Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling

1.   Introduction and main results

Let $u(x, t)$ denote the density of criminals, and let $v(x, t)$ represent the abstract so-called attractiveness value. A class of logarithmic Keller-Segel models of the following form

with the parameters $\chi > 0$ and $\kappa > 0$, was introduced in ^[1] to model the propagation of criminal activities, where $\Omega\subset \mathbb R^n$ are bounded and smooth spatial domains. In the model (1.1), the given functions $h_1(x, t)$ and $h_2(x, t)$ describe the density of additional criminals and the source of attractiveness, respectively.

When $+u$ in the second equation in (1.1) is replaced by $+uv$, it arrives at the original Short et al. crime model ^[2,3], which is rewritten as

with the particular value $\chi = 2$. Note that results on related stationary problems, as in ^{[4,5,6,7,8,9,10,11,12]}, strongly support that the model (1.2) is adequate to describe the formation of crime hotspots encountered in reality. As for the corresponding initial-boundary value problems, { the understanding of them are incomplete. The local-in-time classical solution established in ^[13] is global provided that either $n = 1$ ^[14,15] or $n\geq2$ and $\chi < \frac2n$ ^[16,17] or both the initial data and the given functions $h_1$ and $h_2$ are appropriately small ^[18,19].} For larger ranges of $\chi$, global existence results, without imposing smallness on the initial data and on the given functions, are only available for either certain types of weak solutions or certain modified versions which contain additional regularizing ingredients: the globally radial renormalized solution was obtained for $n = 2$ and any $\chi > 0$ ^[20], which was extended to $n = 3$ with restriction that $\chi\in(0, \sqrt{3})$ ^[21]; the global weak solution was established in ^[22] for $n = 2$ and $\chi > 0$ by nonlinear diffusion enhancement (i.e., $\Delta u$ is replaced by $\Delta u^m$ with $m > \frac32$); the global generalized solution was structured in ^[23] for $n = 2$ and $\chi > 0$ by incorporating the logistic source (i.e., $au-bu^2$), which was extended to the case without incorporating the logistic source in ^[24]. Moreover, to suppress the formation of crime hotspots, the effects of law enforcement agents can be incorporated into (1.2) ^{[3,11,25,26,27]}, and we also refer to ^{[28,29,30,31]} for the existence and stability of the related steady states.

Note that, whenever $\kappa = 0$, $h_1\equiv0$ and $h_2\equiv0$, the model (1.1) becomes the celebrated logarithmic Keller-Segel model ^[32]:

in which $u$ and $v$ respectively represent the density of chemotactic cells and the chemoattractant concentration. To motivate our study, we also recall some results on (1.3). As to the global solvability of (1.3), various thresholds of $\chi$ have been introduced. Namely, the initial-boundary value problem possesses a global bounded classical solution for suitably regular initial data $(u_0, v_0)$, provided that either $\chi < \sqrt{\frac{2}{n}}$ ^[33,34], or $n = 2$ and $\chi < \hat{\chi}$ with some $\hat{\chi}\in(1, 2)$ ^[35], or $\chi\leq\frac4n$ ^[18]. Beyond this, the restrictions on $\chi$ have been relaxed within suitably generalized solution frameworks, for instance, $\chi < \sqrt{\frac{n+2}{3n-4}}$ in the weak sense ^[34], $\chi < \frac{n}{n-2}$ in radially symmetric setting ^[36], $\chi < \chi_0$ with $\chi_0 = \infty$ for $n = 2$ and

in the integrable sense ^[37], and $\chi > 0$ in the measure-valued sense ^[38]. In the case that $u_t$ in the first equation in (1.3) is replaced by $\varepsilon u_t$ with appropriately small $\varepsilon$, there exists an unbounded solution for large initial data, provided that $\chi > \frac{n}{n-2}$ with $n\geq3$ ^[39].{ As to the asymptotic stability of constant steady states, for a variant of (1.3) in more general non-normalized parameter settings it was established in ^[40] under the smallness of the domain size $|\Omega|$, and later on, this restriction was removed out in ^[41] by assuming $\chi\leq\frac12$ and the convexity of $\Omega$.} {In addition, when the second equation in (1.3) is replaced by $v_t = \Delta v-uv$, the corresponding model is known as the logarithmic Keller-Segel model with signal absorption, which has also been studied in a series of papers, see for instance ^{[42,43,44,45,46,47,48,49]}.}

Concerning the mathematical analysis, the model (1.1) is expected to have better solution properties than that of the models (1.2) and (1.3). However, to the best of our knowledge, the analysis results on model (1.1) are very sparse: Rodríguez in ^[1] presented that the corresponding initial-boundary value problem admits a global classical solution for the case that $\kappa = 1$, $\chi = 1$ and $n = 2$, which was extended to the case that $\chi\leq\frac4n$, $\kappa\geq0$ and $n\geq2$ in ^[18]; very recently, we showed in ^[50] that such problem possesses globally generalized solutions in the two-dimensional setting for any $\chi > 0$, and investigated the eventual smoothness of these generalized solutions. Compared these to aforementioned results related to (1.3), an appealing problem naturally appears: Does the mixed-type damping term $-\kappa uv$ possess some regularization effect that contributes to enlarging the range of the parameter $\chi$ within which the higher-dimensional initial-boundary value problem of (1.1) admits global solvability at least within some generalized framework?

To reveal it, the purpose of the present work is to explore the regularization effect of the quadratic absorption term $-\kappa uv$ with $\kappa > 0$ in the following initial-boundary value problem related to (1.1):

with the parameters $\chi > 0$ and $\kappa > 0$, where $\Omega\subset \mathbb R^n(n\ge3)$ are bounded and smooth domains, and $\nu$ denotes the exterior normal vector to the boundary $\partial\Omega$.

To specify the setup for our analysis, we assume throughout the sequel that the initial data $(u_0, v_0)$ fulfill that

and the given functions $h_1$ and $h_2$ satisfy that

The first attempt is to show that the initial-boundary value problem (1.5) admits some global generalized solutions for general initial data and arbitrary $\chi < \chi_1$ with some $\chi_1 > \chi_0$, where $\chi_0$ is given in (1.4). For any given $(u_0, v_0, h_1, h_2)$ obeying (1.6) and (1.7), the global generalized solution of the problem (1.5) can be defined as follows:

Definition 1.1. A pair $(u, v)$ is called a global generalized solution to the initial-boundary value problem (1.5) if for any $T > 0$,

(1) it holds that for some $r > 1$, $p\in(0, 1)$ and $q\in(0, 1)$

(2) it holds that for each nonnegative $\varphi(x, t)\in \mathcal{C}_0^\infty(\overline{\Omega}\times[0, T))$

(3) it holds that for any $\varphi(x, t)\in L^\infty(\Omega\times(0, T))\cap L^2(0, T;H^1(\Omega))$ having compact support in $\overline{\Omega}\times[0, T)$ with $\varphi_t\in L^2(\Omega\times (0, T))$

We would like to remark that such concept of generalized solutions resembles those for the (logarithmic) Keller-Segel system with signal absorption used in ^[49,51], but is different from that proposed in ^[37] by Lankeit and Winkler for the model (1.3). The first result on the global existence of such generalized solutions can be stated as follows.

Theorem 1.2. Let (1.6) and (1.7) hold, and let $\chi > 0$ fulfill that

Then the initial-boundary value problem (1.5) possesses at least one global generalized solution $(u, v)$ in the sense of Definition 1.1.

Remark 1.1. Simple computation shows that $\chi_1$ defined in (1.11) is larger than $\chi_0$ provided by (1.4). This reveals in some sense that $-\kappa uv$ with $\kappa > 0$ indeed has some regularization effect on solutions.

With the global existence statement at hand, it is natural to focus on the large-time behavior of generalized solutions. To achieve it, we need the following additional assumptions on $h_1$ and $h_2$:

with some $0\not\equiv h_{2, \infty}\in \mathcal{C}^1(\overline{\Omega})$. The corresponding result can be stated as follows.

Theorem 1.3. Let $\Omega$ be convex and the extra assumptions (1.12)–(1.14) hold. Then, for the global generalized solution of the initial-boundary value problem (1.5) from Theorem 1.2, there exists a null set $\mathcal{N}\subset(0, \infty)$ such that

where $v_\infty$ denotes the solution of the boundary value problem

Let's state of the art and strategy of our proofs:

{The main objective of this paper is to present that $-\kappa uv$ with $\kappa > 0$ has a regularization effect on the solution of the problem (1.5) in the $n$-dimensional settings with $n\geq3$. Precisely, we prove that the initial-boundary value problem (1.5) possesses a global generalized solution for any $\chi < \chi_1$ (given in (1.11)), where $\chi_1$ is greater than $\chi_0$ (given in (1.4)).} Note that the condition that $\chi < \chi_0$ is required in ^[37] to guarantee the global existence of generalized solutions to the initial-boundary value problem of (1.3). Usually, to get the generalized solvability, one should seek an appropriate generalized framework, and thereby obtain the global existence of generalized solutions by an appropriate approximation procedure. Here, our novelty of analysis consists of further developing the generalized framework given in ^[49] by Winkler for the logarithmic Keller-Segel system with signal absorption, and using the coupled quantity $u_\varepsilon^p v_\varepsilon^q$ with some $p, q\in(0, 1)$ introduced in ^[37] by Lankeit and Winkler for the model (1.3) to derive the uniform in $\varepsilon$ bound of

see Lemma 3.1, in which, in contrast with ^[37], we must deal with the additional term

To this end, we need some additional assumption on $p$ to obtain some now uniform in $\varepsilon$ estimates by using the benefit of $-\kappa u_\varepsilon v_\varepsilon$, see Lemma 2.2. After this, by taking advantage of Lemma 2.2 again, we address ourselvers to the uniform in $\varepsilon$ bound of $\|u_\varepsilon\|_{L^r(\Omega\times(0, T))}$ with some $r > 1$, see Lemma 3.3. We would like to remark that, in contrast with ^[37], the additional condition that $q < \frac4n$ will be required in our situation when $n\geq5$, which results in that [37,Lemma 5.1] is no longer in force. Here, our novelty of analysis contains establishing some fragile estimates, by which we can get the core of our requirement (1.11) on $\chi$. Based on the above processes, we can employ the result that $u_\varepsilon\rightarrow u$ a.e. in $\Omega\times(0, T)$ and the Vitali convergence theorem to get the strong convergence of $\{u_\varepsilon\}$ in $L^1(\Omega\times (0, T))$, see Lemma 3.5. Moreover, by establishing a series of uniform a-priori estimates as desired, we can get the global existence of generalized solutions to the initial-boundary value problem (1.5) via passing to limit, and subsequently complete the proof of Theorem 1.2 in Section 3.

The second objective of this paper is to show the large-time behavior of such generalized solutions, under the additional assumptions on $h_1$ and $h_2$. Here, our novelty of analysis consists of tracking the time evolution of the combinational functional of the form

with some $\mu > 0$, where $v_\infty$ is a classical solution of the boundary value problem (1.16). A fragile calculation yields that

see Lemma 4.5 for details. This, together with the Fubini-Tonelli theorem and Fatou's lemma, ensures the desired results in Theorem 1.3, see Section 4 for details.

2.   Global approximate solutions and basic estimates

To structure the generalized solution of the initial-boundary value problem (1.5) by an approximation procedure, for any $\varepsilon\in(0, 1)$ we shall consider the following approximate problem

and then first obtain the following.

Lemma 2.1. Assume that the assumptions (1.6) and (1.7) hold. For each $\varepsilon\in(0, 1)$ and any $\chi > 0$, there exists a unique pair $(u_\varepsilon, v_\varepsilon)$ of positive functions with the properties that for any $T > 0$

such that $(u_\varepsilon, v_\varepsilon)$ solves the approximate problem (2.1) classically in $\Omega\times[0, T)$. Moreover, the following two statements are true:

and

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

Proof. An application of the well-known strategy, as in ^[34,52], implies that there exist time $T_{\max, \varepsilon}\in(0, \infty]$ and a unique pair $(u_\varepsilon, v_\varepsilon)$ of positive functions with the properties that

such that $(u_\varepsilon, v_\varepsilon)$ solves the approximate problem (1.5) classically in $\Omega\times[0, T_{\max, \varepsilon})$. Moreover, if $T_{\max, \varepsilon} < \infty$, then for any $q > n$

To show that $T_{\max, \varepsilon} = \infty$, let us start with the pointwise lower bound for the solution component $v_\varepsilon$ and the bound of $\|u_\varepsilon\|_{L^1}$. Indeed, according to the variation-of-constants formula for $v_\varepsilon$

the comparison principle for the Neumann problem associated with the heat equation and the facts that $h_2\geq0$ and $u_\varepsilon > 0$, we have

To get the bound of $\|u_\varepsilon\|_{L^1}$, we integrate the first equation in (2.1) over $\Omega$ to obtain

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

We now estimate $\|u_\varepsilon\|_{L^\infty}$. Indeed, according to the variation-of-constants formula for $u_\varepsilon$, we can infer from the maximum principle and the nonnegativity of $\kappa u_\varepsilon v_\varepsilon$ that

which, combining with the properties of the Neumann heat semigroup (see ^[53,54]), (2.6) and the nonnegativity of $u_\varepsilon$, leads to that for any $r > n$

By means of the interpolation inequality and (2.8), we obtain

In addition, the application of the properties of the Neumann heat semigroup to (2.5) entails that for any $r > n$

Using (2.10) and letting $K(T): = \sup_{t\in(0, T)}\|u_\varepsilon(\cdot, t)\|_{L^\infty}$ for any $T\in (0, T_{\max, \varepsilon})$, we arrive at

Substituting this and (2.10) into (2.9), we conclude that

Since $0 < 1-\frac1r < 1$, an application of Young's inequality implies that $K(T)\le C_\varepsilon(T)$. Hence for any $T\in(0, T_{\max, \varepsilon})$ we infer that

This, together with (2.6) and (2.11), establishes a contradiction to (2.4) and thereby ensures that actually we must have $T_{\max, \varepsilon} = \infty$.

Moreover, using (2.6) and (2.8) with $T_{\max, \varepsilon} = \infty$, the assertions (2.2) and (2.3) hold as desired.

The estimate (2.3) turns out to be sufficient for the derivation of the bound of $\|v_\varepsilon\|_{L^2}$.

Lemma 2.2. Let $r\in[1, \frac{n}{n-2})$. Assume that $(u_\varepsilon, v_\varepsilon)$ is taken from Lemma 2.1. Then there exists $C = C(r) > 0$, with the property that

Moreover, there is $C > 0$ such that

and

Proof. Recalling (2.5) and invoking the properties of the Neumann heat semigroup (see ^[53,54]), for any $r\in[1, \frac{n}{n-2})$ and $t > 0$ we have

which, using (1.7) and (2.3), leads to (2.12) as desired.

Next, we test the second equation in (2.1) by $v_\varepsilon$ and use the integration by part to get

By means of (1.7), the applications of Young's inequality and Hölder's inequality yield

Invoking this we arrive at

Integrating (2.15) over $[0, t]$ we obtain that

which, combined with (2.3), ensures (2.13) and (2.14).

In comparison to Lemma 2.2, deriving the bound for $\int_0^t\int_\Omega |\nabla u_\varepsilon|^2dxds$ seems to be more delicate, due to the presence of the taxis-type term in the first equation in (2.1). Motivated by ^[49,51], we resort to estimating $\nabla\ln(u_\varepsilon+1)$ instead.

Lemma 2.3. Let $(u_\varepsilon, v_\varepsilon)$ be given in Lemma 2.1. There exists $C > 0$, with the property that

Proof. Multiplying the first equation in (2.1) by $\frac{1}{1+u_\varepsilon}$ and using the integration by parts, we see that

Invoking this, an application of Young's inequality yields that

which, using (2.2) and integrating in time, leads to

This, together with (2.12) and (2.14), ensures there exists $C > 0$ such that

Since $\zeta\geq\ln(1+\zeta)\geq0$ for any $\zeta\geq0$, we obtain that

which, in view of (2.3), entails (2.16).

3.   Global generalized solutions

To obtain the desired integrability for $u$ in Definition 1.1, a crucial step in our analysis will consist of deriving the uniform spatio-temporal integrability of $u_\varepsilon$. To this end, we further develop the framework presented in [37,Lemma 5.1] to get some essential a-priori estimates for (2.1).

Lemma 3.1. Let $n\ge 3$, $p\in(0, 1)$ satisfying $p < \frac{1}{\chi^2}$ and $p < \frac4n$, and $q\in(q_-(p), q_+(p))$ with

Assume that $(u_\varepsilon, v_\varepsilon)$ is given in Lemma 2.1. Then, there exists $C > 0$, with the property that for any $t > 0$ and $\varepsilon\in(0, 1)$

Proof. Using the facts that $u_\varepsilon > 0$ and $v_\varepsilon > 0$, we have

which, using the integration by parts, leads to

Thanks to the nonnegativity of $h_1$, $h_2$, $u_\varepsilon$ and $v_\varepsilon$, we arrive at

A straightforward rearrangement in the first three integrands on the right entails

Invoking this, we obtain

where

We also note that the assumption (3.1) on $q$ warrants that $q < 1-p$ and

which, due to $p\in(0, 1)$, ensures

Similarly, we have

Collecting these, there exist two positive constants $\widehat{c_1}$ and $\widehat{c_2}$, denoted by

such that

Hence, an integration in time shows

Using Hölder's inequality and the fact that $q < 1-p$ again, we have

which, together with (2.3) and (2.13), implies that there exists $C > 0$, independent of $\varepsilon$, such that

Similarly, an application of Hölder's inequality and Young's inequality yields that

Since $q < 1-p$ and $0 < p < 1$, we have $2p+q+1 < 2+p < 3$. In the case that $n = 3$, thanks to $\frac{n}{n-2} = 3$, it follows from (2.12) that there exists $C > 0$ such that

In the case that $n\geq 4$, if $2p+q+1\leq 2$, according to Young's inequality and (2.13) there exists $C > 0$ such that

In addition, if $2 < 2p+q+1\leq\frac{2n}{n-2}$, then we can infer from the Gagliardo-Nirenberg inequality that

which, combined with (2.13), leads to

Recalling the fact that $2 < 2p+q+1$, we have $0 < \frac n2(2p+q-1) < \frac n2p < 2$ due to $p+q < 1$ and $p < \frac4n$, and thereby infer from Young's inequality and (2.14) that

Collecting (3.6)–(3.8), it follows from (3.5) that, whenever $n\ge3$,

Substituting (3.4) and (3.9) into (3.3), we have (3.2) as desired, due to the facts that $\widehat{c_1}, \widehat{c_2} > 0$ and $q < 1$.

Indeed, using Lemma 3.1, we can get the bound of $u_\varepsilon$ in some reflexive $L^r$ spaces. To achieve it, we need to identify the minimal possible choice of an integrability exponent arising in (3.19) below, which will form the core of our requirement (1.11) on $\chi$.

Lemma 3.2. Let $\chi > 0$, and for $p\in\left(0, \min\{1, \frac 4n, \frac{1}{\chi^2}\}\right)$ let $q_{\pm}(p)$ be defined in (3.1). If $n\in\{3, 4\}$, then

and if $n\geq5$, then

Proof. If $n\in\{3, 4\}$, then $p\in(0, \min\{1, \frac{1}{\chi^{2}}\})$. In this case, the assertion (3.11) directly follows from [37,Lemma 5.1].

If $n\geq5$, then we have $p\in(0, \min\{\frac4n, \frac{1}{\chi^{2}}\})$. A straightforward calculation shows that

and that $I(\chi)\geq1$ for any $\chi > 0$. Setting $\xi: = \sqrt{1-p\chi^2}$, we get that

Note that $p = \frac{1-\xi^2}{\chi^2}$, simple computation shows that

Accordingly, we have

As

which implies that $g(\xi)$ is strictly monotonely decreasing in the interval $[0, \chi-1]$ and strictly monotonely increasing in the intrerval $[\chi-1, +\infty)$, correspondingly, we have

Direct calculation shows that

and

moreover,

and

as well as

Thus, it follows from (3.14) that

Note that

and

these together with (3.15) gives us the desired (3.11).

Now under the assumptions on $\chi$ in Theorem 1.2, the interpolation argument, invoking Lemma 3.2, indeed bears fruit of the desired flavour.

Lemma 3.3. Let $p$ and $q$ be taken from Lemma 3.1, and $\chi$ satisfy (1.11). Then, for $(u_\varepsilon, v_\varepsilon)$ given in Lemma 2.1, there exist $r > 1$ and $C > 0$, with the property that

Proof. According to (3.10), we infer that if $n = 3$, then we have, as long as $\chi < 2\sqrt{3}$,

with $q_{-}(p)$ and $q_{+}(p)$ given by Lemma 3.1, which ensures that we can find $p \in(0, \min\{1, \frac{1}{\chi^{2}}\}$ and $q \in\left(q_{-}(p), q_{+}(p)\right)$ such that

Similarly, we can deduce from (3.10) and (3.11) that

for any $n\ge 4$, as long as $\chi < 2\sqrt{1+\frac4n}$, which also guarantees that we can choose $p \in(0, \min\{\frac4n, \frac{1}{\chi^{2}}\}$ and $q \in\left(q_{-}(p), q_{+}(p)\right)$ such that

Fix $p$ and $q$ in (3.17) and (3.18) respectively, utilizing a continuity argument we can further pick $r\in(1, 1+p)$ sufficiently close to $1$ such that

For such $r$, an application of Young's inequality yields that

which, united (3.2), ensures that

with $\beta = \frac{(1-q)r}{1+p-r}$ and $C > 0$ independent of $\epsilon$.

In the case $n = 3$, if $\beta < \frac{n}{n-2}$, i.e, $\beta < 3$, then it follows from (2.12) that

thus we have from (3.19) that

namely, (3.16) is valid. Meanwhile, if $\beta\in[3, 4)$, then $\gamma: = \frac{3\beta-6}2\in[\frac32, 3)$ and $\frac{6(\beta-\gamma)}{6-\gamma} = 2$, the Gagliardo-Nirenberg inequality implies that

which, together with (2.12)–(2.14), gives us

This, combined with (3.19), also entails (3.16) due to $\beta < 4$.

In the case $n\ge 4$, if $\beta\leq 2$, then the Young inequality and (2.13) entail that

This, together with (3.19), also guarantees the validity of (3.16). If $\beta\in(2, 2+\frac4n)$, then $\frac{n(\beta-2)}{2} < 2$. Applications of the Gagliardo-Nirenberg inequality and the Young inequality imply that

In this case, we can infer from (2.13) and (2.14) that

which, combined (3.19), implies that (3.16) is also valid due to $\beta < 2+\frac4n$.

As a final preparation for our limit procedure, we establish some regularity features of the time derivatives in (2.1).

Lemma 3.4. Let $(u_\varepsilon, v_\varepsilon)$ be established in Lemma 2.1. For any $T > 0$, there exists $C(T) > 0$, with the property that for $r > n$

Proof. On the basis of the second equation in (2.1), we obtain from the integration by parts and Hölder's inequality that for any $\varphi\in \mathcal{C}^\infty(\overline{\Omega})$ and $t > 0$

which, combined with the Sobolev embedding theorem, entails that for any $r > n$ there exists $C > 0$ independent of $\varepsilon$ such that for any $t > 0$

This, in view of (2.3), (2.13), (2.14) and (1.7), in turn ensures (3.20).

Next, multiplying the first equation in (2.1) by $\frac{\varphi}{u_\varepsilon+1}$ for any $\varphi\in \mathcal{C}^\infty(\overline{\Omega})$ we have for any $t > 0$

which, by using Hölder's inequality, Young's inequality and Sobolev's inequality, entails that for any $r > n$ there exists $C > 0$ such that for any $t > 0$

After an integration in time, we infer from (1.7), (2.13), (2.14), (2.16) and (2.2) that (3.21) holds as desired.

On the basis of the standard compactness arguments, we can find a candidate $(u, v)$ for a generalized solution.

Lemma 3.5. Let $(u_\varepsilon, v_\varepsilon)$ be taken from Lemma 2.1. Then, for any $T > 0$ there exist functions $u\geq0$ and $v > 0$ defined on $\Omega\times(0, T)$ and a sequence $\{\varepsilon_j\}_{j = 1}^\infty\subset(0, 1)$ such that $\varepsilon_j\rightarrow0$ as $j\rightarrow \infty$, with the properties that as $\varepsilon = \varepsilon_j\rightarrow0$,

where $q < \frac{2n}{n-2}$.

Proof. Since $\frac12\ln^2(1+\zeta)\leq \zeta$ for any $\zeta\geq0$, we infer from the bounds (2.16) and (2.3) that

Based on this, the bound (3.21), combined with the Aubin-Lions compactness theorem ^[55], implies that there exist a subsequence of $\{\varepsilon_j\}_{j = 1}^\infty$ (still expressed as $\{\varepsilon_j\}_{j = 1}^\infty$) and a function $w\in L^2(0, T; H^1(\Omega))$, fulfilling that as $\varepsilon = \varepsilon_j\rightarrow0$,

and thereby

Hence, denoting $u = e^w-1$ and using the bound (2.3) again, we obtain the assertions (3.22)–(3.24). Due to (3.24), according to the uniform integrability property implied by Lemma 3.3 we may apply the Vitali convergence theorem to get that in fact (3.25) also holds. Meanwhile, using (3.24) and invoking Lemma 3.3, we arrive at (3.26).

According to the bounds (2.13), (2.14) and (3.20), and the Sobolev embedding theorem, a standard subsequence extraction procedure resorting to the Aubin-Lions compactness theorem (see ^[55]) entails model (3.27) immediately. Due to (2.2), we have

and also infer from (2.14) that

Since $(v_{\varepsilon }^{-1})_t = -v_\varepsilon^{-2}\left(\Delta v_{\varepsilon}-v_\varepsilon+u_\varepsilon+h_2\right)$, similar to (3.20), using (2.2) again we get

Hence, invoking the Aubin-Lions compactness theorem (^[55]), there exists a subsequence of $\{\varepsilon_j\}_{j = 1}^\infty$ (still expressed as $\{\varepsilon_j\}_{j = 1}^\infty$) such that (3.28) holds as desired, as $\varepsilon = \varepsilon_j\rightarrow0$. Similarly, (3.29) also holds. On the other hand, using the bounds (2.13) and (2.14) again yields the last two assertions in lemma.

Up to now, our knowledge on approximation of $(u, v)$ by $(u_\varepsilon, v_\varepsilon)$ is enough to pass to the limit $\varepsilon = \varepsilon_j\rightarrow0$ in the weak formulation of the second equation in the approximate problem (2.1), which also show that $v$ is indeed a weak solution of the respective sub-problem of (1.5) in the sense of Definition 1.1.

Lemma 3.6. Let $u$ and $v$ be given in Lemma 3.5. For any $T > 0$, the identity (1.10) in Definition 1.1 is valid for any $\varphi(x, t)\in L^\infty(\Omega\times(0, T))\cap L^2(0, T;H^1(\Omega))$ having compact support in $\overline{\Omega}\times[0, T)$ with $\varphi_t\in L^2(\Omega\times (0, T))$.

Proof. For each $\varphi$ from the class indicated in (1.10), it follows from (3.25) and the Lebesgue dominated convergence theorem that there exists a subsequence of $\{\varepsilon_j\}_{j = 1}^\infty$ (still expressed as $\{\varepsilon_j\}_{j = 1}^\infty$) such that for any $T > 0$, as $\varepsilon = \varepsilon_j\rightarrow0$,

Hence, we can take the limit $\varepsilon = \varepsilon_j\rightarrow0$ on the second equation in (2.1) in the weak sense by employing Lemma 3.5. Moreover, the functions $u$ and $v$ obtained in Lemma 3.5 satisfy the identity (1.10) in Definition 1.1.

To take the limit also in the first equation in the approximate problem (2.1) in an appropriate manner, we shall obtain the strongly convergence of $\nabla\ln v_{\varepsilon}$ in $L^2(\Omega\times(0, T))$ for any $T > 0$.

Lemma 3.7. Let $(u_\varepsilon, v_\varepsilon)$ be described in Lemma 2.1, and let $u$ and $v$ be established in Lemma 3.5. Then there exists a subsequence of $\{\varepsilon_j\}_{j = 1}^\infty$ (still expressed as $\{\varepsilon_j\}_{j = 1}^\infty$) such that for any $T > 0$, as $\varepsilon = \varepsilon_j\rightarrow0$,

Proof. We can adopt a strategy similar to [49,Lemma 2.10] to get (3.32) as desired.

Invoking Lemma 3.7, we can present the validity of (1.9) in Definition 1.1.

Lemma 3.8. Let $u$ and $v$ be given in Lemma 3.5. For any $T > 0$, the inequality (1.9) in Definition 1.1 is valid for each nonnegative $\varphi(x, t)\in \mathcal{C}_0^\infty(\overline{\Omega}\times[0, T))$.

Proof. Testing the first equation in (2.1) by $\frac{\varphi}{1+u_\varepsilon}$ with $0\leq\varphi\in C_0^\infty(\Omega\times [0, T))$, we have

We conclude from (3.23) that as $\varepsilon = \varepsilon_j\rightarrow0$,

Since $\frac{u_{\varepsilon} }{(u_{\varepsilon}+1)(1+\varepsilon u_\varepsilon)} \rightarrow \frac{u}{u+1}$ a.e. in $\Omega\times(0, T)$ as $\varepsilon = \varepsilon_j\rightarrow0$, we infer from (3.32) and [51,Lemma A.4] that, as $\varepsilon = \varepsilon_j\rightarrow0$,

which, combined with (3.23), further implies that, as $\varepsilon = \varepsilon_j\rightarrow0$,

and

Similarly, we obtain that $\varepsilon = \varepsilon_j\rightarrow0$,

By using the Lebesgue dominated convergence theorem, we have

Invoking (3.23), an application of the weak lower semicontinuity of the norm implies

Hence, collecting these, (1.9) holds as desired.

We are now in the position to prove Theorem 1.2.

Proof of Theorem 1.2. In fact, we only need to combine Lemma 3.6 with Lemma 3.8.

4.   Large-time behavior

In this section, we will investigate the large-time behavior of the generalized solution $(u, v)$ determined in Theorem 1.2, under the additional assumptions (1.12)–(1.14). To achieve this, we begin with the following pointwise lower bound for the solution component $v_\varepsilon$, which will play a key role in the sequel.

Lemma 4.1. Let $(u_\varepsilon, v_\varepsilon)$ come from Lemma 2.1, and let (1.12) be in force. Under the additional assumption that $\Omega$ is convex, then there exists $c_1 > 0$, independent of $t$ and $\varepsilon$, fulfilling that

Proof. It immediately follows from [50,Corollary 3.1].

Let us state a straightforward consequence of Lemma 4.1.

Lemma 4.2. Let all the assumptions in Lemma 4.1 be fulfilled. Then there exists a positive constant $c_2$, with the property that

Proof. Invoking (2.7) and (4.1), we arrive at

where $c_1$ is given in (4.1). By taking $\lambda: = \frac6\kappa$, this, combined with (2.15), leads to

Setting $y(t): = \int_\Omega\lambda u_\varepsilon+ v_\varepsilon^2 dx$, we get

which, employing a standard ODE argument, warrants that

Using this and integrating (4.4) over $[t, t+1]$, it follows that for any $t > 0$

which, combined with (4.5), evidently ensures (4.2).

To prove the long-time behavior in Theorem 1.3, we shall consider the Helmholtz problem (1.16).

Lemma 4.3. For given $0\not\equiv h_{2, \infty}\in \mathcal{C}^1(\overline{\Omega})$, the problem (1.16) possesses a unique classical solution $v_\infty$ with the property that $v_\infty\in \mathcal{C}^{2+\theta}(\overline{\Omega})$ for some $\theta\in(0, 1)$.

Proof. The assertion directly follows from ^[56].

We are also concerned with the decay in a linear differential inequality (see [50,Lemma 2.5]).

Lemma 4.4. For $\varepsilon\in(0, 1)$, let $y_\varepsilon\in \mathcal{C}^1([0, \infty))$ be non-negative functions. If $y_\varepsilon(0)$ is dependent of $\varepsilon$, and there exist $a > 0$ and the nonnegative function $g(t)\in C([0, \infty))\cap L^\infty([0, \infty))$ which satisfies

such that

then

As a consequence, under the additional assumptions (1.13)–(1.14), a stronger result than Lemma 4.2 can be shown as follows.

Lemma 4.5. Let all the assumptions in Lemma 4.2 hold, and let (1.13)–(1.14) be in force. Then we have

where $v_\infty$ is a unique classical solution of (1.16).

Proof. Set $\widehat{v_\varepsilon}: = v_\varepsilon-v_\infty$ for convenience. Lemmas 2.1 and 4.3 imply that for fixed $u_\varepsilon$ from Lemma 2.1, the initial-boundary value problem

admits a unique classical solution $\widehat{v_\varepsilon}$. We multiply the first equation in (4.8) by $\widehat{v_\varepsilon}$ to get

and thereby obtain from Young's inequality that

By taking $\lambda\geq\max\left\{\frac4\kappa, \frac{4\|v_\infty\|_{L^\infty}+2}{c_1\kappa}\right\}$, this, combined with (4.3), ensures

Setting $g(t): = \int_\Omega (h_2-h_{2, \infty})^2dx+\lambda\int_\Omega h_1dx$ and $y_\varepsilon(t): = \int_\Omega\widehat{v_\varepsilon}^2+\mu u_\varepsilon dx$, we have

By means of (1.13)–(1.14) and Lemma 4.4, the desired (4.6) holds. We now integrate (4.9) over $[t, t + 1]$ to get

This, in view of (1.13), (1.14) and (4.6) again, ensures that (4.7) holds.

Our second result on the large-time behavior of generalized solutions featured in Theorem 1.3 is in fact a by-product of our previous analysis.

Proof of Theorem 1.3. In fact, Lemma 3.5, combining with the Fubini-Tonelli theorem, provides $(\varepsilon_j)_{j\in\mathbb{N}}\subset(0, 1)$ and a null set $\mathcal{N}\subset(0, \infty)$ such that $\varepsilon_j\rightarrow0$ as $j\rightarrow \infty$ and

as $\varepsilon = \varepsilon_j\rightarrow0$. Based on this, Lemma 4.5, together with Fatou's lemma, presents the desired large-time behavior of the generalized solution in Theorem 1.3.

Acknowledgments

The authors are sincerely grateful to the anonymous reviewers for the detailed comments and valuable suggestions which really helped us to make the paper more readable and meaningful. The research of ZW is supported by the National Natural Science Foundation of China (No. 11701304) and Natural Science Foundation of Ningbo Municipality (No. 2019A610041, No. 2021J143). Wang also gratefully acknowledges the support of KC Wong Education Foundation. The research of LX is partially supported by the Chongqing Science and Technology Commission Project (No. sctc2020jcyj-msxmX0560, No. csts2020jcyj-jqX0022), and the Science Technology Research Program of Chongqing Municipal Education Commission (No. KJZD-M202000502, No. CXQT21014).

Conflict of interest

The authors declare there is no conflict of interest.