2.1 Global existence of bounded solutions

In this subsection, we show the existence of solutions to system (1.1) which are bounded on $Q_T$ for all $T>0$.

The following lemma is useful in our analysis.

Next, we prove the

Proposition 2.2 Under the assumptions of theorem 1.1, and picking $T>0$, system (1.1) has a nonnegative (for each component) strong solution which is bounded on $Q_T$. More precisely, there exists $C_T>0$ (depending on $\Omega$, $T$, $n$, $a$, $b$, $\varepsilon _0$, $\beta$, $\delta$, $\|m_0\|_{L^{\infty }(\Omega )}$, $\|c_0\|_{W^{ 1,\infty }(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$), such that

(2.1)\begin{equation} \|m\|_{L^{\infty}(Q_T)}+\|c\|_{L^{\infty}((0,T); W^{1,\infty}(\Omega))}+\|d\|_{L^{\infty}(Q_T)} \leq C_T. \end{equation}

Proof of proposition 2.2 For a given $r>0$, define $h_r \in C^{\infty }(\mathbb {R}_+,[0,1])$ such that $h_r(y) = 1$ for $0\leq y\leq r$, and $h_r(y) = 0$ for $y\geq 2r$. We also use the standard convention $m_+ := \max \{0; m\}$. Consider the system

(2.2)\begin{equation} \left\{ \begin{aligned} & \partial_t m =\Delta m + m_+(1-m_+^{a-1}) - \chi\nabla\cdot(f_r(m_+)\nabla c), & & (x,t)\in Q_T,\\ & \partial_t c = \varepsilon_0 \Delta c + \delta d - c + \beta m_+, & & (x,t)\in Q_T,\\ & \partial_t d = g(m_+)(1-d), & & (x,t)\in Q_T,\\ & \nabla m\cdot \nu = \nabla c\cdot \nu = 0, & & (x,t)\in \partial\Omega\times(0,T),\\ & m(x,0) = m_0(x), \,\, c(x,0) = c_0(x), \,\, d(x,0) = d_0(x), & & x\in\Omega, \end{aligned} \right. \end{equation}

where the function $f_r$ is defined as

\[ f_r(y) := h_r(f(y)). \]

We will prove, thanks to Leray-Schauder's theorem (cf. [Reference Gilbarg and Trudinger8, theorem 10.3]), that (2.2) has a nonnegative (for each component) solution which is bounded in sup-norm independent of $r>0$, and consequently prove the existence of a solution to the original system (1.1) on $Q_T$. To this end, we define

\[ \mathfrak{Y}:= C^{1,0}(\overline\Omega\times [0,T] ) \quad \text{ and } \mathfrak{X} := \mathfrak{Y}\times \mathfrak{Y}, \]

where $C^{1,0}$ is the space of continuous functions such that their gradient (with respect to space) is also continuous, associated with the natural norm (making it a Banach space)

\[ \|u\|_{\mathfrak{Y}} = \sup_{t\in[0,T]}\|u(t)\|_{L^{\infty}(\Omega)} + \sup_{t\in [0,T]}\|\nabla u(t)\|_{L^{\infty}(\Omega)}, \]

and

\[ \mathcal{F}_r: \mathfrak{X} \times [0,1] \to \mathfrak{X} \quad \text{ via } \quad \mathcal{F}_r(m,c, \lambda):= (\lambda\,\mathfrak{M}, \lambda\, \mathfrak{C}), \]

where $(\mathfrak {M},\mathfrak {C})$ is given (in a unique way) by solving first an ODE (for $d$), and then successively (for $\mathfrak {C}$ and $\mathfrak {M}$, in that order) two Neumann problems for the heat equation with a source. More precisely, $(\mathfrak {M},\mathfrak {C})$ is the (unique) solution on $Q_T$ of the system

(2.3)\begin{equation} \left\{ \begin{aligned} & \partial_t \mathfrak{M} = \Delta \mathfrak{M} + m_+\left({1-m_+^{a-1}}\right) - \chi \nabla\cdot(f_r(m_+)\nabla \mathfrak{C}), & & (x,t)\in Q_T,\\ & \partial_t \mathfrak{C} = \varepsilon_0 \Delta \mathfrak{C} + \delta d - \mathfrak{C} + \beta m_+, & & (x,t)\in Q_T,\\ & \partial_t d = g(m_+)(1-d), & & (x,t)\in Q_T,\\ & \nabla\mathfrak{M} \cdot \nu = \nabla \mathfrak{C} \cdot \nu = 0, & & (x,t)\in \partial\Omega\times(0,T),\\ & \mathfrak{M}(x,0) = m_0(x), \mathfrak{C}(x,0) = c_0(x), d(x,0) = d_0(x), & & x\in\Omega. \end{aligned} \right. \end{equation}

Note first that $\mathcal {F}_r(m,c, 0) = 0$. Then, regardless of $m$, we always have that

(2.4)\begin{equation} 0 \leq d(x,t) \leq \mu := \max\left\{{1, \|d_0\|_{L^{\infty}(\Omega)}}\right\}. \end{equation}

Also for any $1< p\le \tilde p$, there exists $C_p>0$ depending on $\|m\|_{\mathfrak {Y}}$ and $r$ (and $\Omega , T, m_0, c_0, \mu$) such that, thanks to maximal regularity results,

(2.5)\begin{equation} \|\mathfrak{C}\|_{W^{2,1}_{p}(Q_T)} \leq C_p, \end{equation}

and

(2.6)\begin{equation} \|\mathfrak{M}\|_{W^{2,1}_{p}(Q_T)} \leq C_p. \end{equation}

Thanks to the embedding in lemma 2.1,

(2.7)\begin{equation} W^{2,1}_p(Q_T) \hookrightarrow \mathfrak{Y} \text{ compactly when } p >n+2. \end{equation}

So, we see that since $\tilde p>n+2$, the map $\mathcal {F}_r$ sends bounded sets in $\mathfrak {X}$ into compact sets of $\mathfrak {X}$. We now show that for any $\lambda \in [0,1]$, $\mathcal {F}_r(\cdot ,\cdot ,\lambda )$ is continuous from $\mathfrak {X}$ to $\mathfrak {X}$. Let $(m_1, c_1), (m_2,c_2)\in \mathfrak {X}$ and $(\lambda \mathfrak {M}_1, \lambda \mathfrak {C}_1)= \mathcal {F}_r(m_1,c_1,\lambda )$, $(\lambda \mathfrak {M}_2,\lambda \mathfrak {C}_2) = \mathcal {F}_r(m_2,c_2,\lambda )$. We have

(2.8)\begin{align} \begin{cases} \partial_t(\mathfrak{M}_1 - \mathfrak{M}_2) &= \Delta(\mathfrak{M}_1 - \mathfrak{M}_2) + \left[{m_{1+}(1-m_{1+}^{a-1}) - m_{2+}(1-m_{2+}^{a-1}) }\right]\\ &\quad + \chi\nabla\cdot\left[{f_r(m_{1+})\nabla \mathfrak{C}_1 - f_r(m_{2+})\nabla \mathfrak{C}_2}\right],\\ \partial_t(\mathfrak{C}_1 - \mathfrak{C}_2) &= \varepsilon_0\Delta (\mathfrak{C}_1 - \mathfrak{C}_2) + \delta(d_1 - d_2) - (\mathfrak{C}_1 - \mathfrak{C}_2) + \beta(m_{1+} - m_{2+}),\\ (d_1 - d_2)_t &= g(m_{1+})(1-d_1) - g(m_{2+})(1-d_2). \end{cases} \end{align}

By rewriting $g(m_{1+})(1-d_1) - g(m_{2+})(1-d_2)$ $= g(m_{1+})(d_2 - d_1) + (g(m_{1+}) - g(m_{2+}))\, (1-d_2)$, and using $g\in C^{1}([0,\infty ))$, we obtain

\[ d_1(x,t) - d_2(x,t) = \int_0^{t}e^{-\int_{s}^{t}g(m_{1+}(r))dr}g'(\theta(x, s))(m_{1+}(s)-m_{2+}(s))(1\,{-}\,d_2(s))\,\textrm{d}s,\]

where $\theta (x,s)$ is between $m_{1+}(x,s)$ and $m_{2+}(x,s)$. Therefore, there exists $C_T>0$ (which can depend on the norm $\|m_{1}\|_{\mathfrak {Y}}$) such that

\[ \|d_1 - d_2\|_{L^{\infty}(Q_T)} \leq C_{T}\|m_{1+} - m_{2+}\|_{L^{\infty}(Q_T)} \leq C_T\|m_{1}-m_{2}\|_{\mathfrak{Y}}. \]

By maximal regularity results and (2.7)

(2.9)\begin{align} \|\mathfrak{C}_1 - \mathfrak{C}_2\|_{\mathfrak{Y}} &\leq C_T\|\mathfrak{C}_1 - \mathfrak{C}_2\|_{W^{2,1}_p(Q_T)}\notag\\ & \leq C_T\left({\|d_1-d_2\|_{L^{p}(Q_T)} + \|m_{1+} - m_{2+}\|_{L^{p}(Q_T}}\right)\notag\\ & \leq C_T\|m_{1} - m_{2}\|_{\mathfrak{Y}}. \end{align}

For the equation satisfied by $\mathfrak {M}_1 - \mathfrak {M}_2$, we write

\[ \left|{m_{1+}(1-m_{1+}^{a-1}) - m_{2+}(1-m_{2+}^{a-1})}\right| \le \left|{m_{1} - m_{2}}\right| + \left|{m_{1} - m_{2}}\right|\, \max(m_{1+}^{a-1}, m_{2+}^{a-1}), \]

and

\begin{align*} & \left|{\nabla\cdot\left[{f_r(m_{1+})\nabla \mathfrak{C}_1 - f_r(m_{2+})\nabla \mathfrak{C}_2}\right]}\right|\\ & \quad\le \left|{f_r(m_{1+})}\right|\left|{\Delta(\mathfrak{C}_1 - \mathfrak{C}_2)}\right| + \left|{f_r(m_{1+})- f_r(m_{2+})}\right|\left|{\Delta \mathfrak{C}_2}\right|\\ &\qquad + \left|{f_r'(m_{1+})}\right|\left|{\nabla m_{1+}}\right|\left|{\nabla(\mathfrak{C}_1-\mathfrak{C}_2)}\right|\\ & \qquad + \left|{f_r'(m_{1+})}\right|\left|{\nabla(m_{1+}-m_{2+})}\right|\left|{\nabla \mathfrak{C}_2}\right| + \left|{f_r'(m_{1+}) - f_r'(m_{2+})}\right|\left|{\nabla m_{2+}}\right|\left|{\nabla \mathfrak{C}_2}\right| . \end{align*}

Using the fact that $f\in C^{2}([0,\infty ))$, estimate (2.9), and maximal regularity results we get

(2.10)\begin{equation} \|\mathfrak{M}_1 - \mathfrak{M}_2\|_{\mathfrak{Y}} \leq C_T\|\mathfrak{M}_1 - \mathfrak{M}_2\|_{W^{2,1}_{\tilde p}(Q_T)} \leq C_T\|m_{1}-m_{2}\|_{\mathfrak{Y}}. \end{equation}

It follows from (2.9) and (2.10) that $\mathcal {F}_r(\cdot ,\cdot ,\lambda )$ is continuous from $\mathfrak {X}$ to $\mathfrak {X}$ for any $\lambda \in [0,1]$.

We now check the last assumption in Leray-Schauder's theorem. We consider therefore

(2.11)\begin{equation} \mathbf{Z}:= \left\{{(m,c)\in \mathfrak{X} \,:\, (m,c) = \mathcal{F}_r(m,c, \lambda) \; \text{ where } \; 0<\lambda \leq 1 }\right\}, \end{equation}

and will show that $\mathbf {Z}$ is bounded in $\mathfrak {X}$.

Note that if $(m,c)\in \mathbf {Z}$, then $(m,c) = (\lambda \mathfrak {M}, \lambda \mathfrak {C})$, where $(\mathfrak {M},\mathfrak {C})$ solves (2.3). Therefore, by multiplying the equations in (2.3) by $\lambda$, we obtain (for $(m,c) \in \mathbf {Z}$)

(2.12)\begin{align} \left\{ \begin{aligned} & \partial_t m = \Delta m + \lambda\left[{m_+(1-m_+^{a-1}) - \chi\nabla\cdot(f_r(m_+)\nabla \mathfrak{C})}\right], & & (x,t)\in Q_T,\\ & \partial_t c = \varepsilon_0\Delta c - c + \lambda(\delta d + \beta m_+), & & (x,t)\in Q_T,\\ & \partial_t d = g(m_+)(1-d), & & (x,t)\in Q_T,\\ & \nabla m\cdot \nu =\nabla c\cdot \nu = 0, & & (x,t)\in \partial\Omega\times(0,T),\\ & m(x,0) = \lambda m_0(x), c(x,0) = \lambda c_0(x), d(x,0) = d_0(x), & & x\in\Omega. \end{aligned} \right. \end{align}

Since $d\geq 0$, $m_+\geq 0$, and $c_0 \geq 0$, we obtain immediately (for $(m,c) \in \mathbf {Z}$) that $c\geq 0$. We now show that $m$ is also nonnegative. Indeed, denote by $m_- = \max \{0,-m\}$, and multiply the equation satisfied by $m$ in (2.12) by $(m_-)^{2}$. Then, an integration by parts and the fact $m_-m_+ = 0$ give for all $t \in [0,T]$,

\[ -\frac 13\int_{\Omega}m_-^{3}(x,t)\,\textrm{d}x - 2\int_0^{t}\int_{\Omega}m_-|\nabla m_-|^{2}\,\textrm{d}x\,\textrm{d}s \geq 0. \]

Therefore, $m_- = 0$ and thus $m\geq 0$. It follows from the nonnegativity of $m$, $c$ and $d$ (and system (2.12)) that $(m,c,d)$ solves the system

(2.13)\begin{equation} \left\{ \begin{aligned} & \partial_t m = \Delta m + \lambda m(1-m^{a-1}) - \chi\nabla\cdot(f_r(m)\nabla c), & & (x,t)\in Q_T,\\ & \partial_t c = \varepsilon_0\Delta c - c + \lambda(\delta d + \beta m), & & (x,t)\in Q_T,\\ & \partial_t d = g(m)(1-d), & & (x,t)\in Q_T,\\ & \nabla m\cdot \nu =\nabla c\cdot \nu = 0, & & (x,t)\in \partial\Omega\times(0,T),\\ & m(x,0) = \lambda m_0(x), c(x,0) = \lambda c_0(x), d(x,0) = d_0(x), & & x\in\Omega. \end{aligned} \right. \end{equation}

In order to show that $\mathbf {Z}$ is bounded, we propose a series of lemmas. We start with the

Lemma 2.3 Under the assumptions of theorem 1.1, and supposing that $\lambda \in ]0,1]$, we consider $(m,c,d)$ a strong, nonnegative (for each component) solution to (2.13). Then there exists $K_1>0$ (depending on $T$, $a, \beta , \delta$ and $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not depending on $\lambda$) such that

(2.14)\begin{equation} \sup_{t\in[0,T]}\left({\|m({\cdot}, t)\|_{L^{1}(\Omega)} + \|c({\cdot}, t)\|_{L^{1}(\Omega)}}\right) \leq K_1, \end{equation}

and if $0\leq \tau < T$ then

\[ \|m\|_{L^{a}(Q_{\tau,T})} \leq \left[{K_1(T-\tau+1)}\right]^{1/a}. \]

Proof. Note that

(2.15)\begin{equation} \frac{\textrm{d}}{\,\textrm{d}t}\int_{\Omega} (m+c)\,\textrm{d}x = \int_{\Omega}\left[{\lambda(\delta d + (\beta+1)m - m^{a}) - c}\right]\,\textrm{d}x. \end{equation}

Since $a>1$, there exists $k_a>0$ such that

\[ (\beta+1)m - m^{a} \leq k_a - m. \]

Applying this and the fact that $0\leq d \leq \mu$ to (2.15), we obtain some $K_1>0$ such that (2.14) holds. In addition, integrating the equation satisfied by $m$ in (2.13) gives

\[ \int_{\Omega}m(x,T)\,\textrm{d}x = \int_\tau^{T}\int_{\Omega}\lambda m(1-m^{a-1})\,\textrm{d}x\,\textrm{d}t + \int_{\Omega}m(x,\tau)\,\textrm{d}x. \]

Consequently,

(2.16)\begin{equation} \|m\|_{L^{a}(Q_{\tau,T})}^{a} \leq K_1(T-\tau+1). \end{equation}

We now turn to the

Lemma 2.4 Under the assumptions of theorem 1.1, and supposing that $\lambda \in ]0,1]$, we consider $(m,c,d)$ a strong, nonnegative (for each component) solution to (2.13). Then there exists $K_2>0$ (depending on $\Omega$, $T$, $n$, $a$, $\varepsilon _0, \beta , \delta$ and $\|c_0\|_{W^{(2-2/a),a}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not depending on $\lambda$) such that

(2.17)\begin{equation} \|c\|_{W^{2,1}_{a}(Q_T)} + \|\nabla c\|_{L^{q_1}(Q_T)} \leq K_2\,K_1\,(T+1), \end{equation}

where

(2.18)\begin{equation} q_1 = \left\{ \begin{aligned} & \frac{(n+2)a}{n+2-a} & & \text{ if } a < n+2,\\ & <{+}\infty \text{ arbitrary} & & \text{ if } a = n+2, \\ & ={+}\infty & & \text{ if } a > n+2 \end{aligned}\right. \end{equation}

We finally prove the

Lemma 2.5 Under the assumptions of theorem 1.1, and supposing that $\lambda \in ]0,1]$, we consider $(m,c,d)$ a strong, nonnegative (for each component) solution to (2.13). Then for any $1 \le p<\infty$, there exists $F_p^{*}>0$ (depending on $\Omega$, $T$, $n$, $p$, $a$, $b$, $\varepsilon _0$, $\beta$, $\delta$, $\|m_0\|_{W^{(2-2/\tilde {p}),\tilde p}(\Omega )}$, $\|c_0\|_{W^{(2-2/\tilde {p}),\tilde p}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not depending on $\lambda$) such that

(2.19)\begin{equation} \|m\|_{L^{p}(Q_T)} \leq F_p^{*}. \end{equation}

Proof. Note that from (1.4), it follows that there exists $s\in (0,1)$ (depending on $a$, $b$, $n$ only, and with $s$ close to $1$ if the inequality in (1.4) is close to being an equality) satisfying

(2.20)\begin{equation} b-s < \frac{a}{n+2}, \quad \text{ and } \quad b-s<\frac{2a}{n+2} - 1. \end{equation}

It follows that

(2.21)\begin{equation} b < s - 1 + \frac{2a}{n+2} \end{equation}

and

(2.22)\begin{equation} (n+2)(b-s)< a. \end{equation}

Note also that

\[ 1 < \frac{2}{n+2} - \frac{b-s}{a}. \]

Since it is sufficient to prove (2.19) for $p$ large enough, we pick any $p'$ such that

(2.23)\begin{equation} 1 < p' < \min\left\{{\frac{n+2}{2}, \frac{2}{n+2} - \frac{b-s}{a}}\right\}. \end{equation}

Let $\theta \in L^{p'}(Q_T)$ with $\theta \geq 0$ and $\|\theta \|_{L^{p'}(Q_T)} \leq 1$. Let $\phi$ be the unique nonnegative solution to the equation

(2.24)\begin{equation} \left\{ \begin{aligned} \partial_t \phi + \Delta \phi & ={-}\theta, & & (x,t)\in Q_T,\\ \nabla\phi \cdot \nu & = 0, & & (x,t)\in \partial\Omega\times(0,T),\\ \phi(x,T) & = 0, & & x\in\Omega. \end{aligned} \right. \end{equation}

To avoid any possible confusion, we remark that (2.24) is a forward heat equation for the function $\psi (x,s)=\phi (x,T-s)$, with respect to the new time variable $s=T-t$. From maximal regularity in [Reference Ladyženskaija, Alekseevich Solonnikov and Uralceva11] and lemma 2.1, we know that there exists $C_{p'}>0$ (depending on $\Omega$, $T$, $n$ and $p$ only) such that

(2.25)\begin{equation} \|\partial_t\phi\|_{L^{p'}(Q_T)} + \|\phi\|_{L^{q_2}(Q_T)} + \|\nabla \phi\|_{L^{q_3}(Q_T)} \leq C_{p'} , \end{equation}

where

(2.26)\begin{equation} q_2 = \frac{(n+2)p'}{n+2-2p'} \quad \text{ and } \quad q_3 = \frac{(n+2)p'}{n+2-p'}. \end{equation}

Set $p = {p'}/({p'-1})$. Then, by using the equation satisfied by $m$ in (2.13) and integrating by parts, we have

(2.27)\begin{align} \int_0^{T}\int_{\Omega}m\theta \,\textrm{d}x\,\textrm{d}t & = \int_0^{T}\int_{\Omega} m (-\partial_t\phi - \Delta \phi)\,\textrm{d}x\,\textrm{d}t\notag\\ & = \lambda\int_{\Omega}m_0\phi(0)\,\textrm{d}x + \int_0^{T}\int_{\Omega}\phi(\partial_t m - \Delta m)\,\textrm{d}x\,\textrm{d}t\notag\\ & \leq \int_{\Omega}m_0\phi(0)\,\textrm{d}x + \lambda\int_0^{T}\int_{\Omega} \phi m \,\textrm{d}x\,\textrm{d}t + \chi\int_0^{T}\int_{\Omega}f_r(m)\nabla \phi \nabla c \,\textrm{d}x\,\textrm{d}t\notag\\ & \leq \int_{\Omega}m_0\phi(0)\,\textrm{d}x + (I) + (II) , \end{align}

where

\[ (I) = \int_0^{T}\int_{\Omega}\phi m \,\textrm{d}x\,\textrm{d}t \quad \text{ and } \quad (II) = \chi\int_0^{T}\int_{\Omega}|f_r(m)||\nabla \phi||\nabla c|\,\textrm{d}x\,\textrm{d}t. \]

For the first term on the right-hand side of (2.27), we use

\[ \|\phi(0)\|_{L^{p'}(\Omega)}^{p'} = \int_{\Omega}\left|\int_0^{T}\partial_t\phi \,\textrm{d}t\right|^{p'}\,\textrm{d}x \leq T^{\frac{1}{p-1}}\|\partial_t\phi\|_{L^{p'}(Q_T)}^{p'} \leq T^{\frac{1}{p-1}}C_{p'}^{p'} \]

thanks to (2.25), in order to estimate

(2.28)\begin{equation} \int_{\Omega}m_0\phi(0)\,\textrm{d}x \leq \|m_0\|_{L^{p}(\Omega)}\|\phi(0)\|_{L^{p'}(\Omega)} \leq \|m_0\|_{L^{p}(\Omega)}T^{\frac 1p}C_{p'}. \end{equation}

For the treatment of $(I)$, we consider two cases.

• • Case 1. $a\geq ({n+2})/{2}$. Note that

  \[ a \geq \frac{n+2}{2} \Longrightarrow a > \frac{(n+2)p'}{(n+2)(p'-1) + 2p'} \Longrightarrow \frac{a}{a-1} < \frac{(n+2)p'}{n+2-2p'} = q_2. \]
  Therefore, by using Hölder's inequality, we have (for some $C_T$ depending only on $\Omega$, $T$, $a$, $n$ and $p$)
  (2.29)\begin{align} (I) &\leq \|\phi\|_{L^{\frac{a}{a-1}}(Q_T)}\|m\|_{L^{a}(Q_T)} \leq C_T\, \|\phi\|_{L^{q_2}(Q_T)}\|m\|_{L^{a}(Q_T)}\notag\\ & \leq C_T\,C_{p'}\, [K_1\,(T+1)]^{1/a}, \end{align}
  thanks to (2.16) and (2.25).

• • Case 2. $a< ({n+2})/{2}$. Define $\vartheta = 1 - {2a}/({n+2})$ and note that $0<\vartheta <1$. From Hölder's inequality, there exists $R_{T,|\Omega |}>0$ (depending only on $\Omega$, $T$, $a$, $n$ and $p$) such that

  (2.30)\begin{equation} \begin{aligned} (I) & = \int_0^{T}\int_{\Omega} \phi m^{1-\vartheta}m^{\vartheta}\,\textrm{d}x\,\textrm{d}t\\ & \leq R_{T,|\Omega|}\left({\int_0^{T}\int_{\Omega}\phi^{p'}m^{p'(1-\vartheta)}}\right)^{\frac{1}{p'}}\|m\|_{L^{p}(Q_T)}^{\vartheta}\\ & \leq R_{T,|\Omega|} \, \|\phi\|_{L^{\frac{(n+2)p'}{n+2-2p'}}(Q_T)}\left({\int_0^{T}\int_{\Omega}m^{\frac{(n+2)(1-\vartheta)}{2}}\,\textrm{d}x\,\textrm{d}t}\right)^{\frac{2}{n+2}}\|m\|_{L^{p}(Q_T)}^{\vartheta}\\ & = R_{T,|\Omega|} \, \|\phi\|_{L^{q_2}(Q_T)}\|m\|_{L^{a}(Q_T)}^{(1-\vartheta)}\|m\|_{L^{p}(Q_T)}^{\vartheta}, \end{aligned} \end{equation}
  using
  (2.31)\begin{equation} q_2 = \frac{(n+2)p'}{n+2-2p'} \quad \text{ and } \quad \frac{(n+2)(1-\vartheta)}{2} = a. \end{equation}
  Therefore,
  (2.32)\begin{equation} (I) \leq R_{T,\Omega} \, C_{p'}\left[{K_1(T+1)}\right]^{\frac{1-\vartheta}{a}}\|m\|_{L^{p}(Q_T)}^{\vartheta} . \end{equation}

From these two cases, or more precisely, from (2.29) and (2.32), we have

(2.33)\begin{equation} (I) \leq C_T^{*}\,\left({1+\|m\|_{L^{p}(Q_T)}^{\vartheta}}\right) \end{equation}

for some $0<\vartheta <1$ (depending on $a,n$), and $C_T^{*}$ (depending on $\Omega$, $T$, $n$, $p$, $a$, $\beta$, $\delta$, $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not depending on $\lambda$).

In order to estimate $(II)$, we write

(2.34)\begin{equation} \begin{aligned} (II) & = \chi\int_0^{T}\int_{\Omega}|f_r(m)||\nabla \phi||\nabla c|\,\textrm{d}x\,\textrm{d}t\\ & \le \chi \gamma \int_0^{T}\int_{\Omega}|m|^{b}|\nabla \phi||\nabla c|\,\textrm{d}x\,\textrm{d}t\\ & \leq \chi \gamma \left({\int_0^{T}\int_{\Omega}{m^{b-s}|\nabla \phi| |\nabla c|}^{p'}\,\textrm{d}x\,\textrm{d}t}\right)^{\frac{1}{p'}}\|m\|_{L^{p}(Q_T)}^{s}\\ & = \chi \gamma \left({\int_0^{T}\int_{\Omega}m^{p'(b-s)}|\nabla \phi|^{p'} |\nabla c|^{p'}\,\textrm{d}x\,\textrm{d}t}\right)^{\frac{1}{p'}}\|m\|_{L^{p}(Q_T)}^{s}. \end{aligned} \end{equation}

Note that $1< p'< n+2$ (see (2.23)). We can use Hölder's inequality to estimate further

\[ \begin{aligned} (II) & \leq \chi\gamma \, \|\nabla \phi\|_{L^{\frac{(n+2)p'}{n+2-p'}}(Q_T)}\left({\int_0^{T}\int_{\Omega}m^{(n+2)(b-s)}|\nabla c|^{n+2}\,\textrm{d}x\,\textrm{d}t}\right)^{\frac{1}{n+2}}\|m\|_{L^{p}(Q_T)}^{s}\\ & \le \chi\gamma \, \|\nabla \phi\|_{L^{q_3}(Q_T)}\|m\|_{L^{a}(Q_T)}^{b-s}\|\nabla c\|_{L^{q_4}(Q_T)}\|m\|_{L^{p}(Q_T)}^{s}, \end{aligned} \]

where

(2.35)\begin{equation} q_4 = \frac{(n+2)a}{a-(n+2)(b-s)} \end{equation}

is well defined since $(n+2)(b-s) < a$ (see (2.20)). Therefore, we can apply (2.25), (2.16) in order to get

(2.36)\begin{equation} (II) \leq \chi \gamma \, C_{p'}\, \left[{K_1(T+1)}\right]^{\frac{b-s}{a}}\|\nabla c\|_{L^{q_4}(Q_T)}\|m\|_{L^{p}(Q_T)}^{s}. \end{equation}

Note that from (2.35) we see that when $a< n+2$,

(2.37)\begin{equation} q_4 = \frac{(n+2)a}{a-(n+2)(b-s)}<\frac{(n+2)a}{n+2-a} = q_1, \end{equation}

and that when $a\geq n+2$, $q_4< q_1$ since $q_1<+\infty$ can be chosen arbitrarily. Therefore (for some $C_T>0$ depending only on $\Omega$, $T$, $a$, $b$, $n$),

\[ \|\nabla c\|_{L^{q_4}(Q_T)} \leq C_T\|\nabla c\|_{L^{q_1}(Q_T)} \leq C_T\, K_2\, K_1\,(T+1), \]

thanks to (2.17). It then follows from (2.36) that

(2.38)\begin{equation} (II) \leq D_T^{*}\, \|m\|_{L^{p}(Q_T)}^{s}, \end{equation}

where $D_T^{*}>0$ only depends on $\Omega$, $T$, $n$, $p$, $a$, $b$, $\beta$, $\delta$, $\varepsilon _0$, $\chi$, $\gamma$, $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{W^{\tilde p, (2-2/\tilde {p})}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but does not depend on $\lambda$.

From (2.27), (2.28), (2.33) and (2.38), we get (for $E^{*} := C_{p'} T^{1/p} + C_T^{*} + D_T^{*}$)

\[ \int_0^{T}\int_{\Omega}m\theta \,\textrm{d}x\,\textrm{d}t \leq E^{*}\, \left[{\|m_0\|_{L^{p}(\Omega)} + 1 + \|m\|_{L^{p}(Q_T)}^{\vartheta} + \|m\|_{L^{p}(Q_T)}^{s}}\right] \]

for all $0\leq \theta \in L^{p'}(Q_T)$ satisfying $\|\theta \|_{L^{p'}(Q_T)} = 1$. Therefore, by duality, we get

\[ \|m\|_{L^{p}(Q_T)} \leq E^{*}\,\left[{\|m_0\|_{L^{p}(\Omega)} + 1 + \|m\|_{L^{p}(Q_T)}^{\vartheta} + \|m\|_{L^{p}(Q_T)}^{s}}\right]. \]

Note that $0 < \vartheta , s < 1$. Thus we can use Young's inequality in order to get

(2.39)\begin{equation} \|m\|_{L^{p}(Q_T)}\leq F_p^{*}\, \left({1+\|m_0\|_{L^{p}(\Omega)}}\right), \end{equation}

where $F_p^{*}$ depends on the same quantities as $D_T^{*}$.

From (2.23), we recall from $p=\frac {p'}{p'-1}$ that $1< p<\infty$ can be chosen arbitrarily. We also insist that the bound in (2.39) does not depend on $\lambda \in (0,1]$.

We now use lemma 2.5 in order to conclude the proof of proposition 2.2.

Since $\tilde p>n+2$, lemma 2.1 guarantees $W^{2,1}_{\tilde p}(Q_T)\hookrightarrow \mathfrak {Y}$ compactly. Applying this to the equation satisfied by $c$ in (2.13), we get (for some constant $C_T>0$ depending only on $\Omega$, $T$, $n$, $p$ and $\varepsilon _0$)

(2.40)\begin{equation} \|c\|_{L^{\infty}(Q_T)} \leq C_T\, \bigg( \|\lambda(\delta d + \beta m)\|_{L^{\tilde p}(Q_T)} + \|c_0\|_{W^{\tilde p,(2-2/\tilde{p})}(\Omega)} \bigg) \leq \S_T, \end{equation}

for some $\S _T>0$ (depending only on the same quantities as $D_T^{*}$, except $\tilde p$).

Applying the heat semigroup property and maximal regularity, we can find $\S _T^{*}$ (depending on the same quantities as $D_T^{*}$) such that

(2.41)\begin{equation} \| \nabla c\|_{L^{\infty}(Q_T)} + \|c\|_{W^{2,1}_{\tilde p}(Q_T)} \le \S_T^{*}, \end{equation}

and therefore

(2.42)\begin{equation} \|c\|_{\mathfrak{Y}} \leq S_T + \S_T^{*}. \end{equation}

We now consider the equation satisfied by $m$ in (2.13):

(2.43)\begin{equation} \begin{aligned} \partial_t m & = \Delta m + \lambda m(1-m^{a-1}) - \chi \nabla\cdot(f_r(m)\nabla c)\\ & = \Delta m -\chi f_r'(m)\nabla c\cdot \nabla m + \lambda m(1-m^{a-1}) - \chi f_r(m)\Delta c. \end{aligned} \end{equation}

From (2.39) with $p=\tilde p$, (2.42), and assumption (1.3), we apply the properties of regularity for parabolic equations (cf [Reference Ladyženskaija, Alekseevich Solonnikov and Uralceva11]) to (2.43), and get

\begin{align*} \|m\|_{W^{2,1}_{\tilde p}(Q_T)} &\leq C_T\left(\left\|\lambda m(1-m^{a-1}) - \chi f_r(m)\Delta c\right\|_{L^{\tilde p}(Q_T)}+\|m_0\|_{W^{\tilde p,(2-2/\tilde{p})}(\Omega)}\right)\\ & \leq \S_T^{**}, \end{align*}

for some $\S _T^{**}>0$ depending on the same parameters as $\S _T$. By the embedding (2.7)

(2.44)\begin{equation} \|m\|_{\mathfrak{Y}} \leq C_T\|m\|_{W^{2,1}_{\tilde p}(Q_T)} \leq C_T\, \S_T^{**}. \end{equation}

Note that the bounds (2.42) and (2.44) do not depend on $\lambda \in (0,1]$ (or $(m,c)$). This means that the set $\mathbf {Z}$ defined in (2.11) is bounded in $\mathfrak {X}=\mathfrak {Y}\times \mathfrak {Y}$. This shows that the last assumption of Leray-Schauder fixed point theorem holds, and therefore that the mapping $\mathcal {F}_r$ has a fixed point, which satisfies as a consequence system (2.13), together with estimates (2.42), (2.44). Since moreover the upper bound $\S _T$ in (2.44) does not depend on $r$, by taking $r\geq \S _T$, we obtain a solution to the original system (1.1) (for all $T>0$). This solution is strong in the sense that all terms appearing in the system are defined a.e., (as $L^{1}(Q_T)$ functions).

2.2 Uniqueness

In this subsection, we write down a stability result for strong solutions of system (1.1), which entails the uniqueness result in theorem 1.1. More precisely, we write the

Proposition 2.6 Let $T>0$ and let $\Omega$ be a smooth ($C^{2+ \alpha }$ for some $\alpha >0$) bounded connected open set of $\mathbb {R}^{n}$ (for $n \in \mathbb {N} - \{0\}$). In addition, assume $a>1$, $\chi , \varepsilon _0, \delta , \beta > 0$, $\gamma , b, l \ge 0$, and $f$ and $g$ satisfying (F) and (G). We consider two sets of nonnegative (for each component), initial data $(m_{01}, c_{01}, d_{01})$ and $(m_{02}, c_{02}, d_{02})$ in $W^{(2-2/{\tilde p}),\tilde p}(\Omega )\times W^{(2-2/{\tilde p}),\tilde p}(\Omega ) \times L^{\infty }(\Omega )$ for some $\tilde p>\max \{n+2,a\}$, and two sets of nonnegative (for each component) strong (in the sense of theorem 1.1) solutions $(m_{1}, c_{1}, d_{1})$ and $(m_{2}, c_{2}, d_{2})$ to system (1.1) (with corresponding initial data) on $Q_T$, satisfying estimate (2.1). We denote $\mu := \|d_2\|_{L^{\infty }(Q_T)}$, $\mu _c := \|\nabla c_1\|_{L^{\infty }(Q_T)}$, $\mu _m := \max ( \|m_1\|_{L^{\infty }(Q_T)}, \|m_2\|_{L^{\infty }(Q_T)})$, $\mu _f := \|f\|_{L^{\infty }([0, \mu _m])}$, $\mu _{f'} := \|f'\|_{L^{\infty }([0, \mu _m])}$, $\mu _{g'} := \|g'\|_{L^{\infty }([0, \mu _m])}$.

Then, for all $t \in [0,T]$,

(2.45)\begin{align} &\int_{\Omega} |m_1({\cdot}, t) - m_2({\cdot}, t)|^{2}\,\textrm{d}x + \frac{\chi^{2}\, \mu_f^{2}}{2\varepsilon_0} \int_{\Omega} |c_1({\cdot}, t) - c_2({\cdot}, t)|^{2}\,\textrm{d}x\notag\\ &\qquad + \int_{\Omega} |d_1({\cdot}, t) - d_2({\cdot}, t)|^{2}\,\textrm{d}x \notag\\ &\quad \le e^{G\,t} \bigg( \int_{\Omega} |m_{10} - m_{20}|^{2}\,\textrm{d}x + \frac{\chi^{2}\, \mu_f^{2}}{2\varepsilon_0} \int_{\Omega} |c_{10} - c_{20}|^{2}\,\textrm{d}x + \int_{\Omega} |d_{10} - d_{20}|^{2}\,\textrm{d}x \bigg), \end{align}

where

\[ G := \max\bigg( 2 + \chi^{2}\,\mu_{f'}^{2} \, \mu_c^{2} + \frac{\chi^{2}\,\beta\,\mu_f^{2}}{2\varepsilon_0} + \mu_{g'}\,(1+\mu) ; \beta + \delta ; \frac{\chi^{2}\,\delta\,\mu_f^{2}}{2\varepsilon_0} + (1+\mu) \bigg) . \]

Proof. Substracting the equations satisfied by $m_1$ and $m_2$, $c_1$ and $c_2$, and finally $d_1$ and $d_2$, and then performing integrations by parts, we end up with the identities:

(2.46)\begin{align} & \frac{1}{2} \,\frac{{\rm d}}{{\rm d}t} \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x + \int_{\Omega} |\nabla(m_1 - m_2)|^{2}\,\textrm{d}x + \int_{\Omega} (m_1^{a} - m_2^{a})\,(m_1 - m_2)\,\textrm{d}x\notag\\ & \quad= \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x + \chi \int_{\Omega} f(m_2)\, \nabla(m_1 - m_2)\cdot \nabla(c_1 - c_2)\,\textrm{d}x \notag\\ & \qquad + \chi \int_{\Omega}(f(m_1) - f(m_2))\, \nabla(m_1 - m_2)\cdot \nabla c_1\,\textrm{d}x , \end{align}

(2.47)\begin{align} &\frac{1}{2}\,\frac{\textrm{d}}{\textrm{d}t} \int_{\Omega} |c_1 - c_2|^{2}\,\textrm{d}x + \varepsilon_0 \int_{\Omega} |\nabla(c_1 - c_2)|^{2}\,\textrm{d}x + \int_{\Omega} |c_1 - c_2|^{2}\,\textrm{d}x\notag\\ & \quad=\delta \int_{\Omega} (c_1 - c_2)\,(d_1 - d_2)\,\textrm{d}x + \beta \int_{\Omega} (c_1 - c_2)\,(m_1 - m_2)\,\textrm{d}x , \end{align}

(2.48)\begin{align} & \frac{1}{2}\,\frac{\textrm{d}}{\textrm{d}t} \int_{\Omega} |d_1 - d_2|^{2}\,\textrm{d}x + \int_{\Omega} g(m_1)\, |d_1 - d_2|^{2}\,\textrm{d}x\notag\\ & \quad={-} \int_{\Omega} (1 - d_2)\,(d_1 - d_2)\,(g(m_1) - g(m_2))\,\textrm{d}x. \end{align}

Then we perform the following estimates (the last one uses Young's inequality):

(2.49)\begin{align} &\frac{\textrm{d}}{\textrm{d}t} \int_{\Omega} |d_1 - d_2|^{2}\,\textrm{d}x \le (1 +\mu)\, \int_{\Omega} |d_1 - d_2|^{2}\,\textrm{d}x + (1 +\mu)\, \mu_{g'}^{2} \int_{\Omega} |m_1 - m_2|^{2} \,\textrm{d}x , \end{align}

(2.50)\begin{align} & \frac{\textrm{d}}{\textrm{d}t} \int_{\Omega} |c_1 - c_2|^{2}\,\textrm{d}x + 2\varepsilon_0 \int_{\Omega} |\nabla(c_1 - c_2)|^{2}\,\textrm{d}x\notag\\ & \quad\le \delta\, \int_{\Omega} |d_1 - d_2|^{2}\,\textrm{d}x + \beta \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x + (\beta + \delta) \int_{\Omega} |c_1 - c_2|^{2}\,\textrm{d}x, \end{align}

(2.51)\begin{align} &\frac{\textrm{d}}{\textrm{d}t} \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x \le (2 + \chi^{2}\,\mu_{f'}^{2} \mu_c^{2}) \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x\notag\\ &\quad + \chi^{2} \mu_f^{2} \int_{\Omega} |\nabla(c_1 - c_2)|^{2}\,\textrm{d}x . \end{align}

Estimate (2.45) is directly obtained from these estimates by an application of Gronwall's lemma to the quantity

\[ \int_{\Omega} |m_1 - m_2|^{2}\,\textrm{d}x + \frac{\chi^{2}\, \mu_f^{2}}{2\varepsilon_0} \int_{\Omega} |c_1 - c_2|^{2}\,\textrm{d}x + \int_{\Omega} |d_1 - d_2|^{2}\,\textrm{d}x . \]

Note that uniqueness in theorem 1.1 is a direct consequence of proposition 2.6. Moreover, since $T$ is arbitrary in propositions 2.2 and 2.6, one can build the unique solution of system (1.1) on $\mathbb {R}_+$ that appears in the conclusion of theorem 1.1 by patching together the solutions defined on finite time intervals.

2.3 Uniform-in-time bounds

In this section, we will conclude the proof of theorem 1.1 by showing that the unique solution to (1.1) obtained in propositions 2.2 and 2.6 is globally (w.r.t. time) bounded. Thanks to proposition 2.2, it is sufficient to show that the sequences $\left \{\|m\|_{L^{\infty }(Q_{\tau ,\tau +1})}\right \}_{\tau =1}^{\infty }$ and $\left \{\|c\|_{L^{\infty }(Q_{\tau ,\tau +1})}\right \}_{\tau =1}^{\infty }$ are bounded. As a consequence, we will obtain estimates in this subsection in which all constants do not depend on $\tau$. In addition, we will see that the $\limsup$ of each sequence is independent of $m_0$ and $c_0$. We first recall the uniform in time bound

\[ \sup_{t \in \mathbb{R}_+} \|d({\cdot}, t)\|_{L^{\infty}(\Omega)} \leq \mu := \max\left\{{1, \|d_0\|_{L^{\infty}(\Omega)}}\right\}. \]

Coming back to the proof of lemma 2.3, and specializing it in the case when $\lambda =1$, we see that

\[ \frac{\textrm{d}}{\,\textrm{d}t} \int_{\Omega} (m+c) \le |\Omega|\,(\delta\,\mu + k_a) - \int_{\Omega} (m+c), \]

so that

(2.52)\begin{equation} \sup_{t\in \mathbb{R}_+}\left({\|m(t)\|_{L^{1}(\Omega)} + \|c(t)\|_{L^{1}(\Omega)}}\right) \leq K_1^{*}, \end{equation}

where $K_1^{*} := \max ( |\Omega |\,(\delta \,\mu + k_a) , \|m_0\|_{L^{1}(\Omega )} + \|c_0\|_{L^{1}(\Omega )})$, and

(2.53)\begin{equation} \limsup_{t\to \infty}\left({\|m(t)\|_{L^{1}(\Omega)} + \|c(t)\|_{L^{1}(\Omega)}}\right) \leq |\Omega|\,(\delta\,\mu + k_a). \end{equation}

We also recall that $k_a$ and $\mu$ only depend on $\beta , a$ and $\|d_0\|_{L^{\infty }(\Omega )}$.

Then, we prove the

Lemma 2.7 Under the assumptions of theorem 1.1, we consider $(m,c,d)$, the unique solution to (1.1) on $\Omega \times \mathbb {R}_+$. Then

(2.54)\begin{equation} \sup_{\tau \in \mathbb{N}}\|m\|_{L^{a}(Q_{\tau,\tau+2})} \leq (3\,K_1^{*})^{\frac 1a}, \end{equation}

and

(2.55)\begin{equation} \limsup_{\tau \to \infty}\|m\|_{L^{a}(Q_{\tau,\tau+2})} \leq (3\,|\Omega|\,(\delta\,\mu + k_a))^{\frac 1a}. \end{equation}

Proof. By integrating the equation satisfied by $m$ on $Q_{\tau ,\tau +2}$, we have

\[ \int_{\Omega}m(x,\tau+2)\,\textrm{d}x = \int_{\Omega}m(x,\tau)\,\textrm{d}x + \int_{\tau}^{\tau+2}\int_{\Omega}m(x,s)(1-m(x,s)^{a-1})\,\textrm{d}x\,\textrm{d}s. \]

Thus

\[ \int_{\tau}^{\tau+2}\int_{\Omega}m(x,s)^{a}\, \,\textrm{d}x\,\textrm{d}s \leq \|m(\tau)\|_{L^{1}(\Omega)} + \int_{\tau}^{\tau+2}\int_{\Omega}m(x,s)\,\textrm{d}x \,\textrm{d}s, \]

which proves both (2.54) and (2.55).

Lemma 2.8 Under the assumptions of theorem 1.1, we consider $(m,c,d)$, the unique solution to (1.1) on $\Omega \times \mathbb {R}_+$. Let $q_1$ be defined as in (2.18). Then, there exists $K_3>0$ (depending on $\Omega$, $n$, $a$, $\varepsilon _0, \beta , \delta$ and $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not on $\tau$) and $\tilde K_3>0$ (depending on $\Omega$, $n$, $a$, $\varepsilon _0, \beta , \delta$ and $\|d_0\|_{L^{\infty }(\Omega )}$, but not on $\tau$) such that

(2.56)\begin{equation} \sup_{\tau\in\mathbb{N} - \{0\} }\|\nabla c\|_{L^{q_1}(Q_{\tau,\tau+1})} \leq K_3, \end{equation}

and

(2.57)\begin{equation} \limsup_{\tau\to\infty }\|\nabla c\|_{L^{q_1}(Q_{\tau,\tau+1})} \leq \tilde K_3. \end{equation}

Proof. We define a smooth cutoff function $\psi \in C^{\infty }(\mathbb {R}; [0,1])$ such that $\psi (s) = 0$ for $s\leq 0$, and $\psi (s) =1$ for $s\geq 1$. Moreover, we assume that $|\psi '(s)| \leq 2$ for all $s\in \mathbb {R}$. For any $\tau \in \mathbb {N}$, the shifted cutoff function is defined by $\psi _\tau (\cdot ) := \psi (\cdot -\tau )$. By multiplying the equation satisfied by $c$ by $\psi _\tau$, we get

(2.58)\begin{equation} \partial_t(\psi_\tau c) = \varepsilon_0 \Delta(\psi_\tau c) + \psi_\tau^{\prime} c - \psi_\tau c + \psi_\tau[\delta d + \beta m], \end{equation}

and $(\psi _\tau c)(x,\tau ) = 0$. By using the semigroup properties of the heat equation (lemma 2.1), we see that (for some $C>0$ depending only on $\Omega$, $n$, $a$, $\varepsilon _0$, but not $\tau$)

(2.59)\begin{align} \|\psi_\tau c\|_{L^{q_5}(Q_{\tau,\tau+2})} &\leq C(\|\psi_\tau^{\prime} c\|_{L^{a}(Q_{\tau,\tau+2})} + \|\psi_\tau c\|_{L^{a}(Q_{\tau,\tau+2})}\notag\\ &\quad + \|\psi_\tau(\delta d + \beta m)\|_{L^{a}(Q_{\tau,\tau+2})}) , \end{align}

where

\[ q_5 = \left\{ \begin{aligned} & \frac{(n+2)a}{n+2-2a} & & \text{ if } a<\frac{n+2}{2},\\ & <{+}\infty \text{ arbitrary} & & \text{ if } a = \frac{n+2}{2},\\ & ={+}\infty & & \text{ if } a > \frac{n+2}{2} . \end{aligned} \right. \]

In any case, we have $a < q_5$. By using the boundedness of $\psi _\tau '$ and $\psi _\tau$, and Hölder's inequality, we have, for $\alpha \in (0,1)$ satisfying $({1}/{a}) = ({1-\alpha }/{1}) + ({\alpha }/{q_5})$,

(2.60)\begin{align} \|\psi_\tau^{\prime} c\|_{L^{a}(Q_{\tau,\tau+2})} &\leq 2\|c\|_{L^{a}(Q_{\tau,\tau+2})} \leq 2\|c\|_{L^{1}(Q_{\tau,\tau+2})}^{1-\alpha}\|c\|_{L^{q_5}(Q_{\tau,\tau+2})}^{\alpha} \notag\\ &\leq 2(2K_1^{*})^{1-\alpha}\,\|c\|_{L^{q_5}(Q_{\tau,\tau+2})}^{\alpha}, \end{align}

and similarly

(2.61)\begin{equation} \|\psi_\tau c\|_{L^{a}(Q_{\tau,\tau+2})} \leq (2K_1^{*})^{1-\alpha}\,\|c\|_{L^{q_5}(Q_{\tau,\tau+2})}^{\alpha} . \end{equation}

From the uniform boundedness of $d$ and (2.54), we see that

(2.62)\begin{equation} \|\psi_\tau(\delta d+\beta m)\|_{L^{a}(Q_{\tau,\tau+2})} \leq (3 K_1^{*})^{\frac 1a}\beta + \delta\,\mu\,(2|\Omega|)^{1/a}. \end{equation}

Inserting (2.60)–(2.62) into (2.59) and using $\psi _{\tau }\equiv 1$ on $(\tau +1,\tau +2)$ yields, for some constant $C_2>0$ (depending only on $\Omega$, $n$, $\delta$, $\beta$, $a$, $\varepsilon _0$, $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not $\tau$)

(2.63)\begin{equation} \|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})} \leq C_2\,\|c\|_{L^{q_5}(Q_{\tau,\tau+2})}^{\alpha} + C_2. \end{equation}

Denote by $\Lambda := \left \{{\tau +1 \in \mathbb {N}\;:\;\|c\|_{L^{q_5}(Q_{\tau ,\tau +1})} \leq \|c\|_{L^{q_5}(Q_{\tau +1,\tau +2})}}\right \}$. Then for all $\tau \in \Lambda$, we get from (2.63) that

\[ \|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})} \leq 2C_2\,\|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})}^{\alpha} + C_2 \leq \frac 12\|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})} + C_3, \]

where $C_3>0$ only depends on the same parameters as $C_2$. Thus

\[ \|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})} \leq 2C_3 \quad \text{ for all } \quad \tau \in \Lambda. \]

By the definition of $\Lambda$, this in fact implies that

\[ \|c\|_{L^{q_5}(Q_{\tau+1,\tau+2})} \leq 2C_3 \quad \text{ for all } \quad \tau \in \mathbb{N}. \]

Since $a< q_5$, we get with $\frac 1a = \frac {1-\alpha }{1} + \frac {\alpha }{q_5}$,

(2.64)\begin{align} \sup_{\tau\in\mathbb{N} - \{0\} }\|c\|_{L^{a}(Q_{\tau,\tau+1})} \leq \sup_{\tau\in\mathbb{N} - \{0\} }\|c\|_{L^{1}(Q_{\tau,\tau+1})}^{1-\alpha}\|c\|_{L^{q_5}(Q_{\tau,\tau+1})}^{\alpha}\leq (K_1^{*})^{1-\alpha}(2C_3)^{\alpha} . \end{align}

We can now use this estimate, the uniform boundedness of $d$, and (2.54), and apply the semigroup properties of the heat kernel to (2.58), and get (for some constant $C>0$ only depending on $\Omega$, $n$, $\varepsilon _0$ and $a$),

\begin{align*} \|\psi_\tau|\nabla c|\|_{L^{q_1}(Q_{\tau,\tau+2})} &\leq C\, \|\psi_\tau' c - \psi_\tau c + \psi_\tau[\delta d + \beta m]\|_{L^{a}(Q_{\tau,\tau+2})}\\ &\le C\, (6(K_1^{*})^{1-\alpha} (2C_3)^{\alpha} + (3 K_1^{*})^{\frac 1a}\beta + \delta\,\mu\, (2|\Omega|)^{1/a} ). \end{align*}

Finally, by using $\psi _\tau \equiv 1$ on $(\tau +1,\tau +2)$, we obtain (2.56). Similarly, using (2.55), we obtain (2.57).

Using lemmas 2.7 and 2.8, we obtain the following result:

Lemma 2.9 Under the assumptions of theorem 1.1, we consider $(m,c,d)$, the unique solution to (1.1) on $\Omega \times \mathbb {R}_+$. For any $1 \le p<\infty$, there exist a constant $K_p>0$ (depending on $\Omega$, $n$, $p$, $a, \beta , \delta$, $\varepsilon _0$ and $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$, but not on $\tau$), and a constant $\tilde K_p>0$ (depending on $\Omega$, $n$, $p$, $a, \beta , \delta$, $\varepsilon _0$ and $\|d_0\|_{L^{\infty }(\Omega )}$, but not on $\tau$) such that

(2.65)\begin{equation} \sup_{\tau\in\mathbb{N} - \{0\} }\|m\|_{L^{p}(Q_{\tau,\tau+1})} \leq K_p, \end{equation}

and

(2.66)\begin{equation} \limsup_{\tau\to\infty }\|m\|_{L^{p}(Q_{\tau,\tau+1})} \leq \tilde K_p. \end{equation}

Proof. The proof of this lemma is similar to that of lemma 2.5, except that we will consider the cylinder $Q_{\tau ,\tau +2}$ instead of $Q_T$. More precisely, by multiplying the equation satisfied by $m$ in (1.1) with $\psi _\tau$, we get

(2.67)\begin{equation} \partial_t(\psi_\tau m) = \Delta(\psi_\tau m) + \psi_\tau' m + \psi_\tau m(1-m^{a-1}) - \chi \psi_\tau \nabla\cdot(f(m)\nabla c), \end{equation}

and $(\psi _\tau m)(x,\tau ) = 0$. Let $s\in (0,1)$ satisfy (2.20), and choose $p'$ as in (2.23). Consider $0\leq \theta \in L^{p'}(Q_{\tau ,\tau +2})$ with $\|\theta \|_{L^{p'}(Q_{\tau ,\tau +2})} = 1$. Let $\phi$ be the unique nonnegative solution to the equation

\[ \begin{cases} \partial_t\phi + \Delta \phi ={-}\theta, & (x,t)\in Q_{\tau,\tau+2},\\ \nabla\phi\cdot \nu = 0, & (x,t)\in \partial\Omega\times(\tau,\tau+2),\\ \phi(x,\tau+2) = 0, & x\in\Omega. \end{cases} \]

From the semigroup properties of the heat equation (lemma 2.1), we know that there exists a constant $L_{p'}>0$ (depending on $\Omega$, $n$, $p$ but not $\tau$) such that

(2.68)\begin{equation} \|\phi\|_{L^{q_2}(Q_{\tau,\tau+2})} + \|\nabla \phi\|_{L^{q_3}(Q_{\tau,\tau+2})} \leq L_{p'} , \end{equation}

with $q_2$ and $q_3$ defined in (2.26). By integration by parts, we estimate similarly to (2.27) and get

(2.69)\begin{align} \int_{\tau}^{\tau+2}\int_{\Omega}(\psi_\tau m)\theta \,\textrm{d}x\,\textrm{d}t & \leq \int_{\tau}^{\tau+2}\int_{\Omega}(\psi_\tau' m + \psi_\tau m)\, \phi \,\textrm{d}x\,\textrm{d}t\notag\\ &\quad + \chi\int_{\tau}^{\tau+2}\int_{\Omega} |f(m)||\nabla \phi| |\nabla c| \,\textrm{d}x\,\textrm{d}t\notag\\ & =: (III) + (IV). \end{align}

We estimate $(III)$ similarly to $(I)$ in (2.27). More precisely, in the case when $a\geq ({n+2})/{2}$, we have (for some $C_4>0$ depending only on $\Omega$, $n$, $p$, $a, \beta , \delta$, $\varepsilon _0$ and $\|m_0\|_{L^{1}(\Omega )}$, $\|c_0\|_{L^{1}(\Omega )}$, $\|d_0\|_{L^{\infty }(\Omega )}$)

(2.70)\begin{equation} (III) \leq 3\|\phi\|_{L^{\frac{a}{a-1}}(Q_{\tau,\tau+2})}\|m\|_{L^{a}(Q_{\tau,\tau+2})} \leq C_4, \end{equation}

and in the case when $a < ({n+2})/{2}$, we have (for some $C_5>0$ depending only on $\Omega$, and some $C_6>0$ depending on the same quantities as $C_4$)

(2.71)\begin{align} (III) &\leq 3\int_{\tau}^{\tau+2}\int_{\Omega}\phi m^{1-\vartheta}m^{\vartheta}\,\textrm{d}x\,\textrm{d}t\notag\\ & \leq C_5\, \|\phi\|_{L^{q_2}(Q_{\tau,\tau+2})}\|m\|_{L^{a}(Q_{\tau,\tau+2})}^{1-\vartheta}\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{\vartheta}\notag\\ & \leq C_6\, \|m\|_{L^{p}(Q_{\tau,\tau+2})}^{\vartheta}, \end{align}

where $\vartheta$ is defined in (2.31). It follows from (2.70) and (2.71) that, in any case, we have the estimate

(2.72)\begin{equation} (III) \leq \max(C_4, C_6)\, \left({1+\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{\vartheta}}\right). \end{equation}

The term $(IV)$ can be estimated similarly to $(II)$ in (2.34) and (2.36). More precisely, it follows from Hölder's inequality that (for some $C_7, C_8>0$ depending only on $\Omega$, and $C_9>0$ depending on the same quantities as $C_4$ and $b, \gamma , \chi$)

\begin{align*} (IV) & \leq \chi\gamma\int_{\tau}^{\tau+2}\int_{\Omega}|m|^{b}|\nabla \phi||\nabla c|\,\textrm{d}x\,\textrm{d}t\\ & \leq \chi\gamma \left({\int_{\tau}^{\tau+2}\int_{\Omega}m^{p'(b-s)}|\nabla \phi|^{p'}|\nabla c|^{p'}\,\textrm{d}x\,\textrm{d}t}\right)^{\frac{1}{p'}}\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s}\\ & \leq C_7\, \chi \gamma\|\nabla \phi\|_{L^{q_3}(Q_{\tau,\tau+2})}\|m\|_{L^{a}(Q_{\tau,\tau+2})}^{b-s}\|\nabla c\|_{L^{q_4}(Q_{\tau,\tau+2})}\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s} \\ &\qquad \text{(}q_4 \,\,\text{is defined in (2.35))}\\ & \leq C_8\, \chi \gamma L_{p'}\|m\|_{L^{a}(Q_{\tau,\tau+2})}^{b-s}\|\nabla c\|_{L^{q_1}(Q_{\tau,\tau+2})}\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s} \\ &\qquad \text{(since}\, q_4< q_1, \text{see (2.37)})\\ & \leq C_9\,\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s}. \end{align*}

Inserting this and (2.72) into (2.69), we get

\[ \int_{\tau}^{\tau+2}\int_{\Omega}(\psi_\tau m)\theta \,\textrm{d}x\,\textrm{d}t \leq \max(C_4, C_6, C_9)\, \left({1+\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{\vartheta} + \|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s}}\right) \]

for all $0\leq \theta \in L^{p'}(Q_{\tau ,\tau +2})$ satisfying $\|\theta \|_{L^{p'}(Q_{\tau ,\tau +2})} = 1$. Therefore, by duality, we get (for $C_{10} := \max (C_4, C_6, C_9))$

(2.73)\begin{equation} \|\psi_\tau m\|_{L^{p}(Q_{\tau,\tau+2})} \leq C_{10}\, \left({1+\|m\|_{L^{p}(Q_{\tau,\tau+2})}^{\vartheta} + \|m\|_{L^{p}(Q_{\tau,\tau+2})}^{s}}\right). \end{equation}

Define $\Theta := \left \{{\tau +1 \in \mathbb {N}\,:\,\|m\|_{L^{p}(Q_{\tau ,\tau + 1})} \leq \|m\|_{L^{p}(Q_{\tau + 1,\tau +2})}}\right \}$. Then for any $\tau \in \Theta$, we have

\[ \begin{aligned} \text{RHS of (2.73)} & \leq 2\,C_{10}\, \left({1+ \|m\|_{L^{p}(Q_{\tau+1,\tau+2})}^{\vartheta}+ \|m\|_{L^{p}(Q_{\tau+1,\tau+2})}^{s}}\right)\\ & \leq C_{11} + \frac 12 \|m\|_{L^{p}(Q_{\tau+1,\tau+2})}, \end{aligned} \]

(where $C_{11}$ depends on the same quantities as $C_9$) thanks to the fact that $0<\vartheta , s < 1$. From this, (2.73), and $\psi _\tau \equiv 1$ on $(\tau +1,\tau +2)$, we get

\[ \|m\|_{L^{p}(Q_{\tau+1,\tau+2})} \leq C_{11} + \frac 12 \|m\|_{L^{p}(Q_{\tau+1,\tau+2})} \quad \text{ for all } \quad \tau\in \Theta, \]

and consequently

\[ \|m\|_{L^{p}(Q_{\tau+1,\tau+2})} \leq 2\, C_{11} \quad \text{ for all } \quad \tau\in \Theta. \]

Thanks to the definition of $\Theta$, we finally obtain the desired estimate (2.65). In a similar manner, we obtain (2.66).

We are now ready to show the uniform-in-time boundedness of the solution to (1.1).

The uniform-in-time boundedness of $d$ is already shown. It remains to show it for $m$ and $c$. We start by rewriting equation (2.58) as

\[ \partial_t(\psi_\tau c) = \varepsilon_0\, \psi_\tau \,\Delta c + (\psi_\tau' - \psi_\tau)c + H , \quad \text{ with } H = \psi_\tau[\delta d + \beta m]. \]

Thanks to the lemma 2.9, we know that for any $1\le p<\infty$, $\sup _{\tau \in \mathbb {N}}\|H\|_{L^{p}(Q_{\tau ,\tau +2})} \leq \beta \, K_p + \delta \,(2 |\Omega |)^{1/p}$. Proceeding as in lemma 2.8, we see that for any $1\leq p<\infty$, there exists $K_p^{*}>0$ (depending on the same quantities as $K_p$) such that $\sup _{\tau \in \mathbb {N}}\left \|{c}\right \|_{L^{p}(Q_{\tau ,\tau +2})} \leq K_p^{*}$. By maximal regularity, recalling that $(\psi _\tau c)(x,\tau ) = 0$, we have (for some $C>0$ depending on $\Omega$, $p$, $n$, $\varepsilon _0$, but not $\tau$)

\[ \|\psi_\tau c\|_{W^{2,1}_{p}(Q_{\tau,\tau+2})} \leq C\|(\psi_\tau' - \psi_\tau)c + H\|_{L^{p}(Q_{\tau,\tau+2})} , \]

so that picking $p>({n+2})/{2}$, we can use the semigroup properties of the heat kernel (lemma 2.1) in order to get

\[ \|\psi_\tau c\|_{L^{\infty}(Q_{\tau,\tau+2})} \leq C\|\psi_\tau c\|_{W^{2,1}_p(Q_{\tau,\tau+2})} \leq C_{[\frac{n+2}2] + 1}. \]

Due to $\psi _\tau \equiv 1$ on $(\tau +1,\tau +2)$, we finally obtain

\[ \|c\|_{L^{\infty}(Q_{\tau+1,\tau+2})} \leq C_{[\frac{n+2}2] + 1} . \]

Moreover, by picking $p>n+2$, we obtain also the uniform estimate for $\nabla c$, i.e. (for some $C>0$ depending on $\Omega$, $p$, $n$, $\varepsilon _0$, but not on $\tau$)

(2.74)\begin{equation} \|\nabla c\|_{L^{\infty}(Q_{\tau+1,\tau+2})} \leq \|\psi_\tau|\nabla c|\|_{L^{\infty}(Q_{\tau,\tau+2})} \leq C\|c\|_{W^{2,1}_p(Q_{\tau,\tau+2})} \leq C_{n + 3}. \end{equation}

This concludes the proof of boundedness of $c$ (and $\nabla c$).

For the uniform boundedness of $m$, we observe that from (2.67), one can write the Duhamel formula for all $t\in (\tau ,\tau +2)$

\[ \psi_\tau m(t) = \int_{\tau}^{t}S(t-\tau-s)\left[{(\psi_\tau'm) + \psi_\tau m(1-m^{a-1}) - \chi\psi_\tau \nabla\cdot(f(m)\nabla c)}\right] (s)\, \,\textrm{d}s \]

where $\{S(t)\}$ is the semigroup of the heat kernel with homogeneous Neumann boundary condition. We have the following estimates of $\{S(t)\}$, (cf [Reference Winkler16]), for $p$ and $q$ well chosen,

\begin{align*} &\|S(t)f\|_{L^{\infty}(\Omega)} \leq C^{*}\left({1+t^{-\frac{n}{2p}}}\right)\|f\|_{L^{p}(\Omega)} \; \text{ and}\\ &\|S(t)\nabla f\|_{L^{\infty}(\Omega)} \leq C^{*}\left({1+t^{-\frac 12 - \frac{n}{2q}}}\right)\|f\|_{L^{q}(\Omega)}, \end{align*}

where $C^{*}$ only depends on $\Omega$, $n$, $p$ and $q$.

Using these estimates into the above Duhamel formula, we get

\begin{align*} &\sup_{t\in(\tau,\tau+2)} \|\psi_\tau m(t)\|_{L^{\infty}(\Omega)}\\ & \quad \leq 3 \int_\tau^{\tau+2}\|[S(t-\tau-s)m](s)\|_{L^{\infty}(\Omega)}\,\textrm{d}s \notag\\ &\qquad + \int_{\tau}^{\tau+2}\|[S(t-\tau-s)m^{a}](s)\|_{L^{\infty}(\Omega)}\,\textrm{d}s\\ & \qquad + \chi \int_{\tau}^{\tau+2}\|[S(t-\tau-s)\nabla \cdot(f(m)\nabla c)](s)\|_{L^{\infty}(\Omega)}\,\textrm{d}s\\ & \quad \leq 3C^{*}\, \|m\|_{L^{p}(Q_{\tau,\tau+2})}\int_{\tau}^{\tau+2}\left({1+(t-\tau-s)^{-\frac{n}{2p}}}\right)\,\textrm{d}s\\ & \qquad + C^{*}\,\|m\|_{L^{ap}(Q_{\tau,\tau+2})}^{a}\int_\tau^{\tau+2}\left({1+(t-\tau-s)^{-\frac{n}{2p}}}\right)\,\textrm{d}s\\ & \qquad + C^{*}\, \gamma\, \chi\,\|\nabla c\|_{L^{\infty}(Q_{\tau,\tau+2})}\|m\|_{L^{bp}(Q_{\tau,\tau+2})}^{b} \int_{\tau}^{\tau+2}\left({1+(t-\tau-s)^{- \frac 12 -\frac{n}{2p}}}\right)\,\textrm{d}s. \end{align*}

By applying lemma 2.9 and (2.74), we can choose $p>n$, which implies that the last three integrals converge and are bounded w.r.t $\tau$. We finally obtain the estimate (for a constant $C>0$ depending on the parameters stated in theorem 1.1)

\[ \sup_{t\in(\tau,\tau+2)}\|\psi_\tau m\|_{L^{\infty}(\Omega)} \leq C. \]

That is, (1.5) is proved. The result in (1.6) is obtained in a similar manner, by applying the earlier $\limsup$ estimates and parabolic regularity, concluding the proof of theorem 1.1.

3.1 Nonlinear stability

Proof of theorem 1.3. Thanks to theorem 1.1, we recall that there exist $K_m, K_c, K_d>0$ such that for all $t>0$,

(3.4)\begin{equation} \left\|{\tilde m(t)}\right\|_{L^{\infty}(\Omega)} \leq K_m, \quad \left\|{\tilde c(t)}\right\|_{W^{1,\infty}(\Omega)} \leq K_c, \quad \left\|{\tilde d(t)}\right\|_{L^{\infty}(\Omega)} \leq K_d. \end{equation}

It follows from the condition (1.7) that there exist $\alpha \in (0,1)$ and $\vartheta _1 > 0$ such that

(3.5)\begin{equation} \vartheta_1 < \frac{4\alpha(a-1)}{\beta^{2}} \quad \text{ and } \quad \chi < \frac{2\sqrt{\vartheta_1\varepsilon_0}}{f(1)}. \end{equation}

Now we choose

(3.6)\begin{equation} \vartheta_2 > \frac{\delta^{2} \vartheta_1}{4g(1)(1-\alpha)}, \end{equation}

and define a function

\[ \varphi(t) = \frac 12\left({ \|\tilde{m}\|_{L^{2}(\Omega)}^{2} + \vartheta_1\|\tilde{c}\|_{L^{2}(\Omega)}^{2} + \vartheta_2\|\tilde{d}\|_{L^{2}(\Omega)}^{2}}\right). \]

Direct calculations give

(3.7)\begin{align} \varphi'(t) & ={-}\left\|{\nabla \tilde{m}}\right\|_{L^{2}(\Omega)}^{2} + \chi f(1)\left\langle{\nabla \tilde m},{\nabla \tilde c}\right\rangle - \vartheta_1\varepsilon_0\left\|{\nabla \tilde c}\right\|_{L^{2}(\Omega)}^{2}\notag\\ & \quad + (1-a)\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{2} - \vartheta_1\left\|{\tilde c}\right\|_{L^{2}(\Omega)}^{2} - \vartheta_2 g(1)\|\tilde d\|_{L^{2}(\Omega)}^{2} + \beta \vartheta_1\left\langle{\tilde m},{\tilde c}\right\rangle\notag\\ &\quad + \vartheta_1\delta \left\langle{\tilde d},{\tilde c}\right\rangle + \left\langle{R_1(\tilde X)},{\tilde m}\right\rangle + \left\langle{R_3(\tilde X)},{\vartheta_2\tilde d}\right\rangle. \end{align}

From the relation between $\chi$ and $\vartheta _1$ in (3.5), there exists $\omega _1>0$ such that

\begin{align*} &-\left\|{\nabla \tilde{m}}\right\|_{L^{2}(\Omega)}^{2} + \chi f(1)\left\langle{\nabla \tilde m},{\nabla \tilde c}\right\rangle - \vartheta_1\varepsilon_0\left\|{\nabla \tilde c}\right\|_{L^{2}(\Omega)}^{2}\\ &\quad \leq{-}\omega_1\left({\left\|{\nabla \tilde m}\right\|_{L^{2}(\Omega)}^{2} + \left\|{\nabla \tilde c}\right\|_{L^{2}(\Omega)}^{2}}\right). \end{align*}

It also follows from the condition of $\vartheta _1$ in (3.5) that there exists $\omega _2>0$ satisfying

\[ (1-a)\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{2} + \beta \vartheta_1 \left\langle{\tilde m},{\tilde c}\right\rangle - \alpha \vartheta_1\left\|{\tilde c}\right\|_{L^{2}(\Omega)}^{2} \leq{-}\omega_2\left({\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{2} + \left\|{\tilde c}\right\|_{L^{2}(\Omega)}^{2}}\right) , \]

where $\alpha$ is chosen in (3.5). Finally, from (3.6), there exists $\omega _3>0$ such that

\[ -(1-\alpha)\vartheta_1\left\|{\tilde c}\right\|_{L^{2}(\Omega)}^{2} + \vartheta_1\delta \left\langle{\tilde d},{\tilde c}\right\rangle - \vartheta_2g(1)\|\tilde d\|_{L^{2}(\Omega)}^{2} \leq{-}\omega_3\left({\|\tilde c\|_{L^{2}(\Omega)}^{2} + \|\tilde d\|_{L^{2}(\Omega)}^{2}}\right). \]

Note that these three estimates mean that the linear part of (3.1), after the rescaling by $\mathrm {diag}(1,\vartheta _1, \vartheta _2)$, has a negative spectral gap. Inserting all these estimates into (3.7), we obtain for some $\omega _4,\, \omega _5 > 0$ that

(3.8)\begin{align} \varphi'(t) \leq{-}\omega_4\left({\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2} + \left\|{\tilde c}\right\|_{H^{1}(\Omega)}^{2}}\right) - \omega_5\varphi(t) + \left\langle{R_1(\tilde X)},{\tilde m}\right\rangle + \left\langle{R_3(\tilde X)},{\vartheta_2 \tilde d}\right\rangle. \end{align}

To deal with the two last terms on the right-hand side of (3.8), we first use integration by parts to have

(3.9)\begin{align} & \left\langle{R_1(\tilde X)},{\tilde m}\right\rangle + \left\langle{R_3(\tilde X)},{\vartheta_2 \tilde d}\right\rangle\notag\\ & \quad ={-}\int_{\Omega}\tilde{m}\left[{(1+\tilde m)^{a} - 1 - a\tilde m}\right]\,\textrm{d}x + \chi \int_{\Omega}\left[{f(1+\tilde m) - f(1)}\right]\nabla \tilde c\cdot \nabla \tilde m \,\textrm{d}x\notag\\ & \qquad - \vartheta_2\int_{\Omega}\left[{g(1+\tilde m)-g(1)}\right]\tilde{d}^{2}\,\textrm{d}x =: (A) + (B) + (C). \end{align}

Now we fix a constant $\kappa = \kappa (n) \in (0,\min \{a-1,1\})$ small enough such that $H^{1}(\Omega )\hookrightarrow L^{2+2\kappa }(\Omega )$. Thanks to this embedding and interpolation inequality, we have (for some $C(\Omega ,\kappa )$ which only depends on $\Omega$ and $\kappa$)

(3.10)\begin{equation} \left\|{\tilde m}\right\|_{L^{2+\kappa}(\Omega)}^{2+\kappa} \leq \left\|{\tilde m}\right\|_{L^{2+2\kappa}(\Omega)}^{2}\left\|{\tilde m}\right\|_{L^{1+\kappa}(\Omega)}^{\kappa} \leq C(\Omega,\kappa)\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2}\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{\kappa}. \end{equation}

To estimate $(A)$, we use Taylor's expansion and the fact that $\tilde m$ is uniformly bounded in $L^{\infty }(\Omega )$ to get

(3.11)\begin{equation} \begin{aligned} |(A)| & \leq\int_{\Omega}|\tilde{m}| |(1+\tilde{m})^{a} - 1 - a\,\tilde{m}|\,\textrm{d}x\\ & \leq C(a, K_m)\int_{\Omega}|\tilde m|^{1+a}\,\textrm{d}x\\ & \leq C(a,K_m)K_m^{a-1-\kappa}\int_{\Omega}|\tilde m|^{2+\kappa}\,\textrm{d}x \quad (\text{since }\kappa < a-1)\\ & \leq C(a,K_m,\kappa)\left\|{\tilde m}\right\|_{L^{2+\kappa}(\Omega)}^{2+\kappa}\\ & \leq C(a,K_m,\kappa)\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2}\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{\kappa} \quad (\text{using } (3.10)). \end{aligned} \end{equation}

Similarly for $(B)$ and $(C)$

(3.12)\begin{equation} \begin{aligned} |(C)| & \leq \vartheta_2\int_{\Omega} |g(1 +\tilde{m}) - g(1)|\tilde{d}^{2}\,\textrm{d}x \le C(K_m) \int_{\Omega}|\tilde{m}| |\tilde{d}|^{2}\,\textrm{d}x \quad (\text{since }g\in C^{1,0})\\ & \leq C(K_m, K_d)\int_{\Omega}|\tilde m||\tilde d|^{\frac{2(1+\kappa)}{2+\kappa}}\,\textrm{d}x\\ & \leq C(K_m,K_d,\omega_5)\int_{\Omega}|\tilde m|^{2+\kappa}\,\textrm{d}x + \frac{\omega_5\vartheta_2}{4}\int_{\Omega}|\tilde d|^{2}\,\textrm{d}x\quad (\text{by Young's inequality})\\ & \leq C(K_m,K_d,\omega_5,\kappa)\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2}\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{\kappa} + \frac{\omega_5}{2}\varphi(t) \quad (\text{using } (3.10)), \end{aligned} \end{equation}

and

(3.13)\begin{align} |(B)| & \leq \chi \int_{\Omega}|f(1+\tilde{m}) - f(1)||\nabla \tilde{c}||\nabla \tilde{m}|\,\textrm{d}x\notag\\ & \le C(K_m) \int_{\Omega}|\tilde{m}| |\nabla \tilde{c}||\nabla \tilde{m}|\,\textrm{d}x \quad (\text{since }f\in C^{1,0})\notag\\ & \leq C(K_m, K_c)\int_{\Omega}|\tilde m||\nabla \tilde c|^{\frac 13}|\nabla \tilde m|\,\textrm{d}x \quad (\text{thanks to } (3.4))\notag\\ & \leq \frac{\omega_4}{2}\int_{\Omega}\left({|\nabla \tilde m|^{2} + |\nabla \tilde c|^{2}}\right)\,\textrm{d}x + C(K_m,K_c,\omega_4)\int_{\Omega}|\tilde m|^{3}\,\textrm{d}x \notag\\ &\qquad (\text{by Young's inequality})\notag\\ & \leq \frac{\omega_4}{2}\left({\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2} + \left\|{\tilde c}\right\|_{H^{1}(\Omega)}^{2}}\right) + C(K_m,K_c,\omega_4)K_m^{1-\kappa}\int_{\Omega}|\tilde m|^{2+\kappa}\,\textrm{d}x\notag\\ & \leq \frac{\omega_4}{2}\left({\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2} + \left\|{\tilde c}\right\|_{H^{1}(\Omega)}^{2}}\right) + C(K_m,K_c,\omega_4,\kappa)\left\|{\tilde m}\right\|_{H^{1}(\Omega)}^{2}\left\|{\tilde m}\right\|_{L^{2}(\Omega)}^{\kappa} \notag\\ &\qquad (\text{using } (3.10)). \end{align}

By inserting (3.11)–(3.12)–(3.13) into (3.8) we obtain

\begin{align*} \varphi'(t) &\leq{-} \frac{\omega_5}{2}\varphi(t) + \left({\|\tilde m\|_{H^{1}(\Omega)}^{2} + \|\tilde c\|_{H^{1}(\Omega)}^{2}}\right)\left[{C\|\tilde m\|_{L^{2}(\Omega)}^{\kappa} - \frac{\omega_4}{2}}\right]\\ &\leq{-} \frac{\omega_5}{2}\varphi(t) + \left({\|\tilde m\|_{H^{1}(\Omega)}^{2} + \|\tilde c\|_{H^{1}(\Omega)}^{2}}\right)\left[{C\varphi(t)^{\kappa/2} - \frac{\omega_4}{2}}\right] . \end{align*}

Therefore, if $\varphi (0)$ is small enough, $t \mapsto \varphi (t)$ is decreasing and consequently

\[ \varphi(t) \leq e^{-\frac{\omega_5}{2}t}\varphi(0) \]

which proves the nonlinear (exponential) stability in $L^{2}$ of $(\bar {m}, \bar {c}, \bar {d})$. Interpolating with the estimate (1.5), we conclude the proof of theorem 1.3.

3.2 Turing instability and Turing patterns

When $\chi > \chi _c$, the equilibrium point $(1, \beta + \delta , 1)$ is linearly unstable, so that a Turing-type instability sets in. In this section, we display a numerical simulation showing the occurrence of this instability on the spatial domain $\Omega =[0;L_x]\times [0;L_y]$.

First, we recall that on rectangular domains defined by $0< x< L_x$ and $0< y< L_y$, the Fourier transformation of solutions to the linearized problem with homogeneous Neumann boundary conditions has the following form:

(3.14)\begin{equation} \tilde{X}=\sum_{p,q \in \mathbb{N} } \textbf{f}_{p,q} e^{\sigma(\lambda_n) t} \cos\left(\frac{p \pi}{L_x}x\right)\cos\left(\frac{q \pi}{L_y}y\right), \end{equation}

where $\textbf {f}_{p,q}$ are the Fourier coefficients of the initial conditions, and $\lambda _n$ are defined as follows:

(3.15)\begin{equation} \lambda_n=\left(\frac{p \pi}{L_x}\right)^{2}+\left(\frac{q \pi}{L_y}\right)^{2}, \text{ with } p, \, q \in \mathbb{N}, \end{equation}

in such a way that they are ordered (that is $\lambda _n \leq \lambda _{n+1}$). The values $\sigma (\lambda _n)$ are derived from the dispersion relations $\sigma ^{2} - \mathrm {Tr} N_n \sigma + \mathrm {Det} N_n = 0$. When $\chi <\chi _c, \, \sigma (\lambda _n)<0, \, \forall n \in \mathbb {N}$, and then the homogeneous state is linearly stable. For $\chi =\chi _c$, the homogeneous solution is marginally stable and $\exists \lambda _{n_{c}}$ such that $\min (\mathrm {Det} N_{n_{c}})=0$, or in other words $\sigma (\lambda _{n_{c}})=0$. For $\chi >\chi _c$, there exists a range $[n_1; n_2]$ such that $\sigma (\lambda _n) > 0$, $\forall n \in [n_1; n_2]$, and therefore a Turing instability occurs.

The numerical simulation has been performed on the square domain $[0;\pi ]\times [0;\pi ]$, i.e. $L_x = L_y = \pi$. For the discretization in space, we have adopted a Fourier spectral scheme with $64 \times 64$ modes. The integration in time has been realized by using the Crank-Nicholson method for the diffusive part and a second-order Runge-Kutta explicit method for the reaction terms.

We have adopted the functions

\[ f(m) = \frac{m}{1+m} \quad \text{ and } \quad g(m) = \frac{rm^{2}}{1+m}, \]

which corresponds to the special case (1.2) (except for parameter $a$), and the following parameter values have been fixed: $r=1, \, a=3, \,\varepsilon _0=0.03125, \, \delta =1, \, \beta =1$. For this set of values, linear stability analysis predicts that $\lambda _{n_{c}}=8$ and $\chi _c=3.125$. In order to obtain a Turing pattern, we have fixed $\chi =3.18$ (which is bigger than $\chi _c$) and for this choice the only unstable $\lambda _n$ allowed by the domain and the boundary conditions is $\lambda _{n_{c}}$, which has multiplicity one, in that $(p, q)=(2,2)$ is the only pair which satisfies the condition (3.15).

We therefore expect that a random perturbation of the homogeneous state evolves towards a stationary Turing pattern of the form $\tilde {X} \propto \cos (2 x)\cos (2 y)$.

The temporal evolution of the numerical solution is shown in Fig. 1. As predicted by the previous analysis, a small random perturbation (see Fig. 1a) of the homogeneous state evolves towards a stationary square patten (Fig. 1d).

Figure 1.

Spatio-temporal evolution of $m$ on the spatial domain $\Omega = [0;\pi ]\times [0;\pi ]$ above the Turing threshold $\chi _c=3.125$. The following parameter values have been fixed: $r=1, \, a=3, \,\varepsilon _0=0.03125, \, \delta =1, \, \beta =1$ and $\chi =3.18.$ The initial condition is a small random perturbation of the homogeneous equilibrium $\bar m = 1$.

On the other hand, when $\chi <\chi _c$, the homogeneous steady state is stable and therefore perturbations of this equilibrium, as shown in Fig. 2, are dampened.

Figure 2.

Spatio-temporal evolution of $m$ on the spatial domain $\Omega = [0;\pi ]\times [0;\pi ]$ below the Turing threshold $\chi _c$. All parameter values, except $\chi$, are as in Fig. 1, and $\chi =3$. The initial condition is a random perturbation of the homogeneous equilibrium $\bar m = 1$.

3.3 Global asymptotic stability

We expect that the convergence to the steady state $(\bar m, \bar c, \bar d) = (1,\beta + \delta , 1)$ is not only local, as proved in § 3.1 when $\chi < \chi _{\mathrm {subcrit}}$, but is in fact global and valid for $\chi < \chi _c$. This is left as an interesting open question. We performed numerical simulations which support this conjecture. We show one of these simulations in Fig. 3: below the Turing threshold $\chi _c$, an initial datum far from the homogeneous equilibrium evolves towards the stable equilibrium point $(\bar m, \bar c, \bar d)$.

Figure 3.

Spatio-temporal evolution of $m$ on the spatial domain $\Omega = [0;\pi ]\times [0;\pi ]$ below the Turing threshold $\chi _c$. All parameter values are as in Fig. 2. The initial condition is far from the homogeneous equilibrium $(\bar m, \bar c, \bar d)$.

Moreover we performed several other simulations adopting initial data further away from equilibrium and in any of these numerical experiments we observed solutions converging to the stable steady state.

Note that the large time behaviour of system (1.1) when $\chi > \chi _c$ is a much more complicated problem: it involves the research of inhomogeneous stable steady states, and the study of the basin of attraction.