Generalised solution to a 2D parabolic-parabolic chemotaxis system for urban crime: Global existence and large-time behaviour

Abstract

We consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter $\chi =2$. The available analytical findings for this urban crime model including $\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any $\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.

1. Introduction and main results

This paper is concerned with a class of chemotaxis systems with singular sensitivity of the following form

(1) \begin{equation} \left \{ \begin{split} &u_t= \Delta u-\chi \nabla \cdot \left (u\nabla \ln v\right )-\kappa uv+h_1,&\ x\in \Omega,\ t\gt 0, \\ &v_t=\Delta v-v+ uv+h_2, &\ x\in \Omega,\ t\gt 0, \end{split} \right . \end{equation}

in a bounded and smooth spatial domain $\Omega \subset \mathbb{R}^n$ , with the parameters $\chi \gt 0$ and $\kappa \gt 0$ , which is proposed in [33] to describe the propagation of urban criminal activities with the particular value $\chi =2$ . Here, $u(x,t)$ and $v(x,t)$ represent the density of criminals and an abstract so-called attractiveness value at location $x$ and time $t$ , respectively; the given source function $h_1$ denotes the introduction of criminal agents into the system, and the given function $h_2$ describes the density of additional criminals, which may exist even in the absence of any criminal agents. We refer to [32, 34] for more details on (1), to [3, 5, 13, 26, 35, 38, 49] for further developments of crime models and to [4, 9] for more comprehensive reviews of related works.

To elucidate our motivation, we first recall some analytical progress related to the system (1). For the classical solvability of the initial-boundary value problem, the local existence and uniqueness of solutions has been obtained in [28] and the global existence of solutions has been established provided that either $n=1$ [29, 41] or $n\geq 2$ with the additional condition that $\chi \lt \frac 2n$ [11, 31], or the initial data and the given functions $h_1$ and $h_2$ are assumed to be small [1, 37]. It has been shown that these restrictions can be relaxed to $\chi \gt 0$ when $n=2$ [44] or to $\chi \in (0,\sqrt{3})$ when $n=3$ [15], in the sense of radial renormalised solvability. In addition, considering interacting individuals may have attempted to avoid competition, the model corresponding to (1) with the linear criminal diffusion (i.e. $\Delta u$ ) replaced by the nonlinear criminal diffusion (i.e. $\nabla \cdot \left (u^{m-1}\nabla u\right )$ with some $m\gt 0$ ) also has been studied by researchers. For this nonlinear model, it is known that its two-dimensional initial-boundary value problem admits a global bounded weak solution provided that either $m\gt \frac 32$ [30] or $m\gt 1$ and $\chi \lt \frac{\sqrt{3}}{2}$ [47]. On the other hand, if the logistic source term, that is, $au-bu^\alpha$ with $a\in \mathbb{R}$ and $b\gt 0$ , is incorporated into (1), then, the resulting system has a global generalised solution for $n=2$ and $\alpha =2$ [14], which is eventually smooth ( [27]), and possesses a global classical solution for either $n=2,3$ and $\alpha \gt 2$ [14, 40] or $n\geq 4$ and $\alpha \gt \frac{n}{4}+1$ [40]. When $\kappa =0$ , $h_1\equiv 0$ , $h_2\equiv 0$ and $u_t$ is replaced by $\tau u_t$ with $\tau \in (0,1)$ , the model (1) arrives at a reduced crime model considered in [12], where a statement on spontaneous emergence of arbitrarily large values of $\|u(\cdot,t)\|_{L^q}$ with $q\gt \frac n2$ has been derived for $\chi \gt 0$ and $n\geq 3$ . As to long-time behaviours of solutions of (1), for $\chi \lt \frac 2n$ with $n\geq 2$ the asymptotic stability of constant steady states has been considered in [31], provided that $h_1\equiv \textit{const.}$ and $h_2\equiv \textit{const.}$ with certain smallness; under the assumptions that $h_1\rightarrow 0$ and $h_2\rightarrow h_{2,\infty }$ in some sense as $t\rightarrow \infty$ , the convergence results $(u,v)\rightarrow (0,v_\infty )$ have been studied in some appropriate senses in [1, 15, 29, 37, 44], where $v_\infty$ denotes the solution to the boundary value problem

(2) \begin{equation} \left \{ \begin{split} &-\Delta v_\infty +v_\infty =h_{2,\infty },&x\in \Omega,\\ &\nabla v_\infty \cdot \nu =0,&x\in \partial \Omega . \end{split} \right . \end{equation}

Furthermore, we refer to [6, 8, 16, 23–25, 32, 39] for the mathematical analytical work on related stationary problems, which reveal the possibility of stably spatial heterogeneous behaviour resembling crime hotspot formation and support that the system (1) is adequate to describe the formation of crime hotspots encountered in reality.

Compared the analytical results on (1) mentioned above, we find that the range of the parameter $\chi$ which guarantees the global existence of classical solutions of the system (1) seems to become larger when either the spatial domain is one or in multidimensional settings, the initial data and the source functions satisfy some smallness or radial symmetric assumption, or the solution concept is considered under proper generalised frameworks. Thus, we are wondering whether or not there exists an appropriate generalised framework within which for any $\chi \gt 0$ the corresponding $n(\geq 2)$ -dimensional initial-boundary value problem of (1) is solvable without imposing these additional conditions on both the initial data and the source functions $h_1$ and $h_2$ .

Motivated by this, the first purpose of the present work is to present that for arbitrary $\chi \gt 0$ , the following initial-boundary value problem:

(3) \begin{equation} \left \{ \begin{split} &u_t= \Delta u-\chi \nabla \cdot \left (u\nabla \ln v\right )-\kappa uv+h_1,&\ x\in \Omega,\ t\gt 0, \\ &v_t=\Delta v-v+uv+h_2, &\ x\in \Omega,\ t\gt 0,\\ &\nabla u\cdot \nu =\nabla v\cdot \nu =0,&\ x\in \partial \Omega,\ t\gt 0,\\ &u(x,0)=u_0(x),\ v(x,0)=v_0(x),&\ x\in \Omega, \end{split} \right . \end{equation}

where $\nu$ denotes the exterior normal vector to the boundary $\partial \Omega$ , possesses solutions in an appropriate generalised framework in the two-dimensional setting, without imposing any assumption of smallness and radial symmetry of both the initial data and the source functions $h_1$ and $h_2$ .

Before going further, we shall be precise about the notion of generalised solutions to the initial-boundary value problem (3) considered in this work.

Definition 1. A pair $(u,v)$ is called a global generalised solution to the initial-boundary value problem (3) if for any $T\gt 0$ ,

(1) it holds that

(4) \begin{equation} \left \{ \begin{split} &u\in L^1(\Omega \times (0,T)),\,\,\,\nabla \ln (1+u)\in L^2(\Omega \times (0,T)),\\ &v\in L^1(\Omega \times (0,T)),\,\,\,\nabla \ln v\in L^2(\Omega \times (0,T)),\\ &uv\in L^1(\Omega \times (0,T)),\\ &u(x,t)\geq 0,\,\,\, v(x,t)\gt 0,\,\,\, a.e.\,\, \mathrm{in}\,\,\Omega \times [0,T]; \end{split} \right . \end{equation}

(2) it holds that

(5) \begin{align} &\int _\Omega u(\cdot,t)dx+\kappa \int _0^t\int _\Omega uvdxds\leq \int _\Omega u_0dx+\int _0^t\int _\Omega h_1dxds,\,\,\, a.e.\,\, \mathrm{in}\,\, [0,T], \end{align}

(6) \begin{align} &\int _\Omega (u+\kappa v)(\cdot,t)dx+\kappa \int _0^t\int _\Omega vdxds\leq \int _\Omega u_0+\kappa v_0dx+\int _0^t\int _\Omega h_1+\kappa h_2dxds,\,\,\, a.e.\,\, \mathrm{in}\,\, [0,T]; \end{align}

(3) it holds that for some $p,q\gt 0$ and each non-negative $\varphi (x,t)\in C_0^\infty (\overline{\Omega }\times [0,T))$

(7) \begin{align} &-\int _\Omega (u_0+1)^{-p}v_0^{-q}\varphi |_{t=0} dx-\int _0^T\int _\Omega (u+1)^{-p}v^{-q}\varphi _tdxdt \nonumber \\ & \quad\le -p(p+1)\int _0^T\int _\Omega (u+1)^{-p-2}v^{-q}\varphi |\nabla u|^2dxdt\nonumber \\ &\qquad+\int _0^T\int _\Omega \left (p(p+1)\chi \frac{u}{u+1}-2pq\right )(u+1)^{-p-1}v^{-q-1}\varphi \nabla u\cdot \nabla v dxdt\nonumber \\ &\qquad+\int _0^T\int _\Omega \left (\chi pq\frac{u}{u+1}-q(q+1)\right )(u+1)^{-p}v^{-q-2}\varphi |\nabla v|^2dxdt\nonumber \\ &\qquad-p\int _0^T\int _\Omega (u+1)^{-p-1}v^{-q}\left (-\kappa uv+h_1\right )\varphi dxdt-q\int _0^T\int _\Omega (u+1)^{-p}v^{-q-1}\left (- v+u v+h_2 \right )\varphi dxdt\nonumber \\ &\qquad+p\int _0^T\int _\Omega (u+1)^{-p-1}v^{-q}\nabla u \cdot \nabla \varphi dxdt-p\chi \int _0^T\int _\Omega (u+1)^{-p-1}v^{-q-1}u\nabla v\cdot \nabla \varphi dxdt\nonumber \\ &\qquad+q\int _0^T\int _\Omega (u+1)^{-p}v^{-q-1}\nabla v\cdot \nabla \varphi dxdt; \end{align}

(4) it holds that for each non-negative $\varphi (x,t)\in C_0^\infty (\overline{\Omega }\times [0,T))$

(8) \begin{align} \int _0^T\int _{\Omega }\left (-\ln v\varphi _t-\frac{|\nabla v|^2\varphi }{v^2}+\frac{\nabla v\cdot \nabla \varphi }{v}+\varphi -u\varphi -\frac{\varphi h_2}{v}\right ) dxdt\geq \int _\Omega \ln v_0\varphi |_{t=0}dx. \end{align}

We here give a remark on the constants $p$ and $q$ appeared in (7).

Remark 1. For given $\chi \gt 0$ , the admissible $(p,q)$ is that $p\geq 1$ and $q\geq 2$ , fulfilling that

(9) \begin{align} \frac{p(p+1)\chi ^2}{4}-q-\frac{q^2}{p+1}\lt 0, \end{align}

in our subsequent analyses (see Lemma 3.6 below).

To state the first result on the global existence of such generalised solutions, the initial data $(u_0,v_0)$ are throughout assumed to satisfy

(10) \begin{equation} \left \{ \begin{split} &u_0\in C^0(\overline{\Omega })\,\,\,\, \mathrm{with}\,\,\,\, u_0\geq 0\,\,\,\,\mathrm{and}\,\,\,\,u_0\not \equiv 0,\\ &v_0\in W^{1,\infty }(\overline{\Omega })\,\,\,\, \mathrm{with}\,\,\,\, \inf \limits _{x\in \overline{\Omega }}v_0\gt 0, \end{split} \right . \end{equation}

and the source functions $h_1$ and $h_2$ are supposed to fulfil

(11) \begin{align} 0\leq h_i\in C^1(\overline{\Omega }\times [0,\infty ))\cap L^\infty (\Omega \times (0,\infty )),\quad i=1,2. \end{align}

With Definition 1 and the assumptions (10)–(11) at hand, the first result reads as follows.

Theorem 1.1. Let $\Omega \subset \mathbb{R}^2$ be a bounded and smooth spatial domain, and (10)–(11) hold. Then, for any $\chi \gt 0$ , the initial-boundary value problem (3) possesses at least one global generalised solution in the sense of Definition 1.

Remark 2. By a slight adaptation of [45, Lemma 2.1], we can show that if

\begin{equation*}u\in C(\overline {\Omega }\times [0,\infty ))\cap C^{2,1}(\overline {\Omega }\times (0,\infty ))\quad \mathrm {and}\quad v\in C(\overline {\Omega }\times [0,\infty ))\cap C^{2,1}(\overline {\Omega }\times (0,\infty )) \end{equation*}

such that $u\geq 0$ and $v\gt 0$ in $\overline{\Omega }\times (0,T)$ and $(u,v)$ is a solution of (3) in the sense of Definition 1, then $(u,v)$ also solves (3) in the classical sense. This also supports the interest in our concept of generalised solutions.

Going beyond the global existence statement, we naturally focus on the large-time behaviour of generalised solutions. To achieve it, we additionally assume that

(12) \begin{align} \inf _{t\gt 0}\int _\Omega h_2(x,t)dx\gt 0, \end{align}

(13) \begin{align} \int _t^{t+1}\int _\Omega h_1(\cdot,s)dxds\rightarrow 0,\,\,\, \mathrm{as}\,\, t\rightarrow \infty, \end{align}

(14) \begin{align} \int _t^{t+1}\int _\Omega |h_2(\cdot,s)-h_{2,\infty }(\!\cdot\!)|dxds\rightarrow 0,\,\,\, \mathrm{as}\,\, t\rightarrow \infty \end{align}

with some $h_{2,\infty }\in C^1(\overline{\Omega })$ . The second result on the asymptotic behaviour of the generalised solution established in Theorem 1.1 can be stated as follows.

Theorem 1.2. Let all assumptions in Theorem 1.1 be satisfied, and let (12)–(14) be fulfilled. Under the additional assumption that $\Omega$ is convex, for the global generalised solution of the initial-boundary value problem (3) taken from Theorem 1.1, there exists a null set $\mathcal{N}\subset (0,\infty )$ such that

(15) \begin{align} \int _\Omega u(\cdot,t)dx+\int _t^{t+1}\int _\Omega |v(\cdot,s)-v_\infty (\!\cdot\!)|dxds\rightarrow 0,\quad \mathrm{as}\,\,\,(0,\infty )\setminus \mathcal{N}\ni t\rightarrow \infty, \end{align}

where $v_\infty$ denotes the solution of the boundary value problem (2).

Remark 3. In comparison with [20, Theorem 1.2] and [22, Theorem 1.3], where the long-time behaviour of generalised solution to the system that with $u$ instead of $uv$ in the second equation of (3) was obtained in two-dimensional setting and higher dimensional settings, respectively, our result in Theorem 1.2 is weaker due to the presence of the nonlinear production $+uv$ ; especially, we do not know how to prove the eventual smoothness of the generalised solution established in Theorem 1.1.

1.1. Technical strategy and structure of the article

The first objective of this paper is to present that the initial-boundary value problem (3) possesses a global generalised solution. Usually, to this end, one should seek an appropriate generalised framework and thereby obtain the global existence of generalised solutions via an approximation procedure. Here, our novelty of analysis consists of structuring an appropriate generalised framework, in which the difficulty is to define the solution component $v$ adequately. Although our definition of the solution component $u$ is inspired by the generalised framework introduced in [18, 48] for the logarithmic Keller-Segel system with linear production, our definition of the solution component $v$ is completely different from that in [18, 48] due to the presence of the nonlinear production $+uv$ in the second equation in (3), or, more precisely, we structure the generalised definition of the solution component $v$ by respectively defining the generalised subsolution and supersolution, see (6) and (8) in Definition 1. After this, to get the generalised solution via an appropriate approximation procedure, the key steps are to establish a series of uniform a priori estimates, see Lemmas 3.1 and 3.2.

The second objective of this paper is to perform the large-time behaviour of the generalised solution $(u,v)$ determined in Theorem 1.1, under the additional assumptions (12)–(14). To achieve this, we start to present that for any $\varepsilon \in (0,1)$

(16) \begin{align} \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon (\cdot,s) dxds \rightarrow 0\quad \mathrm{as}\,\,\, t\rightarrow \infty \quad \mathrm{uniformly\,\, in}\,\,\, \varepsilon, \end{align}

see Lemma 4.4. Taking advantage of this and an appropriate approximation procedure invoking the Beppo Levi theorem, for any $t\gt 0$ we get the key estimate:

\begin{align*} &\int _\Omega |v_\varepsilon -v_\infty |(\cdot,t+1)dx-\int _\Omega |v_\varepsilon -v_\infty |(\cdot,t)dx+\int _t^{t+1}\int _\Omega |v_\varepsilon -v_\infty | dxds\\ & \quad\leq \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+1}\int _\Omega |h_2-h_{2,\infty }|dxds, \end{align*}

where $v_\infty$ denotes the solution of the boundary value problem (2), see the proof of Lemma 4.5 for details. In terms of it, setting $z_\varepsilon (t)\;:\!=\;\int _t^{t+1}\int _\Omega |v_\varepsilon -v_\infty | dxds$ we have

\begin{equation*} z'_\varepsilon (t)+z_\varepsilon (t)\leq \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+1}\int _\Omega |h_2-h_{2,\infty }|dxds, \end{equation*}

by which an application of the ODE techniques (see Lemma 4.2) invoking (16) presents the desired decay in Theorem 1.2.

The rest of this paper is arranged as follows. In the following section, we present the global well-posedness for the approximate problem (17). In Section 3, the global existence of generalised solutions to the initial-boundary value problem (3) is established. Section 4 is devoted to showing the large-time behaviour desired in Theorem 1.2 via an appropriate approximation procedure invoking the Beppo Levi theorem.

2. Preliminaries

To construct a generalised solution of the initial-boundary value problem (3) by an approximation procedure, for each $\varepsilon \in (0, 1)$ we shall consider the following approximate problem

(17) \begin{equation} \left \{ \begin{split} &u_{\varepsilon t}=\Delta u_{\varepsilon }-\chi \nabla \cdot \left (u_{\varepsilon }\nabla \ln v_{\varepsilon }\right )-\kappa u_\varepsilon v_\varepsilon +h_1, & x \in \Omega, \,t\gt 0,\\ &v_{\varepsilon t}=\Delta v_{\varepsilon }-v_\varepsilon +\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2, & x \in \Omega,\, t\gt 0,\\ &\frac{\partial u_{\varepsilon }}{\partial \nu }=\frac{\partial v_{\varepsilon }}{\partial \nu }=0, & x \in \partial \Omega, \,t\gt 0,\\ &u_{\varepsilon }(x, 0)=u_{0}(x), \quad v_{\varepsilon }(x, 0)=v_{0}(x), & x \in \Omega . \end{split} \right . \end{equation}

An application of the well-known strategy harnessing the contraction mapping principle and the well-known pointwise positivity property of the Neumann heat semigroup, as in [2, 11, 28, 43], ensures the global existence of classical solution to the approximate problems (17).

Lemma 2.1. Let the assumptions (10)–(11) hold. For each $\varepsilon \in (0,1)$ and any $\chi \gt 0$ , there exists a unique pair $(u_\varepsilon,v_\varepsilon )$ of positive functions with the properties that for any $T\gt 0$

\begin{equation*} \left \{ \begin {split} &u_\varepsilon \in C^0\big (\overline {\Omega }\times [0,T]\big )\cap C^{2,1}\big (\overline {\Omega }\times (0,T]\big ),\\ &v_\varepsilon \in \bigcap \limits _{p\gt 2}C^0\big ([0,T];\; W^{1,p}(\overline {\Omega })\big )\cap C^{2,1}\big (\overline {\Omega }\times (0,T]\big ), \end {split} \right . \end{equation*}

such that $(u_\varepsilon,v_\varepsilon )$ solves the approximate problem (17) classically in $\Omega \times [0,\infty )$ . Moreover, we have

(18) \begin{align} v_\varepsilon (\cdot, t)\geq e^{-t}\inf _{x \in \overline{\Omega }} v_{0}(x),\quad t\gt 0 \end{align}

and

(19) \begin{align} \|u_\varepsilon (\cdot,t)\|_{L^1}+ \int _0^t\int _\Omega u_\varepsilon v_\varepsilon (\cdot,s)dxds\leq C(1+t), \quad t\gt 0, \end{align}

(20) \begin{align} \|v_\varepsilon (\cdot,t)\|_{L^1}+ \int _0^t\int _\Omega v_\varepsilon (\cdot,s)dxds \leq C(1+t), \quad t\gt 0 \end{align}

for some $C\gt 0$ , independent of $\varepsilon$ .

Proof. Similar to [2, 11, 28, 43], the well-known strategy invoking the contraction mapping principle presents that for each $\varepsilon \in (0,1)$ and any $\chi \gt 0$ , there exist a time $T_{\max,\varepsilon }\in (0,\infty ]$ and a unique pair $(u_\varepsilon,v_\varepsilon )$ of positive functions with the properties that for any $p\gt 2$

\begin{equation*} \left \{ \begin {split} &u_\varepsilon \in C^0\big (\overline {\Omega }\times [0,T_{\max,\varepsilon })\big )\cap C^{2,1}\big (\overline {\Omega }\times (0,T_{\max,\varepsilon })\big ),\\ &v_\varepsilon \in C^0\big ([0,T_{\max,\varepsilon });\; W^{1,p}(\overline {\Omega })\big )\cap C^{2,1}\big (\overline {\Omega }\times (0,T_{\max,\varepsilon })\big ) \end {split} \right . \end{equation*}

such that $(u_\varepsilon,v_\varepsilon )$ solves the approximate problem (17) classically in $\Omega \times [0,T_{\max,\varepsilon })$ . Moreover, if $T_{\max,\varepsilon }\lt \infty$ , then for any $p\gt 2$

(21) \begin{align} \limsup \limits _{t\rightarrow T_{\max,\varepsilon }}\big (\|u_\varepsilon (\cdot,t)\|_{L^\infty }+\|\nabla v_\varepsilon (\cdot,t)\|_{L^p}+\|v_\varepsilon ^{-1}(\cdot,t)\|_{L^\infty }\big )=\infty . \end{align}

To show that $T_{\max,\varepsilon }=\infty$ , let us start with the pointwise lower bound for the solution component $v_\varepsilon$ . Indeed, we can apply the comparison principle for the Neumann problem associated with the heat equation to the variation-of-constants formula for $v_\varepsilon$ , namely

(22) \begin{align} v_\varepsilon (\cdot, t) &=e^{t(\Delta -1)} v_{0}+\int _{0}^{t} e^{(t-s)(\Delta -1)}\left (\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+ h_2\right )(\cdot, s) d s, \end{align}

and get

(23) \begin{equation} v_\varepsilon (\cdot, t)\geq e^{t(\Delta -1)} v_{0} \geq e^{-t}\inf _{x \in \Omega } v_{0}(x),\quad t\in (0,T_{\max,\varepsilon }), \end{equation}

due to the facts that $h_2\geq 0$ and $u_\varepsilon, v_\varepsilon \gt 0$ . We can also apply the properties of the Neumann heat semigroup (cf. [42, Lemma 1.3], [7, Lemma 2.1]) to (22) and get that for any $q\gt 2$

\begin{align*} \|\nabla v_\varepsilon (\cdot,t)\|_{L^q}\le &C\|\nabla v_0\|_{L^{q}}+C\int _0^t\left (1+(t-s)^{-\frac 12-(\frac 12-\frac 1q)}\right )e^{-(t-s)}\left \|\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon } +h_2\right \|_{L^2}ds. \end{align*}

Since $\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }\leq \varepsilon ^{-1}$ and $-\frac 12-(\frac 12-\frac 1q)=-1+\frac 1q\gt -1$ , it follows from (11) that for any $q\gt 2$

(24) \begin{align} \|\nabla v_\varepsilon (\cdot,t)\|_{L^q} \le &C \|\nabla v_0\|_{L^{q}}+C_\varepsilon,\quad t\in (0,T_{\max,\varepsilon }). \end{align}

We now establish the bound of $\|u_\varepsilon \|_{L^1}$ by integrating the first equation in (17) over $\Omega$

(25) \begin{align} \frac{d}{dt}\int _\Omega u_\varepsilon dx+\kappa \int _\Omega u_\varepsilon v_\varepsilon dx=\int _\Omega h_1dx, \end{align}

which, integrating over $[0,t]$ , implies that

(26) \begin{align} \|u_\varepsilon (\cdot,t)\|_{L^1}+\kappa \int _0^t \int _\Omega u_\varepsilon v_\varepsilon dx ds\leq \|u_0\|_{L^1}+\|h_1\|_{L^\infty (\Omega \times (0,\infty ))}t,\quad t\in (0,T_{\max,\varepsilon }). \end{align}

We next estimate $\|u_\varepsilon \|_{L^\infty }$ by applying the properties of the Neumann heat semigroup (cf. [42, Lemma 1.3], [7, Lemma 2.1]) to the variation-of-constants formula for $u_\varepsilon$ , denoted by

(27) \begin{align} u_\varepsilon (\cdot,t)=e^{t(\Delta -1)}u_0+\int _0^te^{(t-s)(\Delta -1)}\left (-\chi \nabla \cdot \left (u_\varepsilon \nabla \ln v_\varepsilon \right )-\kappa u_\varepsilon v_\varepsilon +u_\varepsilon +h_1\right ) ds, \end{align}

and, due to the maximum principle and the non-negativity of $\kappa u_\varepsilon v_\varepsilon$ , conclude that for $r\in (2,q)$

(28) \begin{align} \|u_\varepsilon (\cdot,t)\|_{L^\infty } &\le \|u_0\|_{L^\infty }+C \int _0^t\left (1+(t-s)^{-\frac 12-\frac 1r}\right )e^{-(t-s)}\|u_\varepsilon \nabla \ln v_\varepsilon \|_{L^r}ds\nonumber \\ &\quad +C\int _0^t\left (1+(t-s)^{-\frac{q-r}{qr}}\right )e^{-(t-s)}\|u_\varepsilon +h_1\|_{L^{\frac{qr}{q-r}}}ds. \end{align}

In view of Hölder’s inequality, (23) and (24), we have

(29) \begin{align} \|u_\varepsilon \nabla \ln v_\varepsilon \|_{L^r}\leq \|u_\varepsilon \|_{L^{\frac{qr}{q-r}}}\|v_\varepsilon ^{-1}\|_{L^\infty }\|\nabla v_\varepsilon \|_{L^{q}}\leq C_\varepsilon e^t\|u_\varepsilon \|_{L^{\frac{qr}{q-r}}},\quad t\in (0,T_{\max,\varepsilon }). \end{align}

By means of the interpolation inequality and (26), we obtain

(30) \begin{align} \|u_\varepsilon \|_{L^{\frac{qr}{q-r}}}\leq \|u_\varepsilon \|_{L^1}^{\frac{q-r}{qr}}\|u_\varepsilon \|_{L^\infty }^{1-\frac{q-r}{qr}}\leq C(1+t^{\frac{q-r}{qr}})\|u_\varepsilon \|_{L^\infty }^{1-\frac{q-r}{qr}},\quad t\in (0,T_{\max,\varepsilon }). \end{align}

Substituting (29) and (30) into (28), and using (11) we arrive at for $t\in (0,T_{\max,\varepsilon })$

\begin{align*} \|u_\varepsilon (\cdot,t)\|_{L^\infty } &\le C+C_\varepsilon e^t\left (1+t^{\frac{q-r}{qr}}\right )\int _0^t\left (1+(t-s)^{-\frac 12-\frac{1}{r}}+(t-s)^{-\frac{q-r}{qr}}\right )e^{-(t-s)} \|u_\varepsilon \|_{L^\infty }^{1-\frac{q-r}{qr}}ds. \end{align*}

Letting $K(T)\;:\!=\;\sup _{t\in (0, T)}\|u_\varepsilon (\cdot, t)\|_{L^\infty }$ for any $T\in (0, T_{\max,\varepsilon })$ , it follows that

\begin{align*} K(T)\leq C+C_\varepsilon e^T \left (1 +T^{\frac{1}{2r}}\right )K^{1-\frac{q-r}{qr}}(T). \end{align*}

Since $0\lt 1-\frac{q-r}{qr}\lt 1$ , an application of Young’s inequality entails that $K(T)\le C_\varepsilon (T)$ . Hence for any $T\in (0,T_{\max,\varepsilon })$ , we infer that

\begin{equation*} \|u_{\varepsilon }(\cdot, t)\|_{L^\infty } \leq C_\varepsilon (T), \quad t \in \left (0, T\right ). \end{equation*}

This, combined with (24) and (23), establishes a contradiction to (21) and thereby ensures that actually we must have $T_{\max,\varepsilon }=\infty$ .

Finally, using (23) and (26) with $T_{\max,\varepsilon }=\infty$ , we can get (18) and (19). After an integration in time, we infer from the second equation in (17) that for any $t\gt 0$

(31) \begin{align} \int _\Omega v_\varepsilon (\cdot,t)dx+ \int _0^t\int _\Omega v_\varepsilon (\cdot,s)dxds=\int _\Omega v_0 dx+\int _0^t\int _\Omega \frac{u_\varepsilon v_\varepsilon (\cdot,s)}{1+\varepsilon u_\varepsilon v_\varepsilon (\cdot,s)}dxds+\int _0^t\int _\Omega h_2dxds, \end{align}

which, together with (11) and (19), ensures (20).

At the end of this section, we also note a useful consequence of the dominated convergence theorem (see [45, Lemma A.4]).

Lemma 2.2. Let $M \subset \mathbb{R}^n$ with $n \geq 1$ be measurable, and suppose that $(w_j)_{j\in N}\subset L^\infty (M)$ and $\left (z_j\right )_{j \in \mathbb{N}}\subset L^2(M)$ are such that $\left |w_j\right | \leq C$ in $M$ for all $j \in \mathbb{N}$ and $w_j \rightarrow w$ a.e. in $M$ as well as

\begin{equation*} z_j \rightarrow z \quad \text { in }\,\,\, L^2(M)\quad \mathrm {as}\,\,\, j \rightarrow \infty \end{equation*}

for some $C\gt 0, w \in L^{\infty }(M)$ , and $z \in L^2(M)$ . Then,

\begin{equation*}w_j z_j \rightarrow w z \quad \mathrm {in} \,\,\,L^2(M) \quad \mathrm {as}\,\,\, j \rightarrow \infty .\end{equation*}

3. Global generalised solutions

To construct the global existence of the generalised solution, we will seek some uniform in $\varepsilon$ estimates on the approximate solutions $(u_\varepsilon,v_\varepsilon )$ given in Lemma 2.1. To this end, we begin with deriving the spatio-temporal integrability of $\nabla \ln v_\varepsilon$ and some regularity features of the time derivatives.

Lemma 3.1. Let $(u_\varepsilon,v_\varepsilon )$ be given in Lemma 2.1. For any $T\gt 0$ , there exists $C(T)\gt 0$ , independent of $\varepsilon$ , with the property that

(32) \begin{align} \int _0^T\|\ln v_\varepsilon (\cdot,s)\|_{H^1}^2ds\leq C(T), \end{align}

(33) \begin{align} \int _0^T\| v_\varepsilon ^{-1}(\cdot,s)\|_{H^1}^2ds\leq C(T), \end{align}

(34) \begin{align} \int _{0}^T\left \|\partial _s \ln v_{\varepsilon }(\cdot, s)\right \|_{\left (H^2\right )^{\star }} d s \leq C(T), \end{align}

(35) \begin{align} \int _{0}^T\left \|\partial _s v_\varepsilon ^{-1}(\cdot,s)\right \|_{\left (H^2\right )^{\star }} d s \leq C(T). \end{align}

Proof. Testing the second equation in (17) by $\frac{\varphi }{v_\varepsilon }$ with $\varphi (x)\in C^\infty (\overline{\Omega })$ , for $t\gt 0$ we have

(36) \begin{align} \int _\Omega \varphi \partial _t\ln v_\varepsilon dx=&\int _\Omega \frac{\varphi }{v_\varepsilon }\left (\Delta v_\varepsilon -v_\varepsilon +\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2\right )\nonumber \\ =&\int _\Omega \frac{\varphi |\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dx-\int _\Omega \frac{\nabla \varphi \cdot \nabla v_\varepsilon }{v_\varepsilon }dx-\int _\Omega \varphi dx+\int _\Omega \frac{u_\varepsilon \varphi }{1+\varepsilon u_\varepsilon v_\varepsilon }dx+\int _\Omega \frac{\varphi h_2}{v_\varepsilon }dx. \end{align}

By taking $\varphi \equiv 1$ in (36) and using $u_\varepsilon, h_2\gt 0$ , we arrive at

\begin{align*} \frac{d}{dt}\int _\Omega \ln v_\varepsilon dx\geq \int _\Omega \frac{|\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dx-|\Omega |,\,\,\,t\gt 0, \end{align*}

which on integration in time implies

\begin{align*} \int _\Omega \ln v_\varepsilon (\cdot,t) dx-\int _\Omega \ln v_0dx\geq \int _0^t\int _\Omega \frac{|\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dxds-|\Omega |t,\,\,\,t\gt 0. \end{align*}

Since $\ln \zeta \leq \zeta$ for any $\zeta \gt 0$ , it follows that

\begin{equation*} \int _0^t\int _\Omega \frac {|\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dxds\le \int _\Omega v_\varepsilon dx-\int _\Omega \ln v_0dx+|\Omega |t, \,\,t\gt 0. \end{equation*}

Note that (10) ensures

\begin{equation*} -\int _\Omega \ln v_0dx\le -|\Omega |\ln \inf _{x\in \overline {\Omega }}v_0\lt \infty, \end{equation*}

Hence, it follows from (20) that

(37) \begin{align} \int _0^t\|\nabla \ln v_\varepsilon (\cdot,s)\|_{L^2}^2ds\leq C(1+t),\,\,\,t\gt 0, \end{align}

with $C\gt 0$ independent of $\varepsilon$ . In addition, let $\eta (t)\;:\!=\;e^{-t}\inf _{x \in \overline{\Omega }} v_{0}(x)$ ; then, we have $\eta ^{-1}(t)v_\varepsilon \geq 1$ owing to (18). Thus, from the fact that $\frac 12\ln ^2\zeta \leq \zeta$ for any $\zeta \geq 1$ and (20) we infer that

\begin{equation*} \int _\Omega \ln ^2\left (\eta ^{-1}(t)v_\varepsilon \right )\le 2\int _\Omega \eta ^{-1}(t)v_\varepsilon \le C e^{t}(1+t),\,\,t\gt 0. \end{equation*}

Consequently, we have

\begin{equation*} \int _\Omega \ln ^2 v_\varepsilon \le 2\left \{\int _\Omega \ln ^2\left (\eta ^{-1}(t)v_\varepsilon \right )+\int _\Omega \ln ^2\eta (t)\right \}\le C e^{t}(1+t),\,\,\,t\gt 0. \end{equation*}

This, combined with (37), entails (32).

Meanwhile, based on (37) and (18), we also have

(38) \begin{align} \left \|v_\varepsilon ^{-1}(\cdot,t)\right \|_{L^\infty }\leq \frac{e^{t}}{\inf _{x \in \overline{\Omega }} v_{0}(x)},\quad t\gt 0 \end{align}

and

\begin{align*} \int _0^t\left \|\nabla v_\varepsilon ^{-1}(\cdot,s)\right \|_{L^2}^2ds=&\int _0^t\left \|-v_\varepsilon ^{-1}(\cdot,s)\nabla \ln v_\varepsilon (\cdot,s)\right \|_{L^2}^2ds\\ \le & \int _0^t\left \|v_\varepsilon ^{-1}(\cdot,s)\right \|_{L^\infty }^2\left \|\nabla \ln v_\varepsilon (\cdot,s)\right \|_{L^2}^2ds\\ \le & C(1+t)e^{2t},\,\,t\gt 0, \end{align*}

these immediately entail (33) as desired.

On the other hand, taking $\varphi \equiv g(x)\in C^\infty (\overline{\Omega })$ with $\|g\|_{H^2}\le 1$ in (36), it follows Hölder’s inequality that

\begin{align*} \left |\int _\Omega g\partial _t\ln v_{\varepsilon }dx\right |\leq &\|\nabla \ln v_\varepsilon \|_{L^2}^2\|g\|_{L^\infty }+\|\nabla \ln v_\varepsilon \|_{L^2}\|\nabla g\|_{L^2}+\|g\|_{L^1}\\ &+\|u_\varepsilon \|_{L^1}\|g\|_{L^\infty }+ \|h_2\|_{L^\infty }\|g\|_{L^1}\|v_\varepsilon ^{-1}\|_{L^\infty }, \end{align*}

which, with the help of Young’s inequality and Sobolev’s embedding theorem, ensures

\begin{align*} \left |\int _\Omega g\partial _t\ln v_{\varepsilon }dx\right |\le C\left (\|\nabla \ln v_\varepsilon \|_{L^2}^2+\|u_\varepsilon \|_{L^1}+1+\|h_2\|_{L^\infty }\|v_\varepsilon ^{-1}\|_{L^\infty }\right )\|g\|_{H^2}. \end{align*}

By means of (37), (19), (38), and (11), it in turn ensures (34). Similarly, in view of

\begin{equation*}(v_{\varepsilon }^{-1})_t=-v_\varepsilon ^{-2}\left (\Delta v_{\varepsilon }-v_\varepsilon +\frac {u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2\right )\end{equation*}

by the same procedure of the proof of (34), we obtain

\begin{align*} \left |\int _\Omega g\partial _t v_{\varepsilon }^{-1}dx\right |=&\left |\int _\Omega -2v_{\varepsilon }^{-1}|\nabla \ln v_{\varepsilon }|^2g+v_{\varepsilon }^{-1}\nabla g\cdot \nabla \ln v_{\varepsilon }+v_{\varepsilon }^{-1}g-\frac{u_\varepsilon v_\varepsilon ^{-1}g}{1+\varepsilon u_\varepsilon v_\varepsilon }- v_\varepsilon ^{-2}h_2g\right |\\ \leq &2\|v_{\varepsilon }^{-1}\|_{L^\infty }\|\nabla \ln v_\varepsilon \|_{L^2}^2\|g\|_{L^\infty }+\|v_{\varepsilon }^{-1}\|_{L^\infty }\|\nabla \ln v_\varepsilon \|_{L^2}\|\nabla g\|_{L^2}+\|v_{\varepsilon }^{-1}\|_{L^\infty }\|g\|_{L^1}\\ &+\|u_\varepsilon \|_{L^1}\|v_{\varepsilon }^{-1}\|_{L^\infty }\|g\|_{L^\infty }+ \|h_2\|_{L^\infty }\|v_{\varepsilon }^{-2}\|_{L^\infty }\|g\|_{L^1},\\ \le &C\|v_{\varepsilon }^{-1}\|_{L^\infty }\left (\|\nabla \ln v_\varepsilon \|_{L^2}^2+1 +\|u_\varepsilon \|_{L^1}+ \|h_2\|_{L^\infty }\|v_{\varepsilon }^{-1}\|_{L^\infty }\right )\|g\|_{H^2}, \end{align*}

which, together with (37), (19), (38) and (11), yields (35) as desired.

Similar to Lemma 3.1, we focus on deriving the spatio-temporal integrability of $\nabla \ln (u_\varepsilon +1)$ .

Lemma 3.2. Let $(u_\varepsilon,v_\varepsilon )$ be given in Lemma 2.1. For any $T\gt 0$ , there exists $C(T)\gt 0$ , independent of $\varepsilon$ , with the property that

(39) \begin{align} \int _0^T\|\ln (u_\varepsilon +1)(\cdot,s)\|_{H^1}^2ds\leq C(T), \end{align}

(40) \begin{align} \int _{0}^T\left \|\partial _s \ln \left (u_{\varepsilon }(\cdot, s)+1\right )\right \|_{\left (H^2\right )^{\star }} d s \leq C(T). \end{align}

Proof. Multiplying the first equation in (17) by $\frac{\varphi }{1+u_\varepsilon }$ with $\varphi (x)\in C^\infty (\overline{\Omega })$ and using the integration by parts, we arrive at

(41) \begin{align} \int _\Omega \varphi \partial _t\ln (1+u_\varepsilon )dx =& \int _{\Omega } \frac{\left |\nabla u_{\varepsilon }\right |^{2}\varphi }{\left (u_{\varepsilon }+1\right )^{2}}dx-\int _{\Omega }\frac{\nabla \varphi \cdot \nabla u_\varepsilon }{1+u_\varepsilon }dx-\chi \int _{\Omega } \frac{u_{\varepsilon }\varphi \nabla u_{\varepsilon } \cdot \nabla \ln v_{\varepsilon }}{\left (u_{\varepsilon }+1\right )^{2}}dx\nonumber \\[5pt] &+\chi \int _\Omega \frac{u_\varepsilon \nabla \varphi \cdot \nabla \ln v_\varepsilon }{1+u_\varepsilon }dx-\kappa \int _\Omega \frac{\varphi u_\varepsilon v_\varepsilon }{1+u_\varepsilon }dx+\int _\Omega \frac{h_1\varphi }{1+u_\varepsilon }dx. \end{align}

By taking $\varphi \equiv 1$ in (41), thanks to $h_1\geq 0$ , an application of Young’s inequality yields that

\begin{align*} \frac 12\int _{\Omega } \frac{\left |\nabla u_{\varepsilon }\right |^{2}}{\left (u_{\varepsilon }+1\right )^{2}}dx\leq &\frac{d}{dt}\int _\Omega \ln (1+u_\varepsilon )dx+\frac{\chi ^2}{2}\int _{\Omega } \frac{u_{\varepsilon }^{2}}{\left (u_{\varepsilon }+1\right )^{2}}\left |\nabla \ln v_{\varepsilon }\right |^{2}dx\\[5pt] &+\int _\Omega \frac{\kappa u_\varepsilon v_\varepsilon }{1+u_\varepsilon }dx-\int _\Omega \frac{h_1}{1+u_\varepsilon }dx\\[5pt] \leq &\frac{d}{dt}\int _\Omega \ln (1+u_\varepsilon )dx+\frac{\chi ^2}{2}\int _{\Omega }\left |\nabla \ln v_{\varepsilon }\right |^{2}dx+\kappa \int _\Omega v_\varepsilon dx. \end{align*}

In view of the non-negativity of $\ln (1+u_\varepsilon )$ , on integration in time gives us

\begin{align*} &\frac 12\int _0^t\int _{\Omega } \frac{\left |\nabla u_{\varepsilon }\right |^{2}}{\left (u_{\varepsilon }+1\right )^{2}}dxds \le \int _\Omega \ln (1+u_\varepsilon )(\cdot,t)dx+\frac{\chi ^2}{2}\int _0^t\int _{\Omega }\left |\nabla \ln v_{\varepsilon }\right |^{2}dxds+\kappa \int _0^t\int _\Omega v_\varepsilon dxds. \end{align*}

Note that $\zeta \geq \ln (1+\zeta )\geq 0$ for any $\zeta \geq 0$ . This, together with (19), leads to

\begin{equation*} \int _\Omega \ln (1+u_\varepsilon (\cdot,t))dx\leq \int _\Omega u_\varepsilon (\cdot,t)dx\leq C(1+t). \end{equation*}

On the basis of this, we infer from (20) and (37) that there exists $C\gt 0$ , independent of $\varepsilon$ , such that for any $t\gt 0$

\begin{align*} \int _0^t\int _{\Omega } \frac{\left |\nabla u_{\varepsilon }\right |^{2}}{\left (u_{\varepsilon }+1\right )^{2}}dxds \le &C(1+t),\quad t\gt 0. \end{align*}

On the other hand, since $\frac 12\ln ^2(1+\zeta )\leq \zeta$ for any $\zeta \geq 0$ , it follows from (19) that there exists $C\gt 0$ , independent of $\varepsilon$ , such that

\begin{equation*}\int _0^t\|\ln (1+u_\varepsilon )\|_{L^2}^2ds\leq 2\int _0^t\| u_\varepsilon \|_{L^1} ds\leq C(1+t),\quad t\gt 0.\end{equation*}

Combining with the above two inequalities, we obtain (39) as desired.

Now, let $\varphi \in C^\infty (\overline{\Omega })$ with $\|\varphi \|_{H^2}\le 1$ in (41), Hölder’s inequality and Young’s inequality imply that

\begin{align*} \left |\int _\Omega \varphi \partial _t\ln (1+u_\varepsilon )dx\right |\leq &\|\varphi \|_{L^\infty }\|\nabla \ln (1+u_\varepsilon )\|_{L^2}^2+\chi \|\nabla \ln (1+u_\varepsilon )\|_{L^2}\|\nabla \ln v_\varepsilon \|_{L^2}\|\varphi \|_{L^\infty } \\[5pt] &+\|\nabla \ln (1+u_\varepsilon )\|_{L^2}\|\nabla \varphi \|_{L^2} +\chi \|\nabla \ln v_\varepsilon \|_{L^2}\|\nabla \varphi \|_{L^2}\\[5pt] &+\kappa \|v_\varepsilon \|_{L^1}\|\varphi \|_{L^\infty }+\|h_1\|_{L^\infty }\|\varphi \|_{L^1}\\[5pt] \leq &C\|\varphi \|_{H^2}\left (\|\nabla \ln (1+u_\varepsilon )\|_{L^2}^2+\|\nabla \ln v_\varepsilon \|_{L^2}^2+\kappa \|v_\varepsilon \|_{L^1}+\|h_1\|_{L^\infty }+1\right ). \end{align*}

After an integration in time, we infer from (11), (20), (32), and (39) that (40) holds as desired.

With the help of Lemmas 2.1, 3.1 and 3.2, we can find a candidate $(u,v)$ for a generalised solution by standard compactness arguments.

Lemma 3.3. Let $(u_\varepsilon,v_\varepsilon )$ be taken from Lemma 2.1. Then, there exist $u\geq 0$ and $v\gt 0$ defined on $\Omega \times (0,T)$ for any $T\gt 0$ and a sequence $\{\varepsilon _j\}_{j=1}^\infty \subset (0,1)$ such that $\varepsilon _j\rightarrow 0$ as $j\rightarrow \infty$ , with the properties that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

(42) \begin{align} &\ln v_\varepsilon \rightarrow \ln v \quad \mathrm{in}\quad L^2\big (0,T;\; L^\sigma (\Omega )\big ),\quad \sigma \lt \infty, \end{align}

(43) \begin{align} &\ln v_\varepsilon \rightharpoonup \ln v \quad \mathrm{in}\quad L^2\big (0,T;\; H^1(\Omega )\big ), \end{align}

(44) \begin{align} &v_\varepsilon \rightarrow v \quad a.e.\,\,\,\mathrm{in}\quad \Omega \times (0,T), \end{align}

(45) \begin{align} &v_\varepsilon \rightarrow v \quad \mathrm{in}\quad L^1\big (\Omega \times (0,T)\big ), \end{align}

(46) \begin{align} &v_\varepsilon ^{-1}\rightarrow v^{-1} \quad \mathrm{in}\quad L^2\big (0,T;\; L^\sigma (\Omega )\big ),\quad \sigma \lt \infty, \end{align}

(47) \begin{align} &\ln (1+u_\varepsilon )\rightarrow \ln (1+u) \quad \mathrm{in}\quad L^2\big (0,T;\; L^\sigma (\Omega )\big ),\quad \sigma \lt \infty, \end{align}

(48) \begin{align} &\ln (1+u_\varepsilon )\rightharpoonup \ln (1+u) \quad \mathrm{in}\quad L^2\big (0,T;\; H^1(\Omega )\big ), \end{align}

(49) \begin{align} &u_\varepsilon \rightarrow u \quad a.e.\,\,\,\mathrm{in}\quad \Omega \times (0,T), \end{align}

(50) \begin{align} &u_\varepsilon \rightarrow u \quad \mathrm{in}\quad L^1\big (\Omega \times (0,T)\big ). \end{align}

Proof. Thanks to (32) and (34), the Aubin-Lions compactness theorem [36] implies that there exist a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) and a function $w\in L^2(0,T;\; H^1(\Omega ))$ , with the property that as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

\begin{equation*} \ln v_{\varepsilon } \rightarrow w, \quad \nabla \ln v_{\varepsilon } \rightharpoonup \nabla w \quad \text { in}\,\,\, L^{2}(\Omega \times (0, T)) \end{equation*}

and by Sobolev’s inequality

\begin{equation*} \ln v_\varepsilon \rightarrow w \quad \mathrm {in}\quad L^2\big (0,T;\; L^q(\Omega )\big ),\quad q\lt \infty, \end{equation*}

which, in particular, ensures

\begin{equation*} \ln v_{\varepsilon } \rightarrow w\quad \mathrm {and}\quad v_\varepsilon \rightarrow e^w\quad \text { a.e. in}\,\,\, \Omega \times (0, T). \end{equation*}

On the basis of these, setting $v=e^w$ , we conclude that (42)–(44) hold as desired. Similarly, according to (33) and (35), we can obtain (46). Furthermore, along the lines of the proof of [46, Lemma 2.8] we can establish the uniform integrability of $\{v_\varepsilon \}_{\varepsilon \in (0,1)}$ . In view of the Vitali convergence theorem, this together with (44) ensures the validity of (45).

Meanwhile, the assertions (47)–(50) immediately follow from the bounds (19), (39), and (40) and the Sobolev embedding theorem by using a standard subsequence extraction procedure and resorting to the Aubin-Lions compactness theorem [36].

Up to now, our knowledge on approximation of $(u,v)$ by $(u_\varepsilon,v_\varepsilon )$ is enough to pass to the limit $\varepsilon =\varepsilon _j\rightarrow 0$ in a manner of (8) in Definition 1.

Lemma 3.4. Let $u$ and $v$ be given in Lemma 3.3. For any $T\gt 0$ , the inequality (8) in Definition 1 is valid for any non-negative $\varphi (x,t)\in C_0^\infty (\overline{\Omega }\times [0,T))$ .

Proof. Note that for any non-negative $\varphi (x,t)\in C_0^\infty (\overline{\Omega }\times [0,T))$ the identity (36) is also valid. Based on it and using the integration by parts, we have

(51) \begin{align} -\int _\Omega \ln v_0\varphi |_{t=0}dx =&\int _0^T\int _\Omega \ln v_\varepsilon \varphi _tdx+\int _0^T\int _\Omega \frac{\varphi |\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dxds-\int _0^T\int _\Omega \frac{\nabla \varphi \cdot \nabla v_\varepsilon }{v_\varepsilon }dxds\nonumber \\[5pt] &-\int _0^T\int _\Omega \varphi dxds+\int _0^T\int _\Omega \frac{u_\varepsilon \varphi }{(1+\varepsilon u_\varepsilon v_\varepsilon )}dxds+\int _0^T\int _\Omega \frac{\varphi h_2}{v_\varepsilon }dxds. \end{align}

It follows from (43) that there exists a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) such that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

\begin{align*} -\int _0^T\int _\Omega \frac{\nabla \varphi \cdot \nabla v_\varepsilon }{v_\varepsilon }dxds\rightarrow -\int _0^T\int _\Omega \frac{\nabla \varphi \cdot \nabla v}{v}dxds \end{align*}

and from (42) that

\begin{align*} \int _0^T\int _\Omega \ln v_\varepsilon \varphi _tdx\rightarrow \int _0^T\int _\Omega \ln v \varphi _tdx. \end{align*}

On the other hand, similar to the proof of [19, Lemma 3.4] we can infer from the Moser-Trudinger inequality that

\begin{align*} \int _\Omega \left (1+\frac{u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon } \right )^2dx\leq & \int _\Omega \left (1+u_\varepsilon \right )^2dx\\[5pt] \leq &C_1\exp \left \{C_2\int _\Omega |\nabla \ln (1+u_\varepsilon )|^2dx+C_3\int _\Omega \ln (1+u_\varepsilon )dx\right \}, \end{align*}

which, in view of (19) and the fact that $\ln (1+\varsigma )\leq \varsigma$ for any $\varsigma \geq 0$ , leads to

\begin{align*} \ln \left \{\frac{1}{|\Omega |}\int _\Omega \left (1+\frac{u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon } \right )^2dx\right \}\leq &\ln \frac{C_1}{|\Omega |}+C_2\int _\Omega |\nabla \ln (1+u_\varepsilon )|^2dx+C_4(1+t),\quad t\gt 0. \end{align*}

Integrating it in time and using (39), for any $T\gt 0$ we can find $\widehat{C}=\widehat{C}(T)$ such that

\begin{align*} \int _0^T\ln \left \{\frac{1}{|\Omega |}\int _\Omega \left (1+\frac{u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon } \right )^2dxds\right \}\leq \widehat{C}. \end{align*}

Invoking this, using (19) again and proceeding along the lines of the proof of [46, Lemma 2.8], for fixed $\eta \gt 0$ , we can find $\delta \gt 0$ suitably small such that given an arbitrary measurable $\mathcal{E}\subset \Omega \times (0,T)$ with $|\mathcal{E}|\lt \delta$ ,

\begin{equation*} \iint _{\mathcal {E}}\frac {u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon } dxdt\lt \eta . \end{equation*}

Since we already know from (44) and (49) that

\begin{equation*} \frac {u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }\rightarrow u \quad \textrm {a.e. in}\quad \Omega \times (0,T)\quad \textrm { as}\quad \varepsilon =\varepsilon _j\rightarrow 0, \end{equation*}

along with the Vitali theorem this shows that in fact

\begin{equation*} \frac {u_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }\rightarrow u \quad \mathrm {in}\quad L^1\big (\Omega \times (0,T)\big )\quad \textrm { as}\quad \varepsilon =\varepsilon _j\rightarrow 0, \end{equation*}

and thereby ensures that

\begin{equation*} \int _0^T\int _\Omega \frac {u_\varepsilon \varphi }{1+\varepsilon u_\varepsilon v_\varepsilon }dxds\rightarrow \int _0^T\int _\Omega u\varphi dxds. \end{equation*}

Similarly, in view of (46), we have

\begin{equation*} \int _0^T\int _\Omega \frac {\varphi h_2}{v_\varepsilon }dxds\rightarrow \int _0^T\int _\Omega \frac {\varphi h_2}{v}dxds. \end{equation*}

Moreover, invoking (32) and (43), the weak lower semicontinuity of the norm ensures

\begin{equation*} \int _0^{T}\int _\Omega \frac {\varphi |\nabla v|^2}{v^2}dxdt\leq \liminf \limits _{\varepsilon =\varepsilon _j\rightarrow 0}\int _0^T\int _\Omega \frac {\varphi |\nabla v_\varepsilon |^2}{v_\varepsilon ^2}dxds. \end{equation*}

Substituting these into (51), the functions $u$ and $v$ obtained in Lemma 3.3 satisfy the inequality (8) in Definition 1.

To show the validity of (7) in Definition 1, we need the following (weak) convergence.

Lemma 3.5. Let $(u_\varepsilon,v_\varepsilon )$ be described in Lemma 2.1 and let $u$ and $v$ be established in Lemma 3.3. Then, for $p\geq 1$ and $q\geq 2$ there exists a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) such that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

(52) \begin{align} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p}\nabla \ln (u_\varepsilon +1)\rightharpoonup v^{-q}(u+1)^{-p}\nabla \ln (u+1) \quad \mathrm{in}\quad L^2\big (\Omega \times (0,T)\big ), \end{align}

(53) \begin{align} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p-1}u_\varepsilon \nabla \ln v_\varepsilon \rightharpoonup v^{-q}(u+1)^{-p-1}u\nabla \ln v \quad \mathrm{in}\quad L^2\big (\Omega \times (0,T)\big ), \end{align}

(54) \begin{align} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p}\nabla \ln v_\varepsilon \rightharpoonup v ^{-q}(u+1)^{-p}\nabla \ln v \quad \mathrm{in}\quad L^2\big (\Omega \times (0,T)\big ). \end{align}

Proof. Thanks to (44) and (49); for any $\alpha,\beta \ge 0$ , there exists a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) such that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

\begin{equation*} v_\varepsilon ^{-\alpha }(u_{\varepsilon }+1)^{-\beta }\rightarrow v^{-\alpha }(u+1)^{-\beta } \quad a.e.\,\,\,\mathrm {in}\quad \Omega \times (0,T), \end{equation*}

which, in particular, ensures that for any $p\geq 1$ and $q\geq 2$

\begin{align*} v_\varepsilon ^{-q+1}(u_{\varepsilon }+1)^{-p}\rightarrow v^{-q+1}(u+1)^{-p} \quad a.e.\,\,\,\mathrm{in}\quad \Omega \times (0,T),\\[5pt] v_\varepsilon ^{-q+1}(u_{\varepsilon }+1)^{-p-1}u_\varepsilon \rightarrow v^{-q+1}(u+1)^{-p-1}u \quad a.e.\,\,\,\mathrm{in}\quad \Omega \times (0,T). \end{align*}

In addition, we can infer from (38) that

\begin{equation*} v^{-1}\in L^\infty, \end{equation*}

which implies that

\begin{equation*} v^{-q+1}(u+1)^{-p}\in L^\infty . \end{equation*}

Hence, invoking (46) with $\sigma =2$ and Lemma 2.2, for $p\geq 1$ and $q\geq 2$ we get that, as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

(55) \begin{align} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p}\rightarrow v^{-q}(u+1)^{-p} \quad \mathrm{in}\quad L^2(\Omega \times (0,T)). \end{align}

This, together with (48), entails that, as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

(56) \begin{align} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p}\nabla \ln (u_\varepsilon +1)\rightharpoonup v^{-q}(u+1)^{-p}\nabla \ln (u+1) \quad \mathrm{in}\quad L^1(\Omega \times (0,T)). \end{align}

Moreover, by means of (38) and (39), for any $T\gt 0$ there exists $C(T)\gt 0$ , independent of $\varepsilon$ , such that

\begin{equation*} \int _0^T\|v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p}\nabla \ln (u_\varepsilon +1)\|_{L^2}^2ds\leq C(T), \end{equation*}

which, combined with (56), implies (52) as desired. Similarly, employing (38), (44), (50), (46), Lemma 2.2, and (32), we conclude that (53) and (54) also hold.

By means of Lemma 3.5, we present the validity of (7) in Definition 1.

Lemma 3.6. Let $u$ and $v$ be given in Lemma 3.3. For $p$ and $q$ taken from Lemma 3.5 which satisfy $\frac{p(p+1)\chi ^2}{4}\lt q+\frac{q^2}{p+1}$ , the inequality ( 7 ) in Definition 1 is valid for any non-negative $\varphi (x,t)\in C_0^\infty (\overline{\Omega }\times [0,T))$ .

Proof. By virtue of (18) and the non-negativity of $u_\varepsilon$ , for $p$ and $q$ taken from Lemma 3.5 and any $0\leq \varphi \in C_0^\infty (\overline{\Omega }\times [0,T))$ , we have

\begin{align*} &\frac{d}{dt}\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi dx-\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi _t dx\\[5pt] &\quad=-p\int _\Omega \varphi (u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q}\partial _tu_\varepsilon dx-q\int _\Omega \varphi (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-1}\partial _tv_\varepsilon dx\\[5pt] &\quad=-p\int _\Omega (u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q}\left (\Delta u_\varepsilon -\chi \nabla \cdot \left ( u_\varepsilon \nabla \ln v_\varepsilon \right )-\kappa u_\varepsilon v_\varepsilon +h_1\right )\varphi dx\\[5pt] &\qquad-q\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-1}\left (\Delta v_\varepsilon - v_\varepsilon +\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2 \right )\varphi dx, \end{align*}

which, by the integration by parts, leads to

\begin{align*} &\frac{d}{dt}\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi dx-\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi _t dx\\[5pt] &\quad=-p(p+1)\int _\Omega (u_\varepsilon +1)^{-p-2}v_\varepsilon ^{-q}\varphi |\nabla u_\varepsilon |^2dx\\[5pt] &\qquad+\int _\Omega \left (p(p+1)\chi \frac{u_\varepsilon }{u_\varepsilon +1}-2pq\right )(u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q-1}\varphi \nabla u_\varepsilon \cdot \nabla v_\varepsilon dx\\[5pt] &\qquad+\int _\Omega \left (\chi pq\frac{u_\varepsilon }{u_\varepsilon +1}-q(q+1)\right )(u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-2}\varphi |\nabla v_\varepsilon |^2dx\\[5pt] &\qquad-p\int _\Omega (u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q}\left (-\kappa u_\varepsilon v_\varepsilon +h_1\right )\varphi dx\\[5pt] &\qquad-q\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-1}\left (- v_\varepsilon +\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2 \right )\varphi dx\\[5pt] &\qquad+p\int _\Omega (u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q}\nabla u_\varepsilon \cdot \nabla \varphi dx-p\chi \int _\Omega (u_\varepsilon +1)^{-p-1}v_\varepsilon ^{-q-1}u_\varepsilon \nabla v_\varepsilon \cdot \nabla \varphi dx \\[5pt] &\qquad+q\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-1}\nabla v_\varepsilon \cdot \nabla \varphi dx\\[5pt] &\quad\;=\!:\;\int _\Omega \sum _{i=1}^8P_i^\varepsilon (\cdot,t)dx. \end{align*}

Integrating it in time arrives at

\begin{align*} &-\int _\Omega (u_0+1)^{-p}v_0^{-q}\varphi |_{t=0} dx-\int _0^T\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi _t dxdt=\int _0^T\int _\Omega \sum _{i=1}^8P_i^\varepsilon (t)dxdt. \end{align*}

A straightforward rearrangement entails

\begin{align*} \sum _{i=1}^3P_i^\varepsilon (x, t)=&-p(p+1)\left |(u_\varepsilon +1)^{-\frac p2-1}v_\varepsilon ^{-\frac{q}{2}}\nabla u_\varepsilon -\frac 12\left (\frac{\chi u_\varepsilon }{u_\varepsilon +1}-\frac{2q}{p+1}\right )(u_\varepsilon +1)^{-\frac{p}{2}}v_\varepsilon ^{-\frac{q}{2}-1}\nabla v_\varepsilon \right |^2\varphi \\[5pt] &+\left (\frac{p(p+1)\chi ^2 u_\varepsilon ^2}{4(u_\varepsilon +1)^2}+\frac{pq^2}{p+1}-q(q+1)\right )(u_\varepsilon +1)^{-p}v_\varepsilon ^{-q-2}|\nabla v_\varepsilon |^2\varphi . \end{align*}

Note that the assumption $\frac{p(p+1)\chi ^2}{4}\lt q+\frac{q^2}{p+1}$ implies that

\begin{align*} \frac{p(p+1)\chi ^2 u_\varepsilon ^2}{4(u_\varepsilon +1)^2}+\frac{pq^2}{p+1}-q(q+1)\lt \frac{p(p+1)\chi ^2 }{4}-q-\frac{q^2}{p+1}\lt 0; \end{align*}

thus, it follows that

\begin{equation*} -\int _0^T\int _\Omega \sum _{i=1}^3P_i^\varepsilon (t)dxdt=\left \|\sqrt {-\sum _{i=1}^3P_i^\varepsilon }\right \|_{L^2(\Omega \times (0, T))}^2. \end{equation*}

Subsequently, based on Lemmas 3.3 and 3.5, a lower semicontinuity argument entails that there exists a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) such that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

\begin{align*} &\liminf _{\varepsilon =\varepsilon _{j} \rightarrow 0} \int _0^T\int _\Omega -\sum _{i=1}^3P_i^\varepsilon dxdt\\[5pt] &\quad\ge p(p+1)\int _0^T\int _\Omega (u+1)^{-p-2}v^{-q}\varphi |\nabla u|^2dxdt\\[5pt] &\qquad-\int _0^T\int _\Omega \left (p(p+1)\chi \frac{u}{u+1}-2pq\right )(u+1)^{-p-1}v^{-q-1}\varphi \nabla u\cdot \nabla v dxdt\\[5pt] &\qquad-\int _0^T\int _\Omega \left (\chi pq\frac{u}{u+1}-q(q+1)\right )(u+1)^{-p}v^{-q-2}\varphi |\nabla v|^2dxdt. \end{align*}

In addition, similar to (55), we can prove that, as $\varepsilon =\varepsilon _j\rightarrow 0$ .

\begin{align*} v_\varepsilon ^{-q+1}(u_{\varepsilon }+1)^{-p-1}u_\varepsilon \rightarrow v^{-q+1}(u+1)^{-p-1}u \quad \mathrm{in}\quad L^2(\Omega \times (0,T)) \end{align*}

and

\begin{align*} v_\varepsilon ^{-q}(u_{\varepsilon }+1)^{-p-1}\rightarrow v^{-q}(u+1)^{-p-1} \quad \mathrm{in}\quad L^2(\Omega \times (0,T)). \end{align*}

Hence, there exists a subsequence of $\{\varepsilon _j\}_{j=1}^\infty$ (still expressed as $\{\varepsilon _j\}_{j=1}^\infty$ ) such that for any $T\gt 0$ , as $\varepsilon =\varepsilon _j\rightarrow 0$ ,

\begin{equation*} \int _0^T\int _\Omega P_4^\varepsilon dxdt\rightarrow -p\int _0^T\int _\Omega (u+1)^{-p-1}v^{-q}\left (-\kappa uv+h_1\right )\varphi dxdt. \end{equation*}

Similarly, we have

\begin{align*} \int _0^T\int _\Omega P_5^\varepsilon dxdt\rightarrow -q\int _0^T\int _\Omega (u+1)^{-p}v^{-q-1}\left (- v+u v+h_2 \right )\varphi dxdt \end{align*}

and

\begin{align*} -\int _0^T\int _\Omega (u_\varepsilon +1)^{-p}v_\varepsilon ^{-q}\varphi _t dxdt\rightarrow -\int _0^T\int _\Omega (u+1)^{-p}v^{-q}\varphi _t dxdt. \end{align*}

Finally, in view of Lemma 3.5, we obtain

\begin{align*} \int _0^T\int _\Omega \sum _{i=6}^8P_i^\varepsilon dxdt\rightarrow &p\int _0^T\int _\Omega (u+1)^{-p-1}v^{-q}\nabla u \cdot \nabla \varphi dxdt\\[5pt] &-p\chi \int _0^T\int _\Omega (u+1)^{-p-1}v^{-q-1}u\nabla v\cdot \nabla \varphi dxdt\\[5pt] &+q\int _0^T\int _\Omega (u+1)^{-p}v^{-q-1}\nabla v\cdot \nabla \varphi dxdt. \end{align*}

Hence, by collecting these (7) holds as desired.

We are now able to proceed to the proof of Theorem 1.1.

Proof of Theorem 1.1. Invoking Lemmas 3.6 and 3.4, we only need to verify the validity of (5) and (6). In fact, according to (25) we have

(57) \begin{align} \int _\Omega u_\varepsilon (\cdot,t) dx+\kappa \int _0^t \int _\Omega u_\varepsilon v_\varepsilon dxds=\int _\Omega u_0dx+\int _0^t\int _\Omega h_1dxds \end{align}

for any $t\gt 0$ and each $\varepsilon \in (0,1)$ . Hence, (5) is a consequence from (44), (49) and Fatou’s lemma. Moreover, combined with (31) and (57), we arrive at

\begin{align*} \int _\Omega (\kappa v_\varepsilon +u_\varepsilon )(\cdot,t)dx+ \int _0^t\int _\Omega \kappa v_\varepsilon (\cdot,s)dxds=\int _\Omega \kappa v_0 +u_0dx+\int _0^t\int _\Omega \kappa h_2+h_1dxds, \end{align*}

which, together with (44), (49) and Fatou’s lemma, ensures that (6) holds as desired. Therefore, $(u,v)$ is a global generalised solution to the initial-boundary value problem (3) in the sense of Definition 1. This finishes the proof of Theorem 1.1.

4. Large-time behaviour

This section is devoted to the large-time behaviour of the generalised solution $(u,v)$ determined in Theorem 1.1, under the additional assumptions (12)–(14). We start with the result on the solvability of the boundary value problem (2), which directly follows from [17].

Lemma 4.1. For any given $h_{2,\infty }\in C^1(\overline{\Omega })$ , the problem (2) possesses a unique classical solution $v_\infty$ fulfilling that $v_\infty \in C^{2+\theta }(\overline{\Omega })$ for some $\theta \in (0,1)$ .

We are also concerned with the decay in a linear differential inequality, which is an extended version of [10, Lemma 4.6] (see also [20, Lemma 2.5], [21, Lemma 2.6]).

Lemma 4.2. Let $\varepsilon \gt 0$ , $y_\varepsilon \in C^1([0,\infty ))$ be non-negative functions satisfying

(58) \begin{equation} y_\varepsilon (0)=m \end{equation}

with some positive constant $m$ independent of $\varepsilon$ . If there exist a positive constant $\mu$ and a non-negative function $g_\varepsilon (t)\in C([0, \infty ))\cap L^\infty ([0,\infty ))$ which satisfy

(59) \begin{align} \lim _{t\rightarrow \infty }\int _t^{t+1}g_\varepsilon (s)ds=0 \quad \textrm{uniformly in}\quad \varepsilon, \end{align}

and

(60) \begin{align} \|g_\varepsilon \|_{L^\infty (0,\infty )}\leq \mu \quad \textrm{uniformly in}\quad \varepsilon, \end{align}

such that for each $\varepsilon \gt 0$ and some $\lambda \gt 0$ ,

(61) \begin{equation} y'_{\!\!\varepsilon}(t)+\lambda y_\varepsilon (t)\leq g_\varepsilon (t)\quad \textrm{for all}\quad t\gt 0, \end{equation}

then

(62) \begin{equation} y_\varepsilon (t)\rightarrow 0 \quad \textrm{as}\quad t\rightarrow \infty \quad \textrm{uniformly in}\quad \varepsilon . \end{equation}

Proof. Based on (58), an integration of (61) shows that

\begin{align*} y_\varepsilon (t)\le y_\varepsilon (0)e^{-at}+e^{-at}\int _0^te^{as}g_\varepsilon (s)ds= me^{-at}+e^{-at}\int _0^te^{as}g_\varepsilon (s)ds,\quad t\gt 0. \end{align*}

Therefore, we only need to show that

(63) \begin{equation} \lim _{t\rightarrow \infty }e^{-at}\int _0^te^{as}g_\varepsilon (s)ds=0. \end{equation}

Similar to [10, Lemma 4.6], thanks to (60), for any $\varsigma \gt 0$ we may fix some $k$ (independent of $\varepsilon$ ) enough large such that $\frac{\mu e^{-ak}}{a}\lt \frac{\varsigma }2$ . For such $k$ , we further take $\sigma \gt 0$ (independent of $\varepsilon$ ) fulfilling that $k\sigma \lt \frac \varsigma 2$ . Subsequently, due to (59) we can find $t_0$ , independent of $\varepsilon$ , sufficiently large in the sense that

\begin{equation*} \int _t^{t+1}g_\varepsilon (s)ds\lt \sigma, \quad t\ge t_0-k. \end{equation*}

Consequently, we have

\begin{align*} e^{-at}\int _0^te^{as}g_\varepsilon (s)ds&=e^{-at}\int _0^{t-k}e^{as}g_\varepsilon (s)ds+e^{-at}\sum _{j=0}^{j=k-1}\int _{t-k+j}^{t-k+j+1}e^{as}g_\varepsilon (s)ds\\[5pt] &\le \frac{\mu e^{-ak}}{a}+k\sigma \\[5pt] &\lt \varsigma, \quad t\gt t_0. \end{align*}

Hence, we have (63) as a desired result and (62) has been established simultaneously.

Next, we focus on the pointwise lower bound for the solution component $v_\varepsilon$ , which plays a key role in the sequel.

Lemma 4.3. Let $(u_\varepsilon,v_\varepsilon )$ come from Lemma 2.1, and let ( 12 ) be in force. Under the additional assumption that $\Omega$ is convex, then there exists $c_1\gt 0$ , independent of $t$ and $\varepsilon$ , fulfilling that

(64) \begin{align} v_\varepsilon (x,t)\geq c_1, \quad x\in \Omega,\,\, t\gt 0. \end{align}

Proof. Thanks to the convexity of $\Omega$ , a slight adaptation of the proof of [20, Corollary 3.1] is easy to show that (64) holds as desired.

A straightforward consequence of Lemma 4.3 is the following $L^1$ -decay on the component $u_\varepsilon$ .

Lemma 4.4. Let (12)–(13) hold, and let all assumptions in Lemma 4.3 be fulfilled. Then, the solution $(u_\varepsilon,v_\varepsilon )$ fulfils that for some $C\gt 0$ independent of $(\varepsilon,t)$

(65) \begin{align} \int _\Omega u_\varepsilon (\cdot,t) dx+\int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon (\cdot,s) dxds\leq C,\quad t\gt 0, \end{align}

and that

(66) \begin{align} \int _\Omega u_\varepsilon (\cdot,t) dx\rightarrow 0\quad \mathrm{as}\,\,\, t\rightarrow \infty \quad \mathrm{uniformly\,\, in}\,\,\, \varepsilon, \end{align}

and

(67) \begin{align} \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon (\cdot,s) dxds \rightarrow 0\quad \mathrm{as}\,\,\, t\rightarrow \infty \quad \mathrm{uniformly\,\, in}\,\,\, \varepsilon . \end{align}

Proof. If (12) holds, invoking (25) and (64), we obtain

(68) \begin{align} \frac{d}{dt}\int _\Omega u_\varepsilon dx+\frac 12\kappa c_1 \int _\Omega u_\varepsilon dx+\frac 12\kappa \int _\Omega u_\varepsilon v_\varepsilon dx\leq \int _\Omega h_1dx. \end{align}

This, invoking (11) and a standard ODE technique, ensures that for some $C\gt 0$ independent of $(\varepsilon,t)$

(69) \begin{align} \int _\Omega u_\varepsilon (\cdot,t) dx\leq C,\quad t\gt 0. \end{align}

In addition, thanks to (13) and Lemma 4.2, using (68) again we infer that the decay (66) holds as desired.

We now integrate (68) over $[t,t+1]$ to get

\begin{align*} \int _\Omega u_\varepsilon (\cdot,t+1)dx+\frac 12\kappa \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds\leq \int _\Omega u_\varepsilon (\cdot,t)dx+\int _t^{t+1}\int _\Omega h_1dxds,\quad t\gt 0. \end{align*}

Recalling (13) and (66), we arrive at (67); moreover, due to (69) and (11), using (69) again, we obtain (65).

In the sequel, we will track the time evolution of $\|v_\varepsilon (\cdot,t)-v_\infty (\!\cdot\!)\|_{L^1}$ , where $v_\infty$ is the classical solution of the boundary value problem (2). For convenience, we set $\widehat{v}_\varepsilon \;:\!=\;v_\varepsilon -v_\infty$ . Thanks to (2) and (17), it is clear that for $(u_\varepsilon,v_\varepsilon )$ given in Lemma 2.1, the initial-boundary value problem

(70) \begin{equation} \left \{ \begin{split} &\widehat{v}_{\varepsilon t}=\Delta \widehat{v}_\varepsilon -\widehat{v}_\varepsilon +\frac{u_\varepsilon v_\varepsilon }{1+\varepsilon u_\varepsilon v_\varepsilon }+h_2-h_{2,\infty },&x\in \Omega,\,\,t\gt 0,\\[5pt] &\nabla \widehat{v}_\varepsilon \cdot \nu =0,&x\in \partial \Omega,\,\,t\gt 0,\\[5pt] &\widehat{v}_\varepsilon (x,0)=v_0(x)-v_\infty (x),&x\in \Omega \end{split} \right . \end{equation}

admits a unique classical solution $\widehat{v}_\varepsilon$ .

Lemma 4.5. Let all assumptions in Theorem 1.2 be in force. Then, we have

(71) \begin{align} \int _t^{t+1}\int _\Omega |\widehat{v}_\varepsilon | dxds\rightarrow 0\quad \mathrm{as}\,\,\, t\rightarrow \infty \quad \mathrm{uniformly\,\, in}\,\,\, \varepsilon, \end{align}

where $\widehat{v}_\varepsilon$ is a unique classical solution of (70).

Proof. For any $\sigma \gt 0$ , we multiply the first equation in (70) by $\frac{\widehat{v}_\varepsilon }{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}$ and integrate by parts to obtain

\begin{align*} &\frac{d}{dt}\int _\Omega \sqrt{\sigma +|\widehat{v}_\varepsilon |^2}dx+\int _\Omega \frac{|\nabla \widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx-\int _\Omega \frac{|\nabla \widehat{v}_\varepsilon |^2|\widehat{v}_\varepsilon |^2}{(\sigma +|\widehat{v}_\varepsilon |^2)^{\frac 32}}dx\\[5pt] &\quad=-\int _\Omega \frac{|\widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx+ \int _\Omega \frac{u_\varepsilon v_\varepsilon \widehat{v}_\varepsilon }{(1+\varepsilon u_\varepsilon v_\varepsilon )\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx+\int _\Omega \frac{\widehat{v}_\varepsilon (h_2-h_{2,\infty })}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx. \end{align*}

Thanks to the non-negativity of $u_\varepsilon v_\varepsilon$ and the fact that

\begin{align*} \int _\Omega \frac{|\nabla \widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx-\int _\Omega \frac{|\nabla \widehat{v}_\varepsilon |^2|\widehat{v}_\varepsilon |^2}{(\sigma +|\widehat{v}_\varepsilon |^2)^{\frac 32}}dx=\sigma \int _\Omega \frac{|\nabla \widehat{v}_\varepsilon |^2}{(\sigma +|\widehat{v}_\varepsilon |^2)^{\frac 32}}dx, \end{align*}

we arrive at

\begin{align*} \frac{d}{dt}\int _\Omega \sqrt{\sigma +|\widehat{v}_\varepsilon |^2}dx+\int _\Omega \frac{|\widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dx\leq \int _\Omega u_\varepsilon v_\varepsilon dx+\int _\Omega |h_2-h_{2,\infty }|dx. \end{align*}

Integrating it over $[t,t+1]$ , we have

\begin{align*} &\int _\Omega \sqrt{\sigma +|\widehat{v}_\varepsilon |^2}(\cdot,t+1)dx-\int _\Omega \sqrt{\sigma +|\widehat{v}_\varepsilon |^2}(\cdot,t)dx+\int _t^{t+1}\int _\Omega \frac{|\widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dxds\\[5pt] &\quad\leq \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+1}\int _\Omega |h_2-h_{2,\infty }|dxds. \end{align*}

Using the Beppo Levi theorem, as $\sigma \rightarrow 0$ , it follows that

\begin{align*} \int _t^{t+1}\int _\Omega \frac{|\widehat{v}_\varepsilon |^2}{\sqrt{\sigma +|\widehat{v}_\varepsilon |^2}}dxds\rightarrow \int _t^{t+1}\int _\Omega |\widehat{v}_\varepsilon |dxds. \end{align*}

A straightforward calculation shows that

\begin{equation*} \left |\sqrt {\sigma +|\widehat {v}_\varepsilon |^2}-|\widehat {v}_\varepsilon |\right |=\frac {\sigma }{\sqrt {\sigma +|\widehat {v}_\varepsilon |^2}+|\widehat {v}_\varepsilon |}\leq \sigma ^{\frac 12}, \end{equation*}

which leads to that, as $\sigma \rightarrow 0$ ,

\begin{align*} \int _\Omega \sqrt{\sigma +|\widehat{v}_\varepsilon |^2}(\cdot,\varsigma )dx\rightarrow \int _\Omega |\widehat{v}_\varepsilon |(\cdot,\varsigma )dx,\quad \varsigma \in \{t,t+1\}. \end{align*}

Collecting these, we conclude that

\begin{align*} &\int _\Omega |\widehat{v}_\varepsilon |(\cdot,t+1)dx-\int _\Omega |\widehat{v}_\varepsilon |(\cdot,t)dx+\int _t^{t+1}\int _\Omega |\widehat{v}_\varepsilon | dxds\\[5pt] &\quad\leq \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+1}\int _\Omega |h_2-h_{2,\infty }|dxds. \end{align*}

By setting $z_\varepsilon (t)\;:\!=\;\int _t^{t+1}\int _\Omega |\widehat{v}_\varepsilon | dxds$ , we have

\begin{equation*}z'_\varepsilon (t)=\int _\Omega |\widehat {v}_\varepsilon |(\cdot,t+1)dx-\int _\Omega |\widehat {v}_\varepsilon |(\cdot,t)dx,\end{equation*}

and thereby obtain

\begin{align*} z'_\varepsilon (t)+z_\varepsilon (t) \leq g_\varepsilon (t)\;:\!=\; \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+1}\int _\Omega |h_2-h_{2,\infty }|dxds. \end{align*}

To use Lemma 4.2, we shall need to verify that $g_\varepsilon (t)$ is uniformly in $(\varepsilon,t)$ bounded and that $\int _t^{t+1}g_\varepsilon (s)ds$ uniformly in $\varepsilon$ converges to $0$ , as $t\rightarrow \infty$ . In fact,

\begin{equation*} g_\varepsilon (t)\leq \int _t^{t+1}\int _\Omega u_\varepsilon v_\varepsilon dxds+\|h_2-h_{2,\infty }\|_{L^\infty (\Omega \times (0,\infty ))}|\Omega |,\quad t\gt 0, \end{equation*}

which, together with (65) and (11), entails

\begin{equation*} g_\varepsilon (t)\leq C,\quad t\gt 0, \end{equation*}

for some $C\gt 0$ independent of $(\varepsilon,t)$ . Thanks to the non-negativity of $g_\varepsilon (t)$ , this is enough to present the uniform in $\varepsilon$ bound of $\|g_\varepsilon (t)\|_{L^\infty (0,\infty )}$ . On the other hand, due to the definition of $g_\varepsilon (t)$ we arrive at

\begin{equation*} \int _t^{t+1}g_\varepsilon (s)ds\leq \int _t^{t+2}\int _\Omega u_\varepsilon v_\varepsilon dxds+\int _t^{t+2}\int _\Omega |h_2-h_{2,\infty }|dxds,\quad t\gt 0, \end{equation*}

which, combined with (14) and (67), implies that $\int _t^{t+1}g_\varepsilon (s)ds\rightarrow 0$ as $t\rightarrow \infty$ uniformly in $\varepsilon \in (0,1)$ . Hence, it follows from Lemma 4.2 that

\begin{equation*} z_\varepsilon (t)\rightarrow 0 \quad \textrm {as}\quad t\rightarrow \infty \quad \textrm {uniformly in}\quad \varepsilon . \end{equation*}

This finishes the proof.

As a consequence of Lemmas 4.4 and 4.5, the large-time behaviour of the generalised solution featured in Theorem 1.2 is now almost immediate.

Proof of Theorem 1.2. On the basis of Lemma 3.3 and the Fubini-Tonelli theorem, there evidently exist $(\varepsilon _j)_{j\in N}\subset (0,1)$ and a null set $\mathcal{N}\subset (0,\infty )$ such that $\varepsilon _j\rightarrow 0$ as $j\rightarrow \infty$ and

\begin{equation*} u_\varepsilon (\cdot,t)\rightarrow u(\cdot,t) \,\,\,\mathrm {and}\,\,\, v_\varepsilon (\cdot,t)\rightarrow (\cdot,t)\,\,\, a.e.\,\, \mathrm {in}\,\,\Omega \,\,\mathrm {for\,\,all}\,\,t\in (0,\infty )\setminus \mathcal {N} \end{equation*}

as $\varepsilon =\varepsilon _j\rightarrow 0$ . This, by virtue of Fatou’s lemma and Lemmas 4.4 and 4.5, gives us the desired large-time behaviour of the generalised solution in Theorem 1.2.