} {u\phi }{\hbox {d}}x{\hbox {d}}t = 0},\\&\quad \iint \limits _{{Q_T}} {{w_t}\psi {\hbox {d}}x{\hbox {d}}t + \delta \iint \limits _{{Q_T}} {vw\psi {\hbox {d}}x{\hbox {d}}t} - \eta \iint \limits _{{Q_T}} {\psi w(1 - w){\hbox {d}}x{\hbox {d}}t} = 0}. \end{aligned}$$
By (A.28), we have \(\nabla {u^m} \in {L^2}({Q_T})\). Then, we also have
$$\begin{aligned}&\int \limits _\Omega {u(x,T)\varphi (x,T){\hbox {d}}x} - \int \limits _\Omega {u(x,0)\varphi (x,0){\hbox {d}}x} - \iint \limits _{{Q_T}} {u{\varphi _t}}{\hbox {d}}x{\hbox {d}}t - \iint \limits _{{Q_T}} {\nabla {u^m}\nabla \varphi {\hbox {d}}x{\hbox {d}}t}\\&\quad - \chi \iint \limits _{{Q_T}} {u\nabla v\nabla \varphi {\hbox {d}}x{\hbox {d}}t}- \xi \iint \limits _{{Q_T}} {u\nabla w\nabla \varphi {\hbox {d}}x{\hbox {d}}t} = \mu \iint \limits _{{Q_T}} {u(1 - u)\varphi }{\hbox {d}}x{\hbox {d}}t + \beta \iint \limits _{{Q_T}} {uv\varphi }{\hbox {d}}x{\hbox {d}}t. \end{aligned}$$
Hence, (u, v, w) is a weak solution of (1.4), and all the estimations are hold by \(\varepsilon \rightarrow 0\).\(\square \)