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Existence of weak solutions to an anisotropic parabolic–parabolic chemotaxis system

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  06 March 2023

Hamid El Bahja Open the ORCID record for Hamid El Bahja [Opens in a new window]
Hamid El Bahja*
Affiliation:
AIMS, Cape Town, South Africa ([email protected])
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Abstract

This work is devoted to the study of the sub-critical case of an anisotropic fully parabolic Keller–Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without restriction on the size of the initial data.

Keywords

Keller–Segel systemNonlinear diffusionWeak solutions

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B33: Critical exponents
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 2 , April 2024 , pp. 424 - 444
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

1. Introduction

In this paper, we consider the following chemotaxis system with anisotropic porous medium-type diffusion:

(1.1)\begin{equation} \begin{cases} \displaystyle u_{t}=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\left(d_{1}\frac{\partial u^{m_{i}}}{\partial \displaystyle x_{i}}-\chi\left(\frac{u}{\gamma+v}\right)^{q_{i}-1}\frac{\partial v}{\partial x_{i}} \right)~~~~~ & \text{in}~\Omega_{T},\\ \displaystyle v_{t}=d_{2}\Delta v-v+u & \text{in }~\Omega_{T},\\ \displaystyle\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0 & \text{on}~\partial\Omega\times(0,T),\\ \displaystyle u(x,0)=u_{0}(x),~v(x,0)=v_{0}(x) & \text{on }~\Omega\times\{0\}, \end{cases} \end{equation}

where $\Omega _{T}=\Omega \times (0,T)$, $T>0$ is a fixed time, $\Omega$ is a bounded domain in $\mathbb {R}^{N}$, $N\geq 3$ with smooth boundary $\partial \Omega$, $q_{i}\geq 2$ and $m^{-}>q_{i}-\frac {2}{N}$ for all $i=1,..,N$, such that

\[ \begin{cases} m^+{=}\underset{1\leq i\leq N}{\max}\{m_{i}\},\text{ and }q^+{=}\underset{1\leq i\leq N}{\max}\{q_{i}\},\\ m^{-}=\underset{1\leq i\leq N}{\min}\{m_{i}\},\text{ and }q^{-}=\underset{1\leq i\leq N}{\min}\{q_{i}\}. \end{cases} \]

The positive constant $\chi$ is called the chemotaxis coefficient, $d_{1},d_{2}>0$ are the diffusion coefficients and $\gamma \geq 1$.

In general, organism or cell moves from a lower concentration towards a higher concentration of the chemo attractant, which is known as positive chemotaxis. In the same way, the opposite movement of the organisms is known as negative chemotaxis. In particular, microorganisms use chemotaxis to position themselves within the optimal portion of their habitats by monitoring the environmental concentration gradients of specific chemical attractant (positive chemotaxis) and repellent ligands (negative chemotaxis). Famous examples of biological species experiencing chemotaxis are the flagellated bacteria Salmonella typhimurium and Escherichia coli, the slime mould amoebae Dictyostelium discoideum and the human endothelial cells (see [Reference Adler1, Reference Bonner4, Reference Eisenbach6]). Theoretical and mathematical modelling of chemotaxis phenomena dates back to the pioneering works of Patlak in 1950s [Reference Patlak20] and Keller–Segel in 1970s [Reference Keller and Segel18]. A general form of Patlak–Keller–Segel model for chemotaxis is given by

(1.2)\begin{equation} \begin{cases} u_{t}=\nabla.\left(\phi(u,v)\nabla u-\psi(u,v)\nabla v\right),\\ \tau v_{t}=d\Delta v+g(u,v)u-h(u,v)v, \end{cases} \end{equation}

where $u$ denotes the density of cell population and $v$ is the chemical attractant concentration. The mobility function $\phi (u,v)$ describes the diffusivity of the cells and $\psi (u,v)$ represents the chemotaxis sensibility, while the functions $g(u,v)$ and $h(u,v)$ are kinetic functions that describe production and degradation of the chemical signal, respectively. When $\phi (u,v)=1$ and $\tau >0$, system (1.2) becomes the classical parabolic–parabolic Keller–Segel system, such system has been extensively studied, see for example [Reference Tao27, Reference Tao and Winkler28, Reference Yan and Li32] and references therein. For the study of the parabolic-elliptic Keller–Segel system of quasilinear type, namely $\tau =0$, with general $\phi (u,v)$ in (1.2), we refer to [Reference Sugiyama23–Reference Sugiyama and Kunii25] and references therein.

Equation (1.1) with $m_i=m$ and $q_i=q$ is sometimes called the equation of isotropic diffusion. In the case of degenerate diffusion, the model $\phi (u,v)=mu^{m-1}$ and $\psi (u,v)=u^{q-1}$ in $\mathbb {R}^{N}$ was studied by several authors. The existence of the weak solutions was shown when $q-m<0$ in [Reference Sugiyama and Kunii25] and when $q-m<\frac {2}{N}$ in [Reference Ishida and Yokota13]. When $q-m\geq \frac {2}{N}$ and the initial data $(u_{0},v_{0})$ is small in some sense, the existence of the weak solutions was proved in [14], whereas, if $q-m>\frac {2}{N}$ then blow-up of solutions as in [Reference Winkler30] was studied in [Reference Ishida, Ono and Yokota15, Reference Ishida and Yokota16].

In the present work, we are interested in the anisotropic case where the diffusion rates differ according to the direction $x_i$. Despite the resemblance with the isotropic cases presented in the previous mentioned works, the properties of the solutions to anisotropic equations are in striking contrast with the properties of the classical isotropic equations. The difficulties brought in by the anisotropy and the inhomogeneity of the diffusion operator are illustrated by the analysis of the self-similar solutions of anisotropic porous medium and $\vec {p}$-Laplacian types [Reference Antontsev and Shmarev2, Reference Antontsev and Shmarev3, Reference Düzgün, Mosconi and Vespri5]. Unlike the isotropic case where the typical geometry is defined in terms of balls in $\mathbb {R}^{N}$, in the anisotropic case it is defined by parallelepipeds with the edge lengths related to the exponents $m_i$ and $q_i$ .

The chemotaxis model with anisotropic porous medium diffusion type is motivated from a biological point of view [Reference Szymanska, Morales-Rodrigo, Lachowicz and Chaplain26]. It is worthy of mentioning that the porous medium type diffusion can represent population pressure in cell invasion models [Reference Rosen21], which initially arises from the ecology literature [Reference Gurney and Nisbet12, Reference Xu, Ji, Jin, Mei and Yin31]. In fact, experimental investigation has shown that the diffusion coefficient depends on the bacterial density [Reference Wakita, Komatsu, Nakahara, Matsuyama and Matsushita29]. In the bacterial experiments done by Ohgiwari, Matsushita and Matsuyama [Reference Ohgiwari, Matsushita and Matsuyama19], they recognized that cells located inside the bacterial colonies move actively, but cells became sluggish at the outermost front with apparently low cell density. This phenomenon indicates that bacteria become active as the cell density $u$ increases. Thus, a natural choice of the bacterial diffusion coefficient is $\phi (u,v)=m_{i}u^{m_{i}-1}$ with $m_{i}>1$ for all $i=1,..,N$, and this porous medium type bacterial diffusivity is based on the degenerate diffusion model proposed by Kawasaki et al. [Reference Kawasaki, Mochizuki, Matsushita, Umeda and Shigesada17]. To our knowledge, Keller–Segel system with anisotropic porous medium diffusion models has not been studied specially and systematically.

2. Preliminary and main result

2.1 Imbedding and technical lemmas

To derive our existence and regularity results, we will need the following

Theorem 2.1 [Reference Esfahani9], Theorem 1.1. Let $N\geq 2$, $\alpha _{j}\in (0,1)$, $1\leq p< q$ and $p_{j}\geq 1$, $j=1,..,N$, be such that $\displaystyle \sum _{j=1}^{N}\frac {1}{\alpha _{j}p_{j}}>1$. Then, for $u\in W^{(\alpha ),(p,\vec {p})}(\mathbb {R}^{N})$ (The fractional Sobolev–Liouville space) the inequality

(2.1)\begin{equation} \displaystyle\|u\|_{L^{q}(\mathbb{R}^{N})}\leq \sigma^{\frac{1}{q}}\left\|u\right\|_{L^{p}(\mathbb{R}^{N})}^{1-\rho}\prod_{j=1}^{N}\left\|D_{x_{j}}^{\alpha_{j}}u\right\|_{L^{p_{j}}(\mathbb{R}^{N})}^{\rho_{j}} \end{equation}

hold provided $M_{N}>0$ and

\[ \displaystyle q< q^{*}=\frac{\sum_{j=1}^{N}\frac{1}{\alpha_{j}}}{\sum_{j=1}^{N}\frac{1}{\alpha_{j}p_{j}}-1}, \]

where

\[ \displaystyle M_{N}=1+\frac{1}{p}\sum_{j=1}^{N}\frac{1}{\alpha_{j}}-\sum_{j=1}^{N}\frac{1}{\alpha_{j}p_{j}},~\rho=\sum_{j=1}^{N}\rho_{j}\text{ and } \rho_{j}=\frac{\frac{1}{p}-\frac{1}{q}}{\alpha_{j}M_{N}}. \]

The following lemma will show that theorem 2.1 holds true even for the case where $0< p<1$ and $p_{j}=2,~\forall j=1,..,N$.

Lemma 2.2 Let $\Omega \subset \mathbb {R}^{N}$ with $N\geq 3$ be a bounded domain with smooth boundary, and $0< p<1\leq q<\frac {2N}{N-2}$. Then, for all $u\in H^{1}(\Omega )$ we have

(2.2)\begin{equation} \displaystyle\|u\|^{q}_{L^{q}(\Omega)}\leq \sigma^{\frac{1}{\beta}}\left\|u\right\|_{L^{p}(\Omega)}^{q(1-N\rho)}\prod_{j=1}^{N}\left\|\frac{\partial u}{\partial x_{j}}\right\|_{L^{2}(\Omega)}^{q\rho}, \end{equation}

where $\rho =\frac {2(q-p)}{q(p(2-N)+2N)}$, and $\frac {1}{\beta }=\frac {(q-p)(2+N)}{(q-1)(p(2-N)+2N)}$.

Proof. By Hölder's inequality , we get that

\begin{align*} \|u\|_{L^{1}(\Omega)}& =\int_{\Omega}\left| u \right|^{\frac{q(1-p)}{q-p}}\left|u \right|^{\frac{(q-1)p}{q-p}}~\,{\rm d}x\\ & \leq \left\|u \right\|^{\frac{(q-1)p}{q-p}}_{L^{p}(\Omega)}\left\|u \right\|^{\frac{q(1-p)}{q-p}}_{L^{q}(\Omega)}. \end{align*}

Then, by using theorem 1 of section 5.4 in [Reference Evans10] and applying theorem 2.1 for $p=1$ and $p_{j}=2,~\forall j=1,..,N$, we obtain

(2.3)\begin{equation} \begin{aligned} \|u\|_{L^{q}(\Omega)} & \leq \sigma^{\frac{1}{q}}\left\|u\right\|_{L^{1}(\Omega)}^{1-N\rho_{0}}\prod_{j=1}^{N}\left\|\frac{\partial u}{\partial x_{j}} \right\|^{\rho_{0}}_{L^{2}(\Omega)}\\ & \leq \sigma^{\frac{1}{q}}\left[ \left\|u \right\|^{\frac{(q-1)p}{q-p}}_{L^{p}(\Omega)}\left\|u \right\|^{\frac{q(1-p)}{q-p}}_{L^{q}(\Omega)}. \right]^{1-N\rho_{0}}\prod_{j=1}^{N}\left\|\frac{\partial u}{\partial x_{j}} \right\|^{\rho_{0}}_{L^{2}(\Omega)}, \end{aligned} \end{equation}

where $\rho _{0}=\frac {1-\frac {1}{q}}{N(\frac {1}{N}+\frac {1}{2})}$. Then from (2.3) we get (2.2) with

\[ \rho=\frac{2(q-p)}{q(p(2-N)+2N)},\text{ and }\frac{1}{\beta}=\frac{(q-p)(2+N)}{(q-1)(p(2-N)+2N)}.\]

□

Possible references on the theory of anisotropic Sobolev spaces are in [Reference El Bahja7, Reference El Bahja8] and references therein. Next, we give some fundamental estimates of solutions to the following Cauchy problem for inhomogeneous linear heat equations:

(2.4)\begin{equation} \begin{cases} z_{t}=\Delta z-z+f & \text{in }\Omega\times(0,T),\\ z(x,0)=z_{0}(x), & x\in\Omega. \end{cases} \end{equation}

The following lemma can be found in [14].

Lemma 2.3 Let $\Omega \subset \mathbb {R}^{N}$ with $N\in \mathbb {N}$ be a bounded domain with smooth boundary, $T>0$, $1\leq p\leq \infty$ and $z_{0}\in L^{p}(\Omega )$. If $f\in L^{1}(0,T; L^{p}(\Omega ))$, then (2.4) has a unique mild solution $z\in C([0,T]; L^{p}(\Omega ))$ given by

(2.5)\begin{equation} z(t)=e^{{-}t}e^{t\Delta}z_{0}+\int_{0}^{t}e^{-(t-s)}e^{(t-s)\Delta}f(s)\,{\rm d}s,~t\in[0,T],\end{equation}

where $(e^{t\Delta }f)(x,t)=(4\pi t)^{-\frac {N}{2}}\int _{\Omega }e^{-\frac {|x-y|^{2}}{4t}}f(y,t)~dy$. Moreover, the following estimates hold.

  • • Let $1\leq q\leq p\leq \infty$ and $\frac {1}{q}-\frac {1}{p}<\frac {1}{N}$. Assume further that $z_{0}\in W^{2,p}(\Omega )$ and $f\in L^{\infty }(0,T; W^{1,q}(\Omega ))$. Then for every $t\in [0,T]$,

    (2.6)\begin{align} & \|z(t)\|_{L^{p}(\Omega)}\leq\|z_{0}\|_{L^{p}(\Omega)}+C_{0}\|f\|_{L^{\infty}(0,T;L^{q}(\Omega))}, \end{align}
    (2.7)\begin{align} & \|\nabla z(t)\|_{L^{p}(\Omega)}\leq\|\nabla z_{0}\|_{L^{p}(\Omega)}+C_{0}\|f\|_{L^{\infty}(0,T;L^{q}(\Omega))}, \end{align}
    (2.8)\begin{align} & \|\Delta z(t)\|_{L^{p}(\Omega)}\leq\|\Delta z_{0}\|_{L^{p}(\Omega)}+C_{0}\|\nabla f\|_{L^{\infty}(0,T;L^{q}(\Omega))}, \end{align}
    where $C_{0}$ is a positive constant depending on $p,q$ and $N$.

  • • Let $1< p<\infty$ and $f\in L^{p}(0,T;L^{p}(\Omega ))$. Then for every $t\in [0,T]$,

    (2.9)\begin{equation} \|\Delta z(t)\|_{L^{p}(0,t;L^{p}(\Omega))}\leq\|\Delta z_{0}\|_{L^{p}(\Omega)}\left(1-e^{{-}pt}\right)^{\frac{1}{p}}+C_{ }\|f\|_{L^{p}(0,T;L^{p}(\Omega))}, \end{equation}
    where $C_{0}$ is a positive constant depending on $N$ and $p$.

2.2 Formulation of the problem and main result

Throughout this paper, we deal with weak solutions of (1.1). Our definition of the weak solutions now reads

Definition 2.4 A pair of nonnegative functions $(u,v)$ is said to be a weak solution of (1.1) if and only if for all $i=1,..,N$ we have

\[ u\in L^{\infty}(\Omega_{T}),~u^{m_{i}}\in L^{2}(0,T;H^{1}(\Omega)),\text{ and }v\in L^{\infty}(0,T; H^{1}(\Omega)), \]

such that $(u,v)$ satisfies the equations in the sense of distribution, i.e., that

\begin{align*} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\biggl\{d_{1}\frac{\partial u^{m_{i}}}{\partial x_{i}}.\frac{\partial \varphi}{\partial x_{i}} -\left(\frac{u}{\gamma+v}\right)^{q_{i}-1}\frac{\partial v}{\partial x_{i}}.\frac{\partial \varphi}{\partial x_{i}}-u\varphi_{t}\biggr\}~\,{\rm d}x{\rm d}t\\ & \quad=\int_{\Omega}u_{0}(x)\varphi(x,0)~\,{\rm d}x,\\ & \quad\int_{0}^{T}\int_{\Omega}\left\{d_{2}\nabla v.\nabla \varphi+v\varphi-u\varphi-v\varphi_{t}\right\}~\,{\rm d}x{\rm d}t=\int_{\Omega}v_{0}(x)\varphi(x,0)~\,{\rm d}x, \end{align*}

for any continuously differentiable function $\varphi$ with compact support in $\Omega \times [0,T)$.

Motivated by the works mentioned in the previous section, our paper extends the results in [14, Reference Sugiyama and Kunii25] to the system (1.1) with anisotropic nonlinear diffusion. Now we state the main result of this paper. To be precise, we will assume the initial data $(u_{0},v_{0})$ to satisfy

(2.10)\begin{equation} \begin{cases} u_{0}\in C^{0}(\overline{\Omega})~\text{with}~u_{0}\geq0~\text{ in }\Omega,\\ v_{0}\in W^{2,\infty}(\Omega),~\text{with}~v_{0}\geq0\text{ in }\Omega. \end{cases} \end{equation}

Theorem 2.5 Let $q_{i}\geq 2$ and $m^{-}>q_{i}-\frac {2}{N}$ for every $i=1,..N$, $\Omega \subset \mathbb {R}^{N}$ for $N\geq 3$ be a bounded domain with smooth boundary. Then for all $(u_{0},v_{0})$ satisfying (2.10), the system (1.1) possesses at last one weak solution in the sense of definition 2.4.

3. Approximated equations

The first equation of (1.1) is a quasilinear parabolic equation of degenerate type. Therefore, we cannot expect the system (1.1) to have a classical solution at the point where $u$ vanishes. In order to prove theorem 2.5, we use a compactness method and introduce the following approximated equation of (1.1):

(3.1)\begin{equation} \begin{cases} \displaystyle u_{\varepsilon,t}=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\biggl(d_{1}m_{i}(u_{\varepsilon}+\varepsilon)^{m_{i}-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}-\chi\frac{(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}}{(\gamma+v_{\varepsilon})^{q_{i}-1}}\frac{\partial v_{\varepsilon}}{\partial x_{i}} \biggr)~~~~~ & \text{in}~\Omega_{T},\\ \displaystyle v_{\varepsilon,t}=d_{2}\Delta v_{\varepsilon}-v_{\varepsilon}+u_{\varepsilon}~~~~~ & \text{in}~\Omega_{T},\\ \displaystyle\frac{\partial u_{\varepsilon}}{\partial\nu}=\frac{\partial v_{\varepsilon}}{\partial\nu}=0~~~~~ & \text{on}~\partial\Omega\times(0,T),\\ \displaystyle u_{\varepsilon}(x,0)=u_{0}(x),~v_{\varepsilon}(x,0)=v_{0}(x)~~~~~ & \text{on}~\Omega\times\{0\}, \end{cases} \end{equation}

where $\varepsilon \in (0,1)$.

3.1 Existence of weak solutions of (3.1)

We are going to give an existence result of (3.1) under the condition that there exists a positive constant $k$ such that

(3.2)\begin{equation} d=\min\{d_{1}m_{i}\varepsilon^{m_{i}-1},d_{2}\}\geq\frac{K}{\gamma^{q_{i}-1}},\quad\forall i=1,..,N.\end{equation}

Theorem 3.1 Assume that (3.2) holds. If $u_{0},v_{0}\in L^{2}(\Omega )$, then (3.1) possesses a nonnegative weak solution $(u_{\varepsilon },v_{\varepsilon })$ such that

\[ u_{\varepsilon},v_{\varepsilon}\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T; H^{1}(\Omega)),~u_{\varepsilon,t},v_{\varepsilon,t}\in L^{2}(0,T; (W^{1,\infty}(\Omega))'), \]

such that $u_{\varepsilon }$ has the conservation law

(3.3)\begin{equation} \|u_{\varepsilon}(t)\|_{L^{1}(\Omega)}=\|u_{0}\|_{L^{1}(\Omega)},\quad t\in[0,T].\end{equation}

Proof. The existence of the weak solution to (3.1) can be obtained by using Schauder's fixed point theorem, a priori estimates and using the compactness results. We start by introducing for a small number $\delta >0$ the following

\begin{align*} F_{\delta}& =\frac{F}{1+\delta F},~\text{with}~F(s,t)={-}t+s, \\ f_{\delta}(s)& =\frac{s^+}{1+\delta s^+},~\text{with}~s^+{=}\max\{0,s\}, \end{align*}

such that we have $0\leq f_{\delta }(s)\leq \min \{s^+,\frac {1}{\delta }\}$ for any $s\in \mathbb {R}$ and $f_{\delta }(s)\longrightarrow s$ pointwise in $\mathbb {R}$ as $\delta \longrightarrow 0$. Therefore, we can conclude that there exists a positive constant $K$ such that

\begin{align*} \frac{(f_{\delta}(u_{\varepsilon})+\varepsilon)^{q_{i}-2}f_{\delta}(u_{\varepsilon})}{(\gamma+f_{\delta}(v_{\varepsilon}))^{q_{i}-1}} & \leq \frac{1}{\gamma^{q_{i}-1}}(\min\{u_{\varepsilon}^+,\delta^{{-}1}\}+1)^{q_{i}-2}\min\{u_{\varepsilon}^+,\delta^{{-}1}\}\\ & \leq \frac{K}{\gamma^{q_{i}-1}}. \end{align*}

Let $\overline {u}_{\varepsilon },\overline {v}_{\varepsilon }\in L^{2}(\Omega _{T})$ be given and consider the linear problem

\[ \begin{cases} u_{\varepsilon,t}-\displaystyle\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\left(a_{11}\frac{\partial u_{\varepsilon}}{\partial x_{i}}+a_{12}\frac{\partial v_{\varepsilon}}{\partial x_{i}}\right)=0,\\ v_{\varepsilon,t}-\displaystyle\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}\left(a_{21}\frac{\partial u_{\varepsilon}}{\partial x_{i}}+a_{22}\frac{\partial v_{\varepsilon}}{\partial x_{i}}\right)=F_{\delta}(\overline{u}_{\varepsilon}^+,\overline{v}_{\varepsilon}^+),\\ \end{cases} \]

where the diffusion matrix $A _{i}=(a_{jk})_{i}$ is given by

\[ A_{i}(\overline{u}_{\varepsilon},\overline{v}_{\varepsilon})= \begin{pmatrix} d_{1}m_{i}(f_{\delta}(\overline{u}_{\varepsilon})+\varepsilon)^{m_{i}-1} & -\chi\frac{(f_{\delta}(\overline{u}_{\varepsilon})+\varepsilon)^{q_{i}-1}f_{\delta}(\overline{u}_{\varepsilon})}{(\gamma+f_{\delta}(\overline{v}_{\varepsilon}))^{q_{i}-1}}\\ 0 & d_{2} \end{pmatrix}. \]

Moreover, the matrix $A _{i}$ is uniformly positive definite, since for any $X=(x,y)\in \mathbb {R}^{2}$ we have

\begin{align*} X^{T}A_{i}X& =x^{2}d_{1}m_{i}(f_{\delta}(\overline{u}_{\varepsilon})+\varepsilon)^{m_{i}-1}+d_{2}y^{2}-\chi\frac{(f_{\delta}(\overline{u}_{\varepsilon})+\varepsilon)^{q_{i}-1} f_{\delta}(\overline{u}_{\varepsilon})}{(\gamma+f_{\delta}(\overline{v}_{\varepsilon}))^{q_{i}-1}}xy\\ & \geq d(x^{2}+y^{2})-\chi\frac{(f_{\delta}(\overline{u}_{\varepsilon})+\varepsilon)^{q_{i}-1} f_{\delta}(\overline{u}_{\varepsilon})}{2(\gamma+f_{\delta}(\overline{v}_{\varepsilon}))^{q_{i}-1}}(x^{2}+y^{2})\\ & \geq (d-\frac{K}{\gamma^{q_{i}-1}})(x^{2}+y^{2})\geq0, \end{align*}

where we used (3.2) and the fact that $xy=\frac {1}{2}(x+y)^{2}-\frac {1}{2}(x^{2}+y^{2})\geq -\frac {1}{2}(x^{2}+y^{2})$ and $-xy=\frac {1}{2}(x-y)^{2}-\frac {1}{2}(x^{2}+y^{2})\geq -\frac {1}{2}(x^{2}+y^{2})$. Hence, the desired existence result is guaranteed by theorem 1 in [Reference Galiano, Garzón and Jüngel11].

3.2 A priori estimates

In order to prove theorem 2.5, we state and prove two key propositions which control $L^{r}-$ and $L^{\infty }-$estimates of the solution $(u_{\varepsilon },v_{\varepsilon })$ of (3.1).

Proposition 3.2 Assume that (2.10), and (3.2) hold. Let $N\geq 3$, $q_{i}\geq 2$ and $m^{-}>q_{i}-\frac {2}{3}$ for all $i=1,..,N$. Then $(u_{\varepsilon },v_{\varepsilon })$ satisfies the following estimates

(3.4)\begin{align} & \underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{r}(\Omega)}\leq C,~\text{for all}~r\in[1,\infty), \end{align}
(3.5)\begin{align} & \underset{0< t< T}{\sup}\|v_{\varepsilon}(t)\|_{W^{1,\infty}(\Omega)}\leq C, \end{align}

where $C$ is a positive constant independent of $\varepsilon$.

Proof. By taking $r\in (1,\infty )$, multiplying the first equation in (3.1) by $u_{\varepsilon }^{r-1}$ and integrating by parts, we get

(3.6)\begin{equation} \begin{aligned} & \frac{1}{r}\frac{\partial}{\partial t}\|u_{\varepsilon}(t)\|^{r}_{L^{r}(\Omega)}=\sum_{i=1}^{N}\biggl[-\int_{\Omega}d_{1}m_{i}(u_{\varepsilon}+\varepsilon)^{m_{i}-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}(r-1)u_{\varepsilon}^{r-2}.\frac{\partial u_{\varepsilon}}{\partial x_{i}}\\ & \qquad+\chi(r-1)\int_{\Omega}\frac{(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}^{r-1}}{(\gamma+v_{\varepsilon})^{q_{i}-1}}\frac{\partial v_{\varepsilon}}{\partial x_{i}}.\frac{\partial u_{\varepsilon}}{\partial x_{i}}\biggr]\\ & \quad\leq \sum_{i=1}^{N}\biggl[-\frac{4d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{i}}u^{\frac{\alpha+r-1}{2}}\right\|^{2}_{L^{2}(\Omega)}\\ & \qquad +\frac{(r-1)\chi}{\gamma^{q^{-}-2}} \int_{\Omega}(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}^{r-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}.\frac{\partial v_{\varepsilon}}{\partial x_{i}}\biggr]\\ & \quad= \sum_{i=1}^{N}\biggl[-\frac{4d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{i}}u^{\frac{\alpha+r-1}{2}}\right\|^{2}_{L^{2}(\Omega)} +\frac{(r-1)\chi}{\gamma^{q^{-}-2}} I_{i}\biggr], \end{aligned} \end{equation}

where

(3.7)\begin{equation} \alpha=\begin{cases} m^{-},\text{ if }(u_{\varepsilon}+\varepsilon)\geq1,\\ m^+,\text{ if }(u_{\varepsilon}+\varepsilon)<1. \end{cases} \end{equation}

Next, we set

\[ F_{i}(s)=\int_{0}^{s}(\tau+\varepsilon)^{q_{i}-2}\tau^{r-1}~d\tau,~s\geq0, \]

such that

\[ F_{i}(s)\leq 2^{q_{i}-2}\left[\frac{s^{r+q_{i}-2}}{r+q_{i}-2}+\frac{s^{r}}{r} \right]. \]

Therefore, $I_{i}$ becomes

(3.8)\begin{equation} \begin{aligned} I_{i} & =\int_{\Omega}\frac{\partial}{\partial x_{i}}[F_{i}(u_{\varepsilon})].\frac{\partial v_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x={-}\int_{\Omega}F_{i}(u_{\varepsilon}).\frac{\partial^{2} v_{\varepsilon}}{\partial^{2} x_{i}}~\,{\rm d}x\\ & \leq\frac{2^{q^+{-}2}}{r+q^{-}-2}\int_{\Omega}u_{\varepsilon}^{r+q_{i}-2}\left| \frac{\partial^{2} v_{\varepsilon}}{\partial^{2} x_{i}}\right|~\,{\rm d}x +\frac{2^{q^+{-}2}}{r}\int_{\Omega}u_{\varepsilon}^{r}\left| \frac{\partial^{2} v_{\varepsilon}}{\partial^{2} x_{i}}\right|~\,{\rm d}x\\ & =I'_{i}+I''_{i}. \end{aligned} \end{equation}

Next, we are going to integrate $I_{i}'$ over $(0,t)$ for $t\in (0,T)$ such that

(3.9)\begin{align} \sum_{i=1}^{N}\int_{0}^{t}I'_{i}(s)~\,{\rm d}s & =\frac{2^{q^+{-}2}}{r+q^{-}-2}\sum_{i=1}^{N}\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q_{i}-2}\left|\frac{\partial^{2}v_{\varepsilon}}{\partial^{2}x_{i}} \right|~\,{\rm d}x{\rm d}s\nonumber\\ & \leq\frac{2^{q^+{-}2}C}{r+q^{-}-2}\biggl\{\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}2}\left|\Delta v_{\varepsilon} \right|~\,{\rm d}x{\rm d}s+ \int_{0}^{t}\int_{\Omega}\left|\Delta v_{\varepsilon} \right|~\,{\rm d}x{\rm d}s\biggr\}\nonumber\\ & \leq \frac{2^{q^+{-}2}C}{r+q^{-}-2}\biggl\{ \left(\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s \right)^{\frac{r+q^+{-}2}{r+q^+{-}1}}\nonumber\\ & \quad \times\left( \int_{0}^{t}\int_{\Omega}\left|\Delta v_{\varepsilon} \right|^{r+q^+{-}1}~\,{\rm d}x{\rm d}s\right)^{\frac{1}{r+q^+{-}1}}\nonumber\\ & \quad+\left\|\Delta v_{\varepsilon} \right\|_{L^{r+q^+{-}1}(0,t;L^{r+q^+{-}1}(\Omega))}\biggr\}\nonumber\\ & \leq\frac{2^{q^+{-}2}C}{r+q^{-}-2}\biggl\{\|\Delta v_{0}\|_{L^{r+q^+{-}1}(\Omega)}\left(1-e^{-(r+q^+{-}1)t}\right)^{\frac{1}{r+q^+{-}1}}\nonumber\\ & \quad \times \left(\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s \right)^{\frac{r+q^+{-}2}{r+q^+{-}1}}\nonumber\\ & \quad+C_{{<}r+q^+{-}1>}\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s\nonumber\\ & \quad +\|\Delta v_{0}\|_{L^{r+q^+{-}1}(\Omega)}\left(1-e^{-(r+q^+{-}1)t}\right)^{\frac{1}{r+q^+{-}1}}\nonumber\\ & \quad+C_{{<}r+q^+{-}1>}\left(\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s\right)^{\frac{1}{r+q^+{-}1}}\biggr\}\nonumber\\ & \leq\frac{2^{q^+{-}2}C}{r+q^{-}-2}\left\{\left\|\Delta v_{0} \right\|^{r+q^+{-}1}_{L^{r+q^+{-}1}(\Omega)}+1 +\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s\right\}, \end{align}

where we used Hölder's inequality, (2.9) and Young's inequality.

Next, we are going to simplify the last integral in the right-hand side of (3.9) by using lemma 2.2. As a consequence, by letting $r>r_{0}\!=\!\max \{\alpha -2q^+\!+\!1,\frac {N}{2}(q^+-\alpha ) -q^++1,\frac {2(N-1)}{N}-\alpha \}$, we have the following

(3.10)\begin{equation} \begin{aligned} \int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{r+q^+{-}1}~\,{\rm d}x{\rm d}s & =\int_{0}^{t}\int_{\Omega}u_{\varepsilon}^{\frac{r+\alpha-1}{2}\frac{2(r+q^+{-}1)}{r+\alpha-1}}~\,{\rm d}x{\rm d}s =\int_{0}^{t}\left\|u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|_{L^{\frac{2(r+q^+{-}1)}{r+\alpha-1}}(\Omega)}^{\frac{2(r+q^+{-}1)}{r+\alpha-1}}~\,{\rm d}s\\ & \leq C\int_{0}^{t}\left\{\left\| u_{\varepsilon} \right\|_{L^{1}(\Omega)}^{(r+q^+{-}1)(1-N\rho)}\prod_{i=1}^{N}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2\rho(r+q^+{-}1)}{r+\alpha-1}}_{L^{2}(\Omega)} \right\}~\,{\rm d}s, \end{aligned} \end{equation}

where

\[ \rho=\frac{\frac{r+\alpha-1}{2}\left(\frac{r+q^+{-}2}{r+q^+{-}1} \right) }{1+\frac{N}{2}(r+\alpha-2)}. \]

We can replace the geometric mean on the right-hand side of (3.10) by an arithmetic mean. Indeed, by the inequality between geometric and arithmetic means we get

(3.11)\begin{align} \prod_{i=1}^{N}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2\rho(r+q^+{-}1)}{r+\alpha-1}}_{L^{2}(\Omega)} & = \prod_{i=1}^{N}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{1}{N}\left(\frac{N(r+q^+{-}2)}{1+\frac{N}{2}(r+\alpha-2)} \right)}_{L^{2}(\Omega)}\nonumber\\ & \leq \frac{1}{N}\sum_{i=1}^{N}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{N(r+q^+{-}2)}{1+\frac{N}{2}(r+\alpha-2)} }_{L^{2}(\Omega)} \end{align}

Since, we took $q_{i}<\frac {2}{N}+m^{-}\leq \frac {2}{N}+\alpha$ for all $i=1,..,N$. Then we get

(3.12)\begin{equation} \rho'=\frac{2N(r+q^+{-}2)}{2+N(r+\alpha-2)}<2. \end{equation}

Therefore, by using (3.11), (3.12), Young's inequality and the mass conservation law (3.3), we obtain

(3.13)\begin{equation} \begin{aligned} \sum_{i=1}^{N}\int_{0}^{t}I'_{i}(s)~\,{\rm d}s & \leq\frac{2^{q^+{-}2}C}{r+q^{-}-2}\biggr[\left\|\Delta v_{0} \right\|^{r+q^+{-}1}_{L^{r+q^+{-}1}(\Omega)}+1\\ & \quad+C(\nu)\int_{0}^{t}\left\|u_{0} \right\|_{L^{1}(\Omega)}^{\frac{2(r+q^+{-}1)(1-N\rho)}{2-\rho'}}\\ & \quad +\nu\sum_{i=1}^{N}\int_{0}^{t}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{2}_{L^{2}(\Omega)}~\,{\rm d}s \biggl]. \end{aligned} \end{equation}

The integral $I''_{i}$ can be controlled by the same way as $I'_{i}$ for all $i=1,..,N$. Consequently, we omit that $\sum _{i}^{N}\int _{0}^{t}I''_{i}~\,{\rm d}s$ satisfy the same estimation as in (3.13). Therefore, by integrating (3.6) over $(0,t)$ and using the previous estimates, we arrive at

(3.14)\begin{equation} \begin{aligned} \|u_{\varepsilon}(t)\|_{L^{r}(\Omega)}^{r} & \leq \|u_{0}\|_{L^{r}(\Omega)}^{r}-\sum_{i=1}^{N}\frac{4d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\int_{0}^{t}\left\|\frac{\partial}{\partial x_{i}}u^{\frac{\alpha+r-1}{2}}\right\|^{2}_{L^{2}(\Omega)}~\,{\rm d}s\\ & \quad+\frac{C2^{q^+{-}2}\chi(r-1)}{\gamma^{q^{-}-2}(r+q^{-}-2)}\biggl[ \left\|\Delta v_{0} \right\|^{r+q^+{-}1}_{L^{r+q^+{-}1}(\Omega)}+\int_{0}^{t}\left\|u_{0}\right\|_{L^{1}(\Omega)}^{\frac{2(r+q^+{-}1)(1-N\rho)}{2-\rho'}}~\,{\rm d}s +1\biggr]\\ & \quad+\sum_{i=1}^{N}\frac{C2^{q^+{-}2}\chi(r-1)\nu}{\gamma^{q^{-}-2}(r+q^{-}-2)}\int_{0}^{t}\left\|\frac{\partial}{\partial x_{i}}u^{\frac{\alpha+r-1}{2}}\right\|^{2}_{L^{2}(\Omega)}~\,{\rm d}s\\ & \leq \|u_{0}\|_{L^{r}(\Omega)}^{r}\!+\!\frac{C2^{q^+{-}2}\chi(r-1)t}{\gamma^{q^{-}-2}(r+q^{-}\!-\!2)}\biggl[ \left\|\Delta v_{0} \right\|^{r+q^+{-}1}_{L^{r+q^+{-}1}(\Omega)}\!+\!\left\|u_{0}\right\|_{L^{1}(\Omega)}^{\frac{2(r+q^+{-}1)(1-N\rho)}{2-\rho'}}\!+\!1 \biggr], \end{aligned} \end{equation}

where we took $\nu =\frac {4d_{1}m^{-}\gamma ^{q^{-}-2}(r+q^{-}-2)}{2^{q^+-2}C\chi (r+\alpha -1)^{2}}$. Moreover, by letting $r>r_{1}=\{r_{0},\beta -q^++1\}$ for $\beta > > 1$, we obtain that

(3.15)\begin{equation} \begin{aligned} \underset{0< t< T}{\sup}\|u_{\varepsilon}\|_{L^{r}(\Omega)} & \leq\biggl\{ \|u_{0}\|_{L^{\infty}(\Omega)}^{r-1}\|u_{0}\|_{L^{1}(\Omega)} +\frac{C2^{q^+{-}2}\chi(r-1)T}{\gamma^{q^{-}-2}(r+q^{-}-2)}\biggl[ \left\|\Delta v_{0} \right\|^{r+q^+{-}1-\beta}_{L^{\infty}(\Omega)}\\ & \quad+ \left\|\Delta v_{0} \right\|^{\beta}_{L^{\beta}(\Omega)}+\left\|u_{0}\right\|_{L^{1}(\Omega)}^{\frac{2(r+q^+{-}1)(1-N\rho)}{2-\rho'}}+1\biggr] \biggr\}^{\frac{1}{r}}=C, \end{aligned} \end{equation}

where $C$ is a positive constant independent of $\varepsilon$.

Now, for the case $1\leq r\leq r_{1}$ we have the following

(3.16)\begin{equation} \|u_{\varepsilon}(t)\|_{L^{r}(\Omega)}\leq\|u_{0}\|_{L^{1}(\Omega)}+\|u_{\varepsilon}(t)\|_{L^{r_{1}}(\Omega)},~~\text{for every}~t\in(0,T),\end{equation}

where we used Hölder's inequality, the mass conservation law (3.3) and Young's inequality. Hence, (3.15) and (3.16) give us the desired estimation for every $r\in [1,\infty )$.

Finally estimation (3.5) is a direct consequence of (2.6), (2.7) and (3.4) with $r=N+1$.

We conclude this section with the proof of $L^{\infty }$-estimates of the approximated solutions.

Proposition 3.3 Let the same assumptions as those in proposition 3.2 hold. Then, there exists a positive constant $C$ independent of $\varepsilon$ such that

(3.17)\begin{equation} \underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{\infty}(\Omega)}\leq C.\end{equation}

Proof. We begin by multiplying the first equation in (3.1) by $u_{\varepsilon }^{r-1}$ such that

(3.18)\begin{equation} \begin{aligned} \frac{1}{r}\frac{\partial}{\partial t}\|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)} & =\sum_{i=1}^{N}\biggl[-\int_{\Omega}d_{1}m_{i}(u_{\varepsilon}+\varepsilon)^{m_{i}-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}(r-1)u_{\varepsilon}^{r-2}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x\\ & \quad+\chi\int_{\Omega}\frac{(u_{\varepsilon}+\varepsilon)^{q_{i}-2}}{(\gamma+v_{\varepsilon})^{q_{i}-1}}\frac{\partial v_{\varepsilon}}{\partial x_{i}}(r-1)u_{\varepsilon}^{r-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x\biggr]\\ & \leq \sum_{i=1}^{N}\biggl[-\frac{4d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\left\|\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}} \right\|^{2}_{L^{2}(\Omega)}+\frac{(r-1)\chi}{\gamma^{q^{-}-2}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|_{L^{\infty}(\Omega)}\\ & \left(\int_{\Omega}u_{\varepsilon}^{q_{i}+r-3}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x +\int_{\Omega}u_{\varepsilon}^{r-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x \right)\biggr], \end{aligned} \end{equation}

where $\alpha$ is defined in (3.7). Next, we are going to simplify the last two integrals in the right-hand side of (3.18). Then, for all $i=1,..,N$ we have

(3.19)\begin{equation} \begin{aligned} & \frac{(r-1)\chi}{\gamma^{q^{-}-2}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|_{L^{\infty}(\Omega)}\left(\int_{\Omega}u_{\varepsilon}^{q_{i}+r-3}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x +\int_{\Omega}u_{\varepsilon}^{r-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x \right)\\ & \quad=\frac{(r-1)\chi}{\gamma^{q^{-}-2}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|_{L^{\infty}(\Omega)}\frac{2}{r+\alpha-1}\biggl[ \int_{\Omega}u_{\varepsilon}^{\frac{r+2q_{i}-\alpha-3}{2}}\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}}~\,{\rm d}x\\ & \quad+\int_{\Omega}u_{\varepsilon}^{\frac{r-\alpha+1}{2}}\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}}~\,{\rm d}x\biggr] \\ & \quad\leq2(r-1)\biggl[\frac{2\nu}{(r+\alpha-1)^{2}}\int_{\Omega}\left|\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}} \right|^{2}~\,{\rm d}x +\frac{C(\nu)\chi^{2}}{\gamma^{2(q^{-}-2)}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\\ & \left(\int_{\Omega}u_{\varepsilon}^{r+2q_{i}-\alpha-3}~\,{\rm d}x+\int_{\Omega}u_{\varepsilon}^{r-\alpha+1}~\,{\rm d}x \right)\biggl]\\ & \quad=\biggl[\frac{2d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\int_{\Omega}\left|\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}} \right|^{2}~\,{\rm d}x +\frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\\ & \left( \left\| u_{\varepsilon} \right\|_{L^{r+2q_{i}-\alpha-3}(\Omega)}^{r+2q_{i}-\alpha-3} +\left\| u_{\varepsilon} \right\|_{L^{r-\alpha+1}(\Omega)}^{r-\alpha+1}\right)\biggl], \end{aligned} \end{equation}

where we used Young's inequality and choose $\nu$ accordingly.

We will deal only with the norm $\left \| u_{\varepsilon } \right \|_{L^{r+2q_{i}-\alpha -3}(\Omega )}^{r+2q_{i}-\alpha -3}$ in the right-hand side of (3.19), because the last norm can be controlled by the same way. Furthermore, we are going to study the following two possible cases.

Case 1: $q_{i}>3-\frac {2}{N},~\forall i=1,..,N.$

Let $l$ be a natural number which is chosen later. Therefore, by applying lemma 2.2, we obtain

(3.20)\begin{equation} \begin{aligned} \left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{r+2q_{i}-\alpha-3}(\Omega)} & =\left\|u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2(r+2q_{i}-\alpha-3)}{r+\alpha-1}}_{L^{\frac{2(r+2q_{i}-\alpha-3)}{r+\alpha-1}}(\Omega)}\\ & \leq\sigma^{\frac{1}{\beta_{i}}}\left\|u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2(r+2q_{i}-\alpha-3)}{r+\alpha-1}(1-N\rho_{i})}_{L^{\frac{2r}{l(r+\alpha-1)}}(\Omega)} \prod_{j=1}^{N}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2(r+2q_{i}-\alpha-3)\rho_{i}}{r+\alpha-1}}_{L^{2}(\Omega)}\\ & \leq \frac{\sigma^{\frac{1}{\beta_{i}}}}{N}\left\|u_{\varepsilon}\right\|^{(r+2q_{i}-\alpha-3)(1-N\rho_{i})}_{L^{\frac{r}{l}}(\Omega)} \sum_{j=1}^{N}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2N\rho_{i}(r+2q_{i}-\alpha-3)}{r+\alpha-1}}_{L^{2}(\Omega)}, \end{aligned} \end{equation}

for $r\!>\!r_{0}=\max \{3\alpha -4q_{i}+5, -\frac {N}{2}(\alpha -1)+\frac {(N-2)}{2}(2q_{i}-\alpha -3),\alpha -2q_{i}+3\}, l\!>\!1$,

\[ \rho_{i}=\frac{\frac{r+\alpha-1}{2}\left(\frac{l(r+2q_{i}-\alpha-3)-r}{r(r+2q_{i}-\alpha-3)} \right) }{1+\frac{N}{2r}(l(r+\alpha-1)-r)} ,~\text{ and} ~\beta_{i}=\frac{(2+N)\left(\frac{2(r+2q_{i}-\alpha-3)}{r+\alpha-1}-\frac{2r}{l(r+\alpha-1)} \right)}{\left(\frac{2(r+2q_{i}-\alpha-3)}{r+\alpha-1}-1 \right)\left(\frac{2r(2-N)}{l(r+\alpha-1)}+2N \right)}. \]

By simple computation, we find that $\frac {2N\rho _{i}(r+2q_{i}-\alpha -3)}{r+\alpha -1}<2$ and $\frac {1}{\beta _{i}}\leq 6$ for every $r>r_{0}$ and $i=1,..,N$. Therefore, by Young's inequality we get

(3.21)\begin{equation} \begin{aligned} & \frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)} \left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{r+2q_{i}-\alpha-3}(\Omega)}\\ & \quad\leq\frac{\chi(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\frac{C}{N}\left\|u_{\varepsilon}\right\|^{(r+2q_{i}-\alpha-3)(1-N\rho_{i})}_{L^{\frac{r}{l}}(\Omega)}\\ & \sum_{j=1}^{N}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2N\rho_{i}(r+2q_{i}-\alpha-3)}{r+\alpha-1}}_{L^{2}(\Omega)}\\ & \quad\leq\frac{1}{N}\sum_{j=1}^{N}(r-1)\nu\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}}\right\|^{2}_{L^{2}(\Omega)}+C(\nu)(r-1)\\ & \left(C\frac{\chi^{2}}{d_{1}m^{-}\gamma^{2(q^{-}-2)}} \left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\right)^{\xi_{i,1}}\left\|u_{\varepsilon} \right\|_{L^{\frac{r}{l}}(\Omega)}^{(r+2q_{i}-\alpha-3)(1-N\rho_{i})\xi_{i,1}}, \end{aligned} \end{equation}

where

\[ \xi_{1,i}=\frac{r+\alpha-1}{(r+\alpha-1)-N\rho_{i}(r+2q_{i}-\alpha-3)}. \]

Next, by taking $\nu =\frac {d_{1}m^{-}}{(r+\alpha -1)^{2}}$, then $\displaystyle C(\nu )=\frac {1}{q_{i}(\nu p_{i})^{\frac {q_{i}}{p_{i}}}}$, where

\[ q_{i}=\frac{r+\alpha-1}{(r+\alpha-1)-N\rho_{i}(r+2q_{i}-\alpha-3)},\text{ and }p=\frac{r+\alpha-1}{N\rho_{i}(r+2q_{i}-\alpha-3)}. \]

Then, (3.21) becomes

(3.22)\begin{equation} \begin{aligned} & \frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)} \left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{r+2q_{i}-\alpha-3}(\Omega)}\\ & \quad\leq\frac{1}{N}\sum_{j=1}^{N}\frac{(r-1)d_{1}m^{-}}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}}\right\|^{2}_{L^{2}(\Omega)}+C(r-1)r^{2\xi_{i,2}}\\ & \left(C\frac{\chi^{2}}{d_{1}m^{-}\gamma^{2(q^{-}-2)}} \left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\right)^{\xi_{i,1}}\left\|u_{\varepsilon} \right\|_{L^{\frac{r}{l}}(\Omega)}^{(r+2q_{i}-\alpha-3)(1-N\rho_{i})\xi_{i,1}}, \end{aligned} \end{equation}

where

\[ \xi_{i,2}=\frac{N\rho_{i}(r+2q_{i}-\alpha-3)}{(r+\alpha-1)-N\rho_{i}(r+2q_{i}-\alpha-3)}. \]

Next, for $l>1$, $r>r_{0}$ and for every $i=1,..N$, we have

(3.23)\begin{equation} N\rho_{i}\longrightarrow\frac{\frac{1}{2}(l-1)}{\frac{1}{2}(l-1)+\frac{1}{N}}\text{ as }r\longrightarrow\infty.\end{equation}

Consequently, we obtain that

(3.24)\begin{equation} \frac{\frac{1}{2}(l-1)-\frac{1}{2N}}{\frac{1}{2}(l-1)+\frac{1}{N}}\leq N\rho_{i}\leq \frac{\frac{1}{2}(l-1)+\frac{1}{2N}}{\frac{1}{2}(l-1)+\frac{1}{N}},~\forall r>r_{0}\text{ and every }i=1,..,N. \end{equation}

Then, by (3.24) we get the following estimations

\[ \xi_{i,1}\leq Nl+2,\text{ and }\xi_{i,2}\leq Nl\text{ for all }r>r_{0}. \]

Therefore, (3.22) becomes

(3.25)\begin{equation} \begin{aligned} & \frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)} \left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{r+2q_{i}-\alpha-3}(\Omega)}\\ & \quad\leq\frac{1}{N}\sum_{j=1}^{N}\frac{(r-1)d_{1}m^{-}}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}}\right\|^{2}_{L^{2}(\Omega)}+Cr^{C}\left\|u_{\varepsilon} \right\|_{L^{\frac{r}{l}}(\Omega)}^{(r+2q_{i}-\alpha-3)(1-N\rho_{i})\xi_{i,1}}. \end{aligned} \end{equation}

Next, we are going to simplify the last term in the right-hand side of (3.25). For this reason, we choose $l$ to verify

\[ l>\max\left\{1,\frac{2\left((q+\frac{2}{N}-3)(\frac{1}{N}-\frac{1}{2})+(\alpha-1)(1-\frac{1}{N}) \right)}{\alpha-q_{i}-\frac{2}{N}+2} \right\},~\text{for all}~i=1,..,N, \]

such that

\[ \frac{q_{i}+\frac{2}{N}-3}{\alpha-1}<\frac{\frac{1}{2}(l-1)-\frac{1}{2N}}{\frac{1}{2}(l-1)+\frac{1}{N}}\leq N\rho_{i},~ \text{ for all } i=1,..,N. \]

Therefore, by taking

\[ r>r_{1}=\max\left\{r_{0},\frac{(N\rho_{i}+1)(\alpha-1)(2q_{i}-\alpha-3)}{(2q_{i}-\alpha-3)-N\rho_{i}(\alpha-1)}\right\},~ \text{ for all } i=1,..,N, \]

we get that

\[ \xi_{i,3}=\frac{r}{(r+2q_{i}-\alpha-3)(1-N\rho_{i})\xi_{i,1}}\geq1, \text{ for all } i=1,..,N. \]

By simple computation, we get also $\xi _{i,3}\leq Nl+2$ for all $i=1,..,N$. To this end, we apply Young's inequality on the last term in the right-hand side of (3.25) such that

(3.26)\begin{equation} \begin{aligned} & \frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)} \left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{r+2q_{i}-\alpha-3}(\Omega)}\\ & \quad\leq\frac{1}{N}\sum_{j=1}^{N}\frac{(r-1)d_{1}m^{-}}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}}\right\|^{2}_{L^{2}(\Omega)}+1+Cr^{C}\left\|u_{\varepsilon} \right\|_{L^{\frac{r}{l}}(\Omega)}^{r}. \end{aligned} \end{equation}

By the same method, we get also that

(3.27)\begin{equation} \begin{aligned} & \frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)} \left\|u_{\varepsilon} \right\|^{r-\alpha+1}_{L^{r-\alpha+1}(\Omega)}\\ & \quad\leq\frac{1}{N}\sum_{j=1}^{N}\frac{(r-1)d_{1}m^{-}}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}}\right\|^{2}_{L^{2}(\Omega)}+1+Cr^{C}\left\|u_{\varepsilon} \right\|_{L^{\frac{r}{l}}(\Omega)}^{r}, \end{aligned} \end{equation}

for every $r>r_{1}$. Then, by putting (3.26) and (3.27) into (3.18) we obtain

(3.28)\begin{equation} \frac{1}{r}\frac{\partial}{\partial t}\left\|u_{\varepsilon} \right\|^{r}_{L^{r}(\Omega)}\leq 2+Cr^{C}\|u_{\varepsilon}\|^{2}_{L^{\frac{r}{l}}(\Omega)}.\end{equation}

Integrating (3.28) from 0 to $t$, we obtain

(3.29)\begin{equation} \underset{0< t< T}{\sup}\left\|u_{\varepsilon} \right\|^{r}_{L^{r}(\Omega)}\leq \|u_{0}\|^{r}_{L^{r}(\Omega)}+2rT+CTr^{C}\underset{0< t< T}{\sup}\|u_{\varepsilon}\|^{r}_{L^{\frac{r}{l}}(\Omega)}.\end{equation}

Since

(3.30)\begin{equation} \|u_{0}\|_{L^{r}(\Omega)}\leq \|u_{0}\|^{\frac{r-1}{r}}_{L^{\infty}(\Omega)}\|u_{0}\|^{\frac{1}{r}}_{L^{1}(\Omega)}\leq C'.\end{equation}

Then,

(3.31)\begin{equation} \underset{0< t< T}{\sup}\left\|u_{\varepsilon} \right\|^{r}_{L^{r}(\Omega)}\leq C(T)^{\frac{1}{r}}r^{\frac{C}{r}}\max\{C',\underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{\frac{r}{l}}(\Omega)}\},~\text{for any } r>r_{1}.\end{equation}

We are now in a position to derive the claimed $L^{\infty }-$estimate. Therefore, we set

(3.32)\begin{equation} \displaystyle \Lambda_{p}=\max\{C',\underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{l^{p}}(\Omega)}\},\text{ for any }p\geq1.\end{equation}

Thereafter, we take $r=l^{p}$ in (3.31) which leads to

(3.33)\begin{equation} \begin{aligned} \Lambda_{p} & =C(T)^{\frac{1}{l^{p}}}l^{\frac{Cp}{2^{p}(\frac{l}{2})^{p}}}\max\{C',\underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{l^{p-1}}(\Omega)}\}\\ & \leq C(T)^{\frac{1}{l^{p}}} l^{\frac{C}{(\frac{l}{2})^{p}}}\Lambda_{p-1}, \end{aligned} \end{equation}

since $p\leq 2^{p}$ for $p\geq 1$. By induction, we get

\[ \Lambda_{p}\leq C(T)^{\sum_{k=1}^{p}l^{{-}k}}l^{C\sum_{k=1}^{p}\left(\frac{l}{2} \right)^{{-}k}}\Lambda_{0}. \]

Then, by using the mass conservation law (3.3), taking $l>2$ and letting $p\longrightarrow \infty$, we arrive at

(3.34)\begin{equation} \underset{0< t< T}{\sup}\|u_{\varepsilon}(t)\|_{L^{\infty}(\Omega)}\leq C(T)l^{c}\Lambda_{0}=C'',\end{equation}

where $C''$ is a positive constant independent of $\varepsilon$.

Case 2: $2\leq q_{i}\leq 3-\frac {2}{N},~\forall i=1,..,N.$

Knowing that $q_{i}<\alpha +\frac {2}{N}$, then $2q_{i}<\alpha +3$ for every $i=1,..,N$. Therefore,

\[ \displaystyle \left\|u_{\varepsilon}\right\|_{L^{r-\alpha+2q_{i}-3}(\Omega)}^{r-\alpha+2q_{i}-3}\leq \|u_{0}\|_{L^{1}(\Omega)}+\|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)}. \]

Then, (3.18) becomes

(3.35)\begin{equation} \begin{aligned} \frac{1}{r}\frac{\partial}{\partial t}\|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)} & \leq \sum_{i=1}^{N}\biggl[-\frac{2d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\left\|\frac{\partial }{\partial x_{i}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}} \right\|^{2}_{L^{2}(\Omega)}\\ & \quad+\frac{(r-1)\chi^{2}}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\left(2\|u_{0}\|_{L^{1}(\Omega)}+2\|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)} \right)\biggr]. \end{aligned} \end{equation}

Thereafter, by applying lemma 2.2 once again we obtain

(3.36)\begin{equation} \begin{aligned} \|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)} & =\left\| u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{\frac{2r}{r+\alpha-1}}_{L^{\frac{2r}{r+\alpha-1}}(\Omega)}\\ & \leq \frac{\sigma^{\frac{1}{\beta}}}{N}\left\|u_{\varepsilon} \right\|_{\frac{r}{4}}^{r(1-N\rho)}\sum_{j=1}^{N}\left\|\frac{\partial }{\partial x_{j}}u_{\varepsilon}^{\frac{\alpha+r-1}{2}} \right\|_{L^{2}(\Omega)}^{\frac{2Nr\rho}{r+\alpha-1}}, \end{aligned} \end{equation}

where

\[ \rho=\frac{3(r+\alpha-1)}{2rN\left(\frac{1}{N}-\frac{1}{2}+\frac{2(r+\alpha-1)}{r} \right)},\text{ and } \beta=\frac{3(r+\alpha-1)}{r(2+3N)+4N(\alpha-1)}<1. \]

It is easy to verify that $\frac {2Nr\rho }{r+\alpha -1}<2$ and $\frac {1}{\beta }\leq 6$ for suitable $r>0$. Then, by using the same method we used to get (3.27), we obtain

(3.37)\begin{equation} \begin{aligned} & \frac{2(r-1)\chi^{2}}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\| \frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega)}\|u_{\varepsilon}\|^{r}_{L^{r}(\Omega)}\\ & \quad\leq \frac{2}{N}\sum_{j=1}^{N}\frac{(r-1)d_{1}m^{-}}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{j}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{2}_{L^{2}(\Omega)}+2Cr^{C}\left\| u_{\varepsilon} \right\|^{r}_{L^{\frac{r}{4}}(\Omega)}+2, \end{aligned} \end{equation}

for suitable $r>0$. Hence, by putting (3.37) into (3.35) and applying similar arguments of the case $q>3-\frac {2}{N}$ we get the desired $L^{\infty }-$estimate.

We complete this section by discussing some uniform estimates (with respect to $\varepsilon$) of $u_{\varepsilon }$ and $v_{\varepsilon }$.

Lemma 3.4 For $q_{i}\geq 2$ and $m^{-}>q_{i}-\frac {2}{N}$ for all $i\!=\!1,..N$, there exists a constant $C$ such that

(3.38)\begin{align} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{\frac{m_{i}+1}{2}} \right|^{2}~\,{\rm d}x{\rm d}t\leq C, \end{align}
(3.39)\begin{align} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}}\right|^{2}~\,{\rm d}x{\rm d}t\leq C, \end{align}
(3.40)\begin{align} & \quad\int_{0}^{T}\left\|\partial_{t}u_{\varepsilon}^{\beta}\right\|_{(W^{1,N+1}(\Omega))'}~dt\leq C, \end{align}

and,

(3.41)\begin{equation} \int_{0}^{T}\left\|\partial_{t}v_{\varepsilon}\right\|_{(W^{1,N+1}(\Omega))'}~dt\leq C,\end{equation}

for each $\varepsilon \in (0,1)$ and $\beta$ a big enough positive constant.

Proof. We multiply the first equation of (3.1) by $u_{\varepsilon }$ and integrate over $\Omega \times (0,T)$ such that

\begin{align*} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\frac{4d_{1}m_{i}}{(m_{i}+1)^{2}}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{\frac{m_{i}+1}{2}} \right|^{2}~\,{\rm d}x{\rm d}t\leq\frac{1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}\\ & \qquad+\sum_{i=1}^{N}\frac{\chi}{\gamma^{q^{-}-2}}\int_{0}^{T}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}\frac{\partial v_{\varepsilon}}{\partial x_{i}}.\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x{\rm d}t\\ & \quad\leq\frac{1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}+C(T)\|\Delta v_{\varepsilon}\|_{L^{\lambda}(0,T;L^{\lambda}(\Omega))}, \end{align*}

where we used the same method we introduced to get (3.8), applying proposition 3.17 and for $\lambda > > 1$. Therefore, by (2.9) we get (3.8). Moreover, we note that

\begin{align*} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}}\right|^{2}~\,{\rm d}x{\rm d}t\\ & \quad \leq C\sum_{i=1}^{N}\left(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}+\varepsilon \right)^{m_{i}-1} \int_{0}^{T}\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{\frac{m_{i}+1}{2}} \right|^{2}~\,{\rm d}x{\rm d}t. \end{align*}

Then, (3.39) follows from (3.38).

Next, taking $\varphi \in C^{\infty }(\Omega _{T})$, multiplying the first equation of (3.1) by $\beta u_{\varepsilon }^{\beta -1}\varphi$, and integrating by parts, we obtain

\begin{align*} \left|\int_{\Omega}\beta u_{\varepsilon}^{\beta-1}\varphi\partial_{t}u_{\varepsilon}~\,{\rm d}x \right|& =\left|\int_{\Omega}\partial_{t}u_{\varepsilon}^{\beta} \varphi~\,{\rm d}x \right|\\ & \leq\sum_{i=1}^{N}\biggl\{\left|\int_{\Omega}d_{1}m_{i}\beta(\beta-1)(u_{\varepsilon}+\varepsilon)^{m_{i}-1}u_{\varepsilon}^{\beta-2}\varphi\left|\frac{\partial u_{\varepsilon}}{\partial x_{i}} \right|^{2}~\,{\rm d}x\right|\\ & \qquad+\left|\int_{\Omega}d_{1}m_{i}\beta(u_{\varepsilon}+\varepsilon)^{m_{i}-1}u_{\varepsilon}^{\beta-1}\frac{\partial u_{\varepsilon}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}~\,{\rm d}x \right|\\ & \qquad+\left|\int_{\Omega}\frac{\beta(\beta-1)\chi(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}^{\beta-1}}{(\gamma+v_{\varepsilon})^{q_{i}-1}} \frac{\partial v_{\varepsilon}}{\partial x_{i}}\varphi\frac{\partial u_{\varepsilon}}{\partial x_{i}}~\,{\rm d}x\right|\\ & \qquad+\left|\int_{\Omega}\frac{\beta\chi(u_{\varepsilon}+\varepsilon)^{q_{i}-2}u_{\varepsilon}^{\beta}}{(\gamma+v_{\varepsilon})^{q_{i}-1}} \frac{\partial v_{\varepsilon}}{\partial x_{i}}\frac{\partial \varphi}{\partial x_{i}}~\,{\rm d}x\right|\biggr\}\\ & \leq C\sum_{i=1}^{N}\biggl\{(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}+\varepsilon)^{\beta-m_{i}-1}\int_{\Omega}|\varphi|\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}} \right|^{2}~\,{\rm d}x\\ & \qquad+\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}^{\beta-1}\int_{\Omega}\left|\frac{\partial \varphi}{\partial x_{i}}\right|\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}} \right|~\,{\rm d}x\\ & \qquad+(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}+\varepsilon)^{q_{i}+\beta-m_{i}-2}\left|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right|\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}} \right||\varphi|~\,{\rm d}x\\ & \qquad+(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}+\varepsilon)\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|_{L^{\infty}(\Omega)}\int_{\Omega}\left|\frac{\partial \varphi}{\partial x_{i}} \right|~\,{\rm d}x\biggr\}\\ & \leq C\sum_{i=1}^{N}\int_{\Omega}\left|\frac{\partial}{\partial x_{i}}(u_{\varepsilon}+\varepsilon)^{m_{i}} \right|^{2}~\,{\rm d}x\Bigg(\|\varphi\|_{L^{\infty}(\Omega)}+ \Bigg(\left\|\frac{\partial\varphi}{\partial x_{i}}\right\|_{L^{\infty}(\Omega)} \Bigg)\\ & \leq C\|\varphi\|_{(W^{1,N+1}(\Omega))'}, \end{align*}

where we used proposition 3.3, the embedding of $W^{1,N+1}(\Omega )$ into $L^{\infty }(\Omega )$, and (3.38). Thus, we get (3.40). Also, by the same method and using (3.5) and (3.17) we get (3.41).

4. Proof of theorem 2.5

The goal of this section is to prove theorem 2.5. In the proof, we need the strong convergence of $u_{\varepsilon }$ and $v_{\varepsilon }$. Then, from (3.18), (3.19), proposition 3.3 and integrating over $(0,T)$, we get

(4.1)\begin{equation} \begin{aligned} & \sum_{i=1}^{N}\frac{2d_{1}m^{-}(r-1)}{(r+\alpha-1)^{2}}\left\|\frac{\partial}{\partial x_{i}}u_{\varepsilon}^{\frac{r+\alpha-1}{2}} \right\|^{2}_{L^{2}(\Omega_{T})}\leq\frac{1}{r}\|u_{0}\|^{r}_{L^{\infty}(\Omega)}\\ & \quad +\sum_{i=1}^{N}\frac{\chi^{2}(r-1)}{d_{1}m^{-}\gamma^{2(q^{-}-2)}}\left\|\frac{\partial v_{\varepsilon}}{\partial x_{i}} \right\|^{2}_{L^{\infty}(\Omega_{T})}\left\{\left\|u_{\varepsilon} \right\|^{r+2q_{i}-\alpha-3}_{L^{\infty}(\Omega_{T})}+\left\|u_{\varepsilon} \right\|^{r-\alpha-1}_{L^{\infty}(\Omega_{T})} \right\}, \end{aligned} \end{equation}

for suitable $r$. Therefore, by taking $r=2\beta -\alpha +1$ in (4.1) and using (3.5) and proposition 3.3 we get that $u_{\varepsilon }^{\beta }\in L^{2}(0,T;H^{1}(\Omega ))$ while $\partial _{t}u_{\varepsilon }^{\beta }$ is bounded in $L^{1}(0,T;(W^{1,N+1}(\Omega ))')$ by lemma 3.4. Since $H^{1}(\Omega )$ is compactly embedded in $L^{2}(\Omega )$ and $L^{2}(\Omega )$ is continuously embedded in $(W^{1,N+1}(\Omega ))'$, it follows from corollary 4 in [Reference Simon22] that $u_{\varepsilon }^{\beta }$ is compact in $L^{2}(0,T;L^{2}(\Omega ))$. Since $u_{\varepsilon }\longmapsto u_{\varepsilon }^{\frac {1}{\beta }}$ is Hölder continuous with exponent $\frac {1}{\beta }$, we get that $u_{\varepsilon }$ is compact in $L^{2\beta }(0,T;L^{2\beta }(\Omega ))$. Thus, there exist a function $u\in L^{2\beta }(0,T;L^{2\beta }(\Omega ))$ and a subsequence $(\varepsilon _{n})_{n\geq 1}$ such that

(4.2)\begin{equation} u_{\varepsilon_{n}}\longrightarrow u~\text{ Strongly in }~L^{2\beta}(0,T;L^{2\beta}(\Omega)).\end{equation}

This gives

(4.3)\begin{equation} u_{\varepsilon_{n}}\longrightarrow u\text{ a.e. in }\Omega_{T}.\end{equation}

On the other hand, by proposition 3.3 we get that

(4.4)\begin{equation} \underset{0< t< T}{\sup}\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq M.\end{equation}

As a consequence, we get that

(4.5)\begin{equation} \int_{\Omega_{T}}|u_{\varepsilon}|^{p}~\,{\rm d}x{\rm d}t\leq C(T)M^{p},\text{ for any }1< p<\infty.\end{equation}

Therefore, by using Lebesgue dominated convergence theorem, (4.3) and (4.5), we obtain

(4.6)\begin{equation} u_{\varepsilon}\longrightarrow u \text{ Strongly in }L^{p}(0,T;L^{p}(\Omega))\text{ for any }1< p<\infty.\end{equation}

By using the following inequality

(4.7)\begin{equation} |X^{m_{i}}-Y^{m_{i}}|\leq m_{i}^{2}\max\{|X|^{2(m_{i}-1)},|Y|^{2(m_{i}-1)}\}|X-Y|^{2},\quad\forall i=1,..,N,\end{equation}

we get

(4.8)\begin{equation} \int_{\Omega_{T}}|u_{\varepsilon_{n}}^{m_{i}}-u^{m_{i}}|^{2}~\,{\rm d}x{\rm d}t\leq C\int_{\Omega_{T}}|u_{\varepsilon_{n}}-u|^{2}~\,{\rm d}x{\rm d}t\longrightarrow0,\end{equation}

where we used (4.6) for $p=2$. Then, we get that

(4.9)\begin{equation} u_{\varepsilon_{n}}^{m_{i}}\longrightarrow u^{m_{i}}\text{ Strongly in }L^{2}(0,T;L^{2}(\Omega)).\end{equation}

Since $\frac {\partial u^{m_{i}}}{\partial x_{i}}$ is bounded in $L^{2}(0,T;L^{2}(\Omega ))$ by (3.39), and using (4.9) we arrive at

(4.10)\begin{equation} \frac{\partial}{\partial x_{i}}(u_{\varepsilon_{n}}+\varepsilon_{n})^{m_{i}}\rightharpoonup \frac{\partial u^{m_{i}}}{\partial x_{i}}~\text{Weakly in}~L^{2}(0,T;L^{2}(\Omega)),\end{equation}

for any $i=1,..,N$. Thereafter, by using (3.5), (3.41) and the same method we used to get (4.6), we obtain

(4.11)\begin{equation} v_{\varepsilon_{n}}\longrightarrow v~\text{Strongly in}~L^{p}(0,T;L^{p}(\Omega)),~\text{for any}~1< p<\infty, \end{equation}

and

(4.12)\begin{equation} \frac{\partial v_{\varepsilon_{n}}}{\partial x_{i}}\rightharpoonup\frac{\partial v}{\partial x_{i}}~\text{Weakly in}~L^{2}(0,T;L^{2}(\Omega)).\end{equation}

Using (4.6), (4.11), (4.7) for $q_{i}-1$ instead of $m_{i}$, and since $q_i\geq 2$ and $\gamma \geq 1$ we get that

(4.13)\begin{equation} \frac{(u_{\varepsilon_{n}}+\varepsilon_{n})^{q_{i}-2}u_{\varepsilon_{n}}}{(\gamma+v_{\varepsilon_{n}})^{q_{i}-1}} \longrightarrow \left(\frac{u}{\gamma+v} \right)^{q_{i}-1}~\text{Strongly in}~L^{2}(0,T;L^{2}(\Omega)).\end{equation}

Integrating (3.1) with respect to $x$ and $t$, we see that $(u_{\varepsilon _{n}},v_{\varepsilon _{n}})$ satisfies

\begin{align*} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\biggl\{ d_{1}\frac{\partial}{\partial x_{i}}(u_{\varepsilon_{n}}+\varepsilon_{n})^{m_{i}}.\frac{\partial \varphi}{\partial x_{i}}-\frac{(u_{\varepsilon_{n}}+\varepsilon_{n})^{q_{i}-2}u_{\varepsilon_{n}}}{(\gamma+v_{\varepsilon_{n}})^{q_{i}-1}}\frac{\partial v_{\varepsilon_{n}}}{\partial x_{i}}.\frac{\partial\varphi}{\partial x_{i}}-u_{\varepsilon_{n}}\varphi_{t}\biggr\}~\,{\rm d}x{\rm d}t\\ & \quad=\int_{\Omega}u_{0}(x)\varphi(x,0)~\,{\rm d}x,\\ & \int_{0}^{T}\int_{\Omega}\biggl\{\nabla v_{\varepsilon_{n}}.\nabla\varphi+v_{\varepsilon_{n}}\varphi-u_{\varepsilon_{n}}\varphi-v_{\varepsilon_{n}}\varphi_{t}\biggr\}~\,{\rm d}x{\rm d}t=\int_{\Omega}v_{0}(x)\varphi(x,0)~\,{\rm d}x, \end{align*}

for any continuously differentiable function $\varphi$ with compact support in $\Omega \times [0,T)$. Wherefore, by using (4.6), (4.9), (4.10), (4.11), (4.12), (4.13) and by the standard convergence argument we obtain

\begin{align*} & \sum_{i=1}^{N}\int_{0}^{T}\int_{\Omega}\biggl\{ d_{1}\frac{\partial u^{m_{i}}}{\partial x_{i}}.\frac{\partial \varphi}{\partial x_{i}}-\left(\frac{u}{\gamma+v} \right)^{q_{i}-1}\frac{\partial v}{\partial x_{i}}.\frac{\partial\varphi}{\partial x_{i}}-u\varphi_{t}\biggr\}~\,{\rm d}x{\rm d}t\\ & \quad=\int_{\Omega}u_{0}(x)\varphi(x,0)~\,{\rm d}x,\\ & \int_{0}^{T}\int_{\Omega}\biggl\{\nabla v.\nabla\varphi+v\varphi-u\varphi-v\varphi_{t}\biggr\}~\,{\rm d}x{\rm d}t=\int_{\Omega}v_{0}(x)\varphi(x,0)~\,{\rm d}x, \end{align*}

where $q_{i}\geq 2$ and $m^{-}>q_{i}-\frac {2}{N}$ for any $i=1,..,N$. Hence, we conclude the proof of theorem 2.5.

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