Understanding the oscillations of an epidemic due to vaccine hesitancy

1.   Introduction

In a series of works ^[1,2], Kim and his collaborators introduced the following chemotaxis model

where $u$ is the bacterial density and $w$ is the concentration of nutrient. $d\geq0$ is the diffusion rate of nutrient. $\kappa(w)\geq0$ is the consumption rate. Typical examples of $\kappa(w)$ include $\kappa(w) = w^m$ with $m\geq0$.

System (1.1) is an alternative model to describe the propagation of traveling band of bacteria observed in the experiment of Adler ^[3]. Compared with the classical Keller-Segel system ^[4], this model is rigorously derived from the notion of "metric of food", and brings the theory of Riemannian geometry to the field of chemotaxis. Choi and Kim ^[1] have proved that system (1.1) with $d = 0$ can generate traveling bands and traveling fronts under various assumptions on $\kappa(w)$. They also generalized their results of ^[1] to the models with porous medium diffusion for the bacterial density, and showed that there exist compactly supported traveling waves for chemotaxis. Ahn, Choi and Yoo ^[5] proved the global existence of strong solutions of Cauchy problem of system (1.1) if the initial value of $w$ has positive lower bound. Very recently, they ^[6] have generalized this result to the case where $w$, with infinite initial mass, can be zero at spatial infinity.

In this paper, we are interested in the existence and stability of spiky patterns to system (1.1). We assume that the consumption rate is linear for simplicity, and write system (1.1) as

We shall consider the system in the half-space ${\mathbb{R}}_+ = [0, \infty)$, with the following initial value

and boundary conditions

where $b > 0$ is a constant. That means we prescribe no-flux boundary condition for the bacterial density and saturated boundary condition for the oxygen. This kind of boundary conditions have also been used in a chemotaxis-fluid model to describe the formation of concentration patterns for aerobic bacteria observed in the experiment of ^[7].

System (1.2) is actually a chemotaxis model with signal-suppressed motility. In other words, the diffusion rate of the bacterial density is monotonically decreasing as the concentration of the signal increases. There are several analytical works for the chemotaxis model of self-aggregation type with signal-suppressed motility in bounded domains. See ^[8,9] for the global existence of classical solutions if the motility function satisfies the power law, ^[10,11] for the existence of critical mass generating blowup if the motility is an exponential function, ^[12] for the formation of spiky patterns. In contrast, system (1.2) is of consumption type. That is the chemical signal $w$ is consumed by the bacteria $u$. It turns out that such two types of chemotaxis model may exhibit different dynamics. Indeed, one can easily verify (following the argument of Proposition 2.1 of ^[13]) that if $b = 0$ or $w$ satisfies homogeneous Neumann boundary in the half space, then system (1.2) only has constant steady states, and no pattern exists. In other words, it is the nonhomogenous boundary condition that generates spiky patterns. Such phenomenon is quite different from the solution structures of chemotaxis model of self-aggregation type for which the intrinsic mechanics of chemotactic interaction generates spiky patterns (see ^[14,15]). Furthermore, Tao ^[16] showed that, under homogeneous Neumann boundary conditions in bounded domains, the multidimensional classical chemotaxis model of consumption type has a unique global bounded solutions under suitable assumptions on the initial data $w_0$ and the chemotactic coefficient. In particular, the global existence or blow-up of solutions is independent of the initial data $u_0$. This study indicates that the chemotaxis model of consumption type posses another different property from the one of self-aggregation type since the latter has the well-known critical mass on $u_0$ for blow-up in dimension 2. This work subsequently led to various generalizations. Baghaei and Khelghati ^[17] improved the results of ^[16] to a larger set of $w_0$ and chemotactic coefficient. Frassu and Viglialoro ^[18] further generalized the works of ^[16] to the models with indirect signal consumption. Recently, Li and Zhao ^[19] and Wang ^[20] proved that under homogeneous Neumann boundary conditions, the chemotaxis-consumption system with regular signal-dependent motility also has global bounded solutions under some assumptions on $w_0$. It is worth mentioning that for the chemotaxis-consumption system with logarithmic sensitivity, Winkler ^[21,22] introduced the notion of renormalized solutions to handle the singularity in the study of global existence of large solutions.

There are some studies on the dynamics of classical chemotaxis model of consumption type with nonhomogeneous boundary conditions. In the one dimensional case, Hong and Wang ^[23] studied the stability of steady state to the minimal model with Dirichlet boundary condition for the nutrient in bounded domains; Carrillo, Li and Wang ^[13] obtained the stability of steady state to the model with constant motility and logarithmic singular sensitivity in the half space. In the multidimensional case, Braukhoff and Lankeit ^[24] proved the existence and uniqueness of steady state to the minimal model with nonhomogeneous Robin boundary condition, while Lee, Wang and Yang ^[25] obtained similar results for the minimal model with Dirichlet boundary condition, and they further analyzed the boundary layer phenomena. Recently, Fuest, Lankeit and Mizukami ^[26] further showed the stability of steady state for the minimal parabolic-elliptic model on the base of the works on the steady state obtained in ^[24].

One can observe from the boundary condition $w(+\infty, t) = 0$ that, in contrast with the models studied in the above mentioned works, system (1.2) is actually a chemotaxis model with singular sensitivity and singular motility. In this paper, we shall develop some new strategies to overcome analytical difficulties caused by the coupling of nonhomogenous boundary condition and singularities. And we obtain the following results:

(1) system (1.2)–(1.4) admits a unique steady state $(U, W)$, and $U\rightarrow \lambda\delta(x)$ as $d\rightarrow0$, where $\delta(x)$ is the Dirac function and $\lambda$ is the initial bacterial density, i.e., $\lambda = \int_0^\infty u_0(x)dx$;

(2) this spiky steady state $(U, W)$ is asymptotically stable in the sense that if the initial data $(u_0, w_0)$ is a small perturbation of $(U, W)$ in some topology, then the solution $(u, w)$ will converge to $(U, W)$ time asymptotically.

Following the argument of ^[13], one can easily show that result (1) holds. The aim of this paper is to show the nonlinear stability of steady state. The main difficulty of the problem is the presence of two types of singularities in the model: one is the logarithmic singularity of the sensitivity function, the other is the inverse square singularity of the signal-dependent motility. As in the arguments of ^{[13,27,28,29]}, we relegate the former singularity by using the Cole-Hopf transformation to a nonlinear nonlocal term. However, this transformation is not powerful enough to settle the latter singularity. We shall develop new ideas to deal with the challenge of inverse square singularity of motility. Indeed, we first reformulate the problem in the perturbation variables using the method of anti-derivative, to classify the strength of singularity. Then we construct an appropriately approximate system, which retains some key structures of the original system, to establish the local well-posedness of the perturbation equations. In this step we will first prove that the approximate system is locally well-posed in a time interval $[0, T]$, where $T$ is independent of the artificial parameter $\varepsilon$; and then pass to the limit $\varepsilon\rightarrow0^+$ by using the Aubin-Lions compactness lemma and a diagonal argument. Finally, to close the a priori estimate that is necessary to obtain the global well-posedness of the perturbation equations (or the asymptotic stability of steady state), we establish a new weighted elliptic estimate upon the weighted energy estimates where the weights are artfully chosen according to the nice structures of the equations.

The paper is organized as follows. In Section 2, we present some elementary calculations and state the main results of this paper. In Section 3, we derive the perturbation equations, and establish the local well-posedness theory. Section 4 is devoted to the proof of nonlinear stability of the spiky steady state.

2.   Preliminaries and main results

In this section, we first show the existence of spiky steady state to the system (1.2)–(1.4). Then we present some elementary calculations and state the main results on the asymptotic stability of such spike profile.

Owing to the zero-flux boundary condition for $u$, the mass of bacterial should be conserved. In other words,

Thus, the steady state of (1.2) satisfies

with boundary conditions

Observe that when $W > 0$, the steady state equations (2.2) and (2.3) is actually the $m = \chi = 1$ case of the chemotaxis model studied in ^[13]. Thus, according to Proposition 2.1 and Theorem 2.1 of ^[13], we have the following result.

Proposition 2.1. (1) The system (2.2) and (2.3) has a unique smooth solution $(U, W)$ satisfying $U' < 0$, $W' < 0$, and

(2) $U$ concentrates as a spike at $x = 0$ as $d \to 0^+$, i.e.,

We next pay attention to the asymptotic stability of $(U, W)$ to system (1.2)–(1.4). Because the chemical concentration $w(x, t)$ has vacuum end state at $x = +\infty$, there are two types of singularities in system (1.2): one is the singular sensitivity $\frac{w_x}{w}$, the other is the singular motility $w^{-2}$. To handle the former singularity, motivated by the works of ^{[13,27,28,29]}, we employ the following Cole-Hopf type transformation

which along with the boundary condition $w(0, t) = b$ gives

Hence we transform system (1.2) into a nonlocal system of viscous conservation laws as follows

where $v_0 = -\frac{w_{0x}}{w_0}$. One may observe that the new system (2.7) still has singular motility near $x = +\infty$ for the bacterial mass $u$. In this paper, we shall develop some novel ideas to solve this challenging problem.

We next determine the boundary conditions of (2.7). The second equation of (1.2) gives $({\mathrm {ln}} w)_t = -d v_x+d v^2-u$. Because $b$ is a constant, for smooth solutions $({\mathrm {ln}} w)_t = 0$ at $x = 0$, it then follows that

Denote by $(U, V)(x)$ the steady state of (2.7), where $U(x)$ is given in (2.4). Then we have

It is easy to see that

Because it is expected that $v(x, t)\to V(x)$ as $t \to \infty$, it is natural to impose $v(+\infty, t) = 0$. Therefore, the boundary conditions of (2.7) are

We also need some notation. $H^k$ denotes the usual Sobolev space whose norm is abbreviated as $\|f\|_k^2: = \sum_{j = 0}^k\|\partial_x^jf\|^2$ with $\|f\|: = \|f\|_{L^2({\mathbb{R}}_+)}$, and $H_{\omega}^k$ is the weighted Sobolev space of measurable function $f$ such that $\sqrt{\omega} \partial_{x}^j f \in L^2({\mathbb{R}}_+)$ with norm $\|f\|_{\omega}: = \|\sqrt{\omega} f\|_{L^2({\mathbb{R}}_+)}$ and $\|f\|_{k, \omega}^2: = \sum_{j = 0}^k \|\sqrt{\omega}\partial_{x}^j f \|^2$ for $0\le j \le k$.

We are now ready to state the main results.

Theorem 2.1 (Local well-posedness). Let $(U, V)$ be the steady state of (2.7) and (2.8). Assume that the initial perturbation around $(U, V)$ satisfies $\phi_0(\infty) = \psi_0(\infty) = 0$, where

Suppose that

Then there is a time $T > 0$, such that the system (2.7) and (2.8) has a unique strong solution $(u, v)$ on ${\mathbb{R}}_+ \times (0, T)$, satisfying

where $\omega_1 = U$ and $\omega_2 = \frac{1}{U}$.

Theorem 2.2 (Global well-posedness). Assume that the conditions of Theorem 2.1 hold, and that there exists a constant $\delta_0 > 0$ such that,

where $\omega_3 = \frac{1}{U^2}$ and $\omega_4 = \frac{1}{U^3}$. Then the system (2.7) and (2.8) has a unique global solution $(u, v)(x, t)$ satisfying

Moreover, the following asymptotic convergence hold:

and

Using the Cole-Hopf transformation (2.5), we transfer Theorem 2 to the original system (1.2)–(1.4).

Theorem 2.3. Let $(U, W)$ be the unique steady state of (1.2)–(1.4). Assume that the initial perturbation satisfies $\phi_0(\infty) = \psi_0(\infty) = 0$, where

Suppose that there is a constant $\delta_0 > 0$ such that

Then the system (1.2)–(1.4) has a unique global solution $(u, w)(x, t)$ satisfying

Moreover, we have the following asymptotic convergence:

and

Remark 2.1. We provide both the pointwise convergence and $L^1$ convergence for the solution. In contrast with the result of ^[6] where it is required infinite initial mass for $w$, our Theorem 2.3 implies that the chemical concentration $w$ carries finite mass for all time.

Remark 2.2. In view of its biological background, it is also interesting to study the stability of traveling waves to system (1.1). However, when we apply our argument to that problem, the perturbation equation involves several unfavorable terms which are sophisticated to estimate. We leave this problem for the future study.

Remark 2.3. We shall remark that the steady state $(U, W)$ obtained in Proposition 2.1 is a smooth solution of system (2.2) and (2.3), and it satisfies $U(x) > 0$ and $W(x) > 0$ for any $x\in[0, +\infty)$. In other words, $U(x)$ only vanishes at the far field, and the singularity only happens at $x = +\infty$. This fact enables us to take $\frac{1}{U}$ as the key weight function, and derive the stability of steady state in specific weighted space.

3.   Local well-posedness

This section is devoted to proving Theorem 2.1, i.e., the local well-posedness of system (2.7) and (2.8). We first reformulate the problem in the perturbation variables using the method of anti-derivative. Because the perturbation system still has a singularity, we have to construct an appropriately approximate system. Then we prove that the approximate system is locally well-posed in a time interval $[0, T]$ where $T$ is independent of the artificial parameter $\varepsilon$. After establishing the uniqueness of solutions in weighted Sobolev space, we finally derive the local well-posedness of system (2.7) and (2.8) by the Aubin-Lions compactness lemma and a diagonal argument.

3.1.   Reformulation of the problem

The steady state $(U, V)$ of system (2.7) and (2.8) satisfies

where the boundary conditions are given by

Integrating (3.1) in $x$, we have

Observing that $(u, v)$ satisfies the zero-flux boundary condition, the perturbation around $(U, V)$ should have the conservation of mass. That is

Then we could adopt the method of anti-derivative to decompose the solution $(u, v)$ as

which implies

Substituting (3.5) into (2.7), integrating the equations in $x$, noting $w = e^{-\psi}W$, and using (3.1), we have

which is equivalent to

The initial value of $(\phi, \psi)$ is given by

with

and the boundary condition satisfies

We shall remark that the anti-derivative for $v$ could remove the nonlocality of the problem, but it can not handle the singularity of the motility. Indeed, to overcome the difficulties caused by the singular motility (or degenerate relaxation), we construct an approximate system of (3.6) as

where $\varepsilon > 0$ is a constant, $W_\varepsilon = W+\varepsilon$ and $V_\varepsilon = \frac{W}{W_\varepsilon}\cdot V$. Here we also approximate $V$ by $V_\varepsilon$ so that system (3.10) retains the key structure of system (3.6):

Indeed, recalling that $V = -\frac{W_x}{W}$, a direct calculation leads to

3.2.   Local well-posedness for the approximate system

Employing the principle of contraction mapping (e.g., see ^[30]), one could easily get the local well-posedness for the approximate system on a time interval that may depend on $\varepsilon$.

Proposition 3.1. Assume that the initial data $(\phi_0, \psi_0)$ satisfies

Then, there exists a constant $T > 0$ depending on $\varepsilon$, $\|\phi_0\|_{H^1}$ and $\|\psi_0\|_{H^1}$ such that the approximate system (3.10) with (3.7)–(3.9) has a unique local strong solution on ${\mathbb{R}}_+ \times [0, T]$ satisfying

Proof.

By Proposition 3.1, there exists a time $T_1 > 0$ such that the system (3.10) with (3.7)–(3.9) has a unique solution $(\phi, \psi)$ on $(0, T_1)$ that satisfies $\phi\in C([0, T];H^1)\cap L^2((0, T);H^2).$ Starting at $T_1$, applying Proposition 3.1 again, we can extend the solution $(\phi, \psi)$ to another time $T_2 = T_1+t_1$, where $t_1 > 0$ depends on $\varepsilon$, $\|\phi(T_1)\|_{H^1}$ and $\|\psi(T_1)\|_{H^1}$. Continuing this procedure, we get two sequences $\{t_j\}_{j = 1}^{\infty}$ and $\{T_j\}_{j = 1}^{\infty}$, where $t_j$ depends on $\varepsilon$, $\|\phi(T_j)\|_{H^1}$ and $\|\psi(T_j)\|_{H^1}$, such that the solution $(\phi, \psi)$ exists on the time interval $(0, T_j)$, and satisfies

Take the maximal existing time $T^*$ as $T^* = T_1+\sum_{j = 1}^{\infty} t_j$. Then the solution can be extended to $(0, T^*)$ and satisfies

for any $T\in(0, T^*)$. Clearly, if $T^* < \infty$, then

However, one can not use Proposition 3.1 to derive the local well-posedness of system (3.6) by directly passing to the limit $\varepsilon\rightarrow0$, since the time interval $[0, T]$ obtained in Proposition 3.1 depends on $\varepsilon$. In the following, we have to establish appropriate a priori estimates that are independent of $\varepsilon$.

Proposition 3.2. Assume that $(\phi_0, \psi_0)$ satisfies

Then there exists a constant $T_{0} > 0$ independent of $\varepsilon$, such that the approximate system (3.10) with (3.7)–(3.9) has a unique solution on ${\mathbb{R}}_+\times [0, T_0]$, which satisfies

and

Proof. Thanks to (3.12), it suffices to establish a priori estimate in the weighted Sobolev space that is independent of $\varepsilon$. To achieve this, we multiply the first equation of (3.10) by $\frac{\phi}{W_\varepsilon}$, and integrate the resultant equation over $(0, t)\times(0, +\infty)$ to get

By (3.11) we have

It follows from Young's inequality that

Using the inequality

we get

Combining (3.17) and (3.19), one obtains

Multiplying the second equation of (3.10) by $\psi$, we get

By Young's inequality, we have

and

Moreover, using (3.18) yields

Now substituting (3.22) and (3.23) into (3.21), we have

Combining (3.20) with (3.24) yields

Multiplying the first equation of (3.10) by $\frac{\phi_{xx}}{W_\varepsilon}$, one gets

where we have used the first equation of (3.10) in the second equality. As in (3.19),

From Young's inequality, it follows that

and

Substituting (3.27)–(3.30) into (3.26) leads to

Multiplying the second equation of (3.10) by $\frac{\psi_{xx}}{W_\varepsilon}$, one gets

By Young's inequality,

Moreover, integration by parts leads to

The boundary term can be estimated as

where $\delta$ is a small constant. It follows from (3.18) that

In view of the second equation of (3.10), it holds:

where

and by (3.18),

Then choosing $\delta\ll1$, by (3.31), we get

Combining (3.31) with (3.33), we have

Multiplying (3.25) by $K\gg 1$ and combing the resultant inequality with (3.34), we have

which further gives

Set $H(t): = \int_0^{\infty}(W_\varepsilon\phi^2+\psi^2+W_\varepsilon\phi_x^2 +\frac{\psi_x^2}{W_\varepsilon})dx$. Noting

we are led to

where $H_0 = \int_0^{\infty}(W_\varepsilon\phi_0^2+\psi_0^2+W_\varepsilon\phi_{0x}^2 +\frac{\psi_{0x}^2}{W_\varepsilon})dx$. It is easy to verify that when $\overline{T_0}$ satisfies

then

Indeed, consider

where $M = Ce^{12\sqrt{H_0}}$, then

Since $(1-x)^{-\frac{1}{2}} < 1+x$ for $x\in(0, \frac{1}{2})$, it holds

for $t$ small. Thus, when we take $\overline{T_0}$ satisfying (3.36), we have (3.37).

If we take

then $H_1 > H_0$ and $H_2 < H_0$. Now we take $T_0$ satisfying

Clearly, $T_0$ is independent of $\varepsilon$, and

Thanks to Proposition 3.1, (3.13) and (3.38), for any $0 < \varepsilon < 1$, system (3.10) with (3.7)–(3.9) has a unique solution $(\phi, \psi)$ on ${\mathbb{R}}_+\times (0, T_0)$ satisfying (3.14). The other desired estimate (3.15) follows from (3.38) and an integration of (3.35) in $t$.

3.3.   Local well-posedness for the singular system

Let us now study the local well-posedness of (3.6)–(3.9). We start with the uniqueness of the solutions.

Proposition 3.3. Let $(\phi_1, \psi_1)$ and $(\phi_2, \psi_2)$ be two solutions of system (3.6)–(3.9) satisfying

for $i = 1, 2$. Then $(\phi_1, \psi_1)\equiv (\phi_2, \psi_2)$ on ${\mathbb{R}}_+ \times [0, T]$.

Proof. Define $(\phi, \psi)$ by

Then $(\phi, \psi)$ satisfies

Multiplying the first equation of (3.39) by $\frac{\phi}{W}$, and the second one by $\psi$, summing the resultant equations up, one gets after an integration by parts that

By Young's inequality, we have

and

where $\delta > 0$ is a small constant. A direct calculation yields $-\frac{V}{W}+(\frac{1}{W})_x = 0$, and then we get

Integration by parts leads to

As in (3.42),

As in (3.41),

and

Now substituting (3.41)–(3.48) into (3.40), we arrive at

We next present the estimate for $\int_0^{\infty}\frac{\psi_x^2}{W}dx$. Multiplying the second equation of (3.39) by $\frac{\psi_{xx}}{W}$, we get

Using the second equation of (3.39), we get

and

Thus,

Similarly,

and

Substituting these inequalities into (3.50), we get

Multiplying (3.49) by $K\gg 1$ and combing the resultant inequality with (3.51), we have

It then follows from the Gronwall's inequality that

Therefore, $\phi\equiv0$ and $\psi\equiv0$. We complete the proof.

We are now ready to prove the local existence of solutions to system (3.6)–(3.9).

Proposition 3.4. Assume that $(\phi_0, \psi_0)$ satisfies

Then there exists a constant $T > 0$, such that the system (3.6)–(3.9) has a unique solution $(\phi, \psi)$ on ${\mathbb{R}}_+ \times (0, T)$, which satisfies

where $C$ is a constant independent of $T$.

Proof. Owing to Proposition 3.2, there exists a constant $T > 0$ independent of $\varepsilon > 0$ such that the approximate system (3.10), subject to (3.7)–(3.9), has a unique solution $(\phi_{\epsilon}, \psi_{\epsilon})$ satisfying

where $C$ is a constant independent of $\varepsilon$. Owing to (3.53) and (3.15), passing to the limit $\varepsilon\rightarrow0^+$, applying the Banach-Alaoglu theorem and the diagonal argument, we know that there is a subsequence, still denoted by $(\phi_{\varepsilon}, \psi_{\varepsilon})$, such that for any $r\in(0, \infty)$

Noting $H^2(0, r)$ and $H^1(0, r)$ compactly embed into $H^1(0, r)$ and $L^\infty(0, r)$, respectively, for any $r > 0$, we obtain from the Aubin-Lions compactness lemma that

Observing that $W_\varepsilon\rightarrow W$ and $V_\varepsilon\rightarrow V$ in $C[0, r]$, one can see that the nonlinear terms in (3.10), $e^{2\psi_{\varepsilon}}\phi_{\varepsilon x}\psi_{\varepsilon x}$ and $d\psi_{\varepsilon x}^2$ converge strongly in $L^2((0, T); L^2(0, r))$ to $e^{2\psi}\phi_x\psi_x$ and $d \psi_{x}^2$, respectively. Then one can take the limit as $\varepsilon \to 0$ in (3.10) to derive that $(\phi, \psi)$ satisfies (3.6) in the sense of distribution. Moreover, it follows from the weakly lower semi-continuity of the norms, the first inequality of (3.53) that

Therefore, (3.52) holds, and the proof is complete.

Proof. [Proof of Theorem 2.1] It is a consequence of Propositions 3.3 and 3.4.

4.   Nonlinear stability

In this section, we prove the global well-posedness of strong solutions to the system (3.6)–(3.9), which also implies the nonlinear stability of spiky steady state to the original chemotaxis system (1.2)–(1.4). We construct global solutions of system (3.6)–(3.9) in the more regular space:

for $T\in (0, +\infty]$, where $\omega_2 = \frac{1}{U}$, $\omega_3 = \frac{1}{U^2}$ and $\omega_4 = \frac{1}{U^3}$. Set

Since $U(x)\le \frac{\lambda^2}{6d}$, the Sobolev embedding theorem implies

Moreover, noting

we have

Similarly,

which implies

For system (3.6)–(3.9), we have the following results.

Proposition 4.1. There exists a constant $\delta_1 > 0$, such that if $N(0)\le \delta_1$, then the system (3.6)–(3.9) has a unique global solution $(\phi, \psi)\in X(0, \infty)$ satisfying

for any $t\in [0, \infty)$.

Thanks to the local well-posedness established in Propositions 3.3 and 3.4, we only need to derive the following a priori estimates to prove Proposition 4.1.

Proposition 4.2. Assume that the conditions of Proposition 4.1 hold, and that $(\phi, \psi)\in X(0, T)$ is a solution of system (3.6)–(3.9) for some constant $T > 0$. Then there is a constant $\varepsilon > 0$, independent of T, such that if $N(t)\le \varepsilon$ for any $0 < t\le T$, then $(\phi, \psi)$ satisfies (4.5) for any $0\le t\le T$.

To establish the a priori estimate, we need the following Hardy inequality (see Lemma 3.4 of ^[13] for the proof).

Lemma 4.1. (Hardy inequality) If $f\in H_0^1(0, \infty)$, then for $j\not = -1$, it holds that

where $k > 0$ is a constant.

We start with the $L^2$ estimate.

Lemma 4.2. If $N(t)\ll 1$, then there exists a constant $C > 0$ such that

Proof. We rewrite (3.6) as

Multiplying the first equation of (4.8) by $\frac{\phi}{U}$, the second one by $\psi$, and integrating the resulting equations on $(0, t)\times(0, +\infty)$, we have

By (2.4) and Hardy inequality, we get

Owing to (3.3), it is easy to compute that

which gives

By (4.2) and Young's inequality, we derive that

Similarly, since $\|\psi (\cdot, t)\|_{L^{\infty}} \le N(t)$ and

we have

and

where we have used the Taylor expansion

Now substituting (4.10)–(4.15) into (4.9), noting $V_x < 0$, and using Hardy inequality, we get

Thus, we obtain (4.7) provided $N(t)\ll 1$.

We next establish the $H^1$ estimate.

Lemma 4.3. If $N(t)\ll 1$, then the solution of (3.6)–(3.9) satisfies

Proof. Multiplying the first equation of (4.8) by $\frac{\phi_{xx}}{W^2}$, we get

In view of (2.4), it holds

It then follows from Young's inequality that

Moreover, by (4.13),

Using (4.2), it is easy to see that

By (4.16), the fact that $\|\psi(\cdot, t)\|_{L^{\infty}} \le N(t)$ and (4.2) again, one has

Substituting (4.20)–(4.23) into (4.18), we get

Thus, by (4.19) and Lemma 4.2, when $N(t)\ll1$, we arrive at

Multiplying the second equation of (4.8) by $\frac{\psi_{xx}}{U}$, we have

By Young's inequality,

Moreover, (4.13) gives

By (3.3) and (4.13),

which in combination with (4.2) leads to

and

Now substituting (4.26)–(4.30) into (4.25), we derive that

Then by Lemma 4.2, when $N(t)\ll1$, we have

Combining (4.31) and (4.24), we get (4.17).

The $H^2$ estimate is as follows.

Lemma 4.4. If $N(t)\ll 1$, then it holds

Proof. Differentiating the first equation of (4.8) with respect to $t$ leads to

Multiplying (4.33) by $\frac{\phi_{t}}{U}$ and integrating it in $x$ and $t$, we get

where we have used

By Young's inequality, Hardy's inequality, (4.2) and (4.4), we get

and

Moreover, integration by parts leads to

Using (4.16), (4.2) and (4.4) again, a simple calculation yields

and

Similarly,

and

Now substituting (4.35)–(4.41) into (4.34), we arrive at

By (4.2), (3.6), and Lemmas 4.2 and 4.3, we estimate the last term of (4.42) as

We next estimate $\int_0^t\int_0^{\infty}\frac{\psi_{tx}^2}{U}$. Differentiating the second equation of (3.6) with respect to $t$ leads to

Multiplying (4.44) by $\frac{\psi_t}{U}$, we have

Owing to Young's inequality, we have

and

By the boundedness of $U(x)$ and $V(x)$, we arrive that

Moreover, the fact that $\|\psi_x(\cdot, t)\|_{L^{\infty}}\le C N(t)$ leads to

Substituting (4.46)–(4.49) into (4.45) gives

Combing (4.42) and (4.50), by (4.43), we get

Squaring (3.6) and multiplying the resultant equations by $\frac{1}{U}$, owing to (4.51) and Lemma 4.3, we obtain

and

Differentiating the first equation of (3.6) in $x$ yields

Squaring (4.54) and multiplying the resultant equations by $\frac{1}{U}$ lead to

By (2.4), (4.51), the boundedness of $U(x)$, and Hardy inequality, we have

and

Using (4.16) and Lemmas 4.2 and 4.3, with the fact that $\|\psi(\cdot, t)\|_{L^\infty} \le N(t)$, we get

Substituting (4.56)–(4.58) into (4.55), and using (4.52) and (4.53), when $N(t)\ll1$, one gets

Similarly, differentiating the second equation of (3.6) in $x$ yields

Then squaring (4.60), multiplying the resultant equations by $\frac{1}{U}$, using (4.51) and Lemmas 4.2 and 4.3, we arrive at

From (4.59), (4.61) and (4.51)–(4.53), we get the desired (4.32).

Notice that the estimate (4.32) requires that the initial data satisfies $\frac{\phi_{0x}^2}{U^2} +\frac{\phi_{0xx}^2}{U^3} < \infty$. Hence, to guarantee the extension procedure works, we further need the following weighted elliptic estimate.

Lemma 4.5. If $N(t)\ll 1$, we have

Proof. By (4.1), (4.17) and (4.53), we have

We write the first equation of (4.8) as

Multiplying (4.64) by $-\frac{\phi_x}{U^2V}$, noting

we get

A direct calculation by (3.3) gives

Thus,

We next estimate the RHS of (4.65). By (4.2),

By (4.16),

Then we get from (4.13) and Young's inequality that

Now substituting (4.66) and (4.67) into (4.65), by Lemmas 4.3 and 4.4, we have

which along with Hardy inequality gives

We next write the first equation of (3.6) as

Squaring this equation and multiplying the resultant equation by $\frac{1}{U^3}$, owing to (4.2) again, we obtain

Thus, by (4.68) and Lemma 4.4,

Using (4.3), (4.68) and (4.70), we get

Therefore, (4.62) follows from (4.63) and (4.68)–(4.71). We complete the proof.

Proof. [Proof of Proposition 4.2] It is a direct consequence of Lemmas 4.2–4.5.

Proof. [Proof of Theorem 2.2] The a priori estimate (4.5) guarantees that if $N(0)$ is small, then $N(t)$ is small for all $t > 0$. Thus, applying the standard extension argument, we obtain the global well-posedness of system (3.6)–(3.9) in $X(0, \infty)$. Owing to the transformation (3.5), system (2.7) and (2.8) has a unique global solution $(u, v)(x, t)$ satisfying (2.9).

We next prove the convergence (2.10). We first show that

It suffices to prove that $\|\phi_x(\cdot, t)\|^2\in W^{1, 1}(0, \infty)$ and $\|\psi_x(\cdot, t)\|^2\in W^{1, 1}(0, \infty)$. By Lemma 4.2, we get

By Lemma 4.5, we have $\|\psi(\cdot, t)\|_{L^{\infty}}\le C$ and $\|\frac{\psi_x(\cdot, t)}{\sqrt{W}}\|_{L^{\infty}}\le C$. In view of the first equation of (3.6), there exists a constant $C$ such that

where we have used (4.13) and (4.19). Then integrating (4.74) with respect to $t$ and using (4.5), we get

which, along with (4.73) leads to $\|\phi_x(\cdot, t)\|^2\in W^{1, 1}(0, \infty)$. Thus

Similarly, one has

Using the second equation of (3.6), there is a constant $C > 0$ such that

Then we integrate (4.77) with respect to $t$ and make use of (4.5) to get

which, along with (4.76) implies $\|\psi_x(\cdot, t)\|^2\in W^{1, 1}(0, \infty)$. Thus

(4.72) then follows from (4.75) and (4.78). By Cauchy-Schwarz inequality and (4.5), we get

This implies

Similarly, we have

Thus, (2.10) holds.

Finally, we prove the $L^1$ convergence. By Lemmas 4.2 and 4.4, we get

A simple calculation gives

Integrating (4.80) with respect to $t$ and using (4.79), we obtain

which, along with the first inequality (4.79) yields $\int_0^{\infty}\frac{\phi_{x}^2(x, t)}{U}dx\in W^{1, 1}(0, \infty)$. And then

Thus, from Hölder inequality and the fact that $\int_0^{\infty}Udx < \infty$, it follows that

This yields the convergence (2.11).

Proof. [Proof of Theorem 2.3.] We just need to pass the results from $v$ to $w$ to complete the proof of Theorem 2.3. The transformation (3.5) and Theorem 2.2 give the regularity of $\frac{w_x}{w}-\frac{W_x}{W}$.

Next, we derive the results of $w-W$. Let $\xi: = w-W$. Owing to (2.5) and (3.5), it is easy to calculate that

Thus, $\xi = W(e^{-\psi}-1)$ and $\xi_x = W_x(e^{-\psi}-1)-We^{-\psi}\psi_x$, which gives the regularity of $w-W$.

It is left to show the convergence. By Cauchy-Schwarz inequality and (4.5), we get

Noting $\|\psi(\cdot, t)\|_{L^{\infty}}\le N(t)\ll 1$, the Taylor expansion gives

Therefore, by (4.78), we get

For the $L^1$ convergence, noting $\int_0^\infty W(x)dx < \infty$, it follows from Hölder inequality and Hardy inequality that

We complete the proof of Theorem 2.3.

5.   Summary and discussion

We are concerned with the existence and stability of spiky patterns to the chemotaxis model (1.1) proposed by Kim and his collaborators ^[1,2]. This model was derived from the notion of "metric of food" which measures the amount of food. It avoids the mysterious assumption that the microscopic scale bacteria sense the macroscopic scale gradient of food. Moreover, this model also admits two types of traveling waves: traveling band and traveling front, under suitable assumptions on the consumption rates. Hence, it can be viewed as an alternative model to describe the propagation of traveling bands of bacteria observed in the experiment of Adler ^[3]. However, since the traveling wave of oxygen $W$ vanishes at far field, one has to encounter the challenge of presence of two types of singularities in the study of stability of traveling waves. As the first step we investigate instead the stability of stationary waves to the model in the half space. In this case the model remains singular at the far field. We successfully find an effective strategy to handle the two types of singularities. In the following studies, we will apply the strategy of this paper to study the stability of traveling waves of the model by modifying some estimates.

The potential biological application of our results is the explanation of formation of a plume pattern for aerobic bacteria observed in the experiment of ^[7], where the bacteria consume oxygen in a water drop. We conjecture that this plume pattern is a superposition of series of one dimensional spikes. However, owing to the lack of effective mathematical tools to handle the stability of biological patterns to a chemotaxis-fluid model, we consider a simplified fluid free chemotaxis model. And we expect that our argument is effective for more general chemotaxis models and even some chemotaxis-fluid models.

Acknowledgments

The authors are grateful to the three referees for their insightful comments and suggestions, which lead to great improvements of our original manuscript. This work is supported by the Natural Science Foundation of Jilin Province (20210101144JC).

Conflict of interest

The authors declare there is no conflict of interest.

Appendix

Proof. [Proof of Proposition 3.1] The local existence can be proved using the principle of contraction mapping. Set

equipped with norm

Define a mapping $\mathcal{Z}$: $(\hat{\phi}, \hat{\psi})\in \mathcal{Y}_T \mapsto \mathcal{Z}(\hat{\phi}, \hat{\psi})$ such that $(\phi, \psi) = \mathcal{Z}(\hat{\phi}, \hat{\psi})$ is a solution of

with the initial and boundary conditions (3.7)–(3.9). Taking a ball

where $M$ is a constant to be determined later. We shall show that there are $M$ and $T$ such that (i) $\mathcal{Z}$ maps $B_{M, T}$ into itself; (ii) $\mathcal{Z}$ is a contraction in $B_{M, T}$.

We first show (i). According to the standard linear parabolic theory, for any $(\hat{\phi}, \hat{\psi})\in \mathcal{Y}_T$, the second equation of (A1) has a unique strong solution $\psi$. Substituting $\psi$ into the first equation, we obtain the existence of strong solution $\phi$. Hence the mapping $\mathcal{Z}$ is well-defined.

We next derive the estimates for $(\phi, \psi)$. Multiplying the second equation of (A1) by $\psi$ gives

where

Then choosing $T\leq\frac{1}{2}$, we get

Similarly, multiplying the second equation of (A1) by $\psi_{xx}$ leads to

where we have used

By (A2) and (A3), if we take $M^2\geq2\int_0^{\infty}(\psi_{0}^2+\psi_{0x}^2)$ and chose $T$ small enough, then

Multiplying the first equation of (A1) by $\phi$ gives

where

and

If $e^{4M}M^4T\leq\frac{\varepsilon^2}{2}$, we get from (A5) that

Similarly, multiplying the first equation of (A1) by $\phi_{xx}$ gives

where

Then choosing $T\leq\frac{\varepsilon^{2}}{2}$, we have

In view of (A2), (A3), (A6) and (A7), we choose $M$ and $T$ satisfying

then $\|(\phi, \psi)\|_{{\mathcal{Y}}_{T}}\le M$, which verifies (i).

We proceed to show (ii). For any $(\hat{\phi}_1, \hat{\psi}_1)$, $(\hat{\phi}_2, \hat{\psi}_2)\in B_{M, T}$, set $(\phi_1, \psi_1) = \mathcal{Z}(\hat{\phi}_1, \hat{\psi}_1)$, $(\phi_2, \psi_2) = \mathcal{Z}(\hat{\phi}_2, \hat{\psi}_2)$ and $(\bar{\phi}, \bar{\psi}): = (\phi_1, \psi_1) -(\phi_2, \psi_2)$. Then $(\bar{\phi}, \bar{\psi})$ satisfies

with zero initial-boundary conditions. Multiplying the second equation of (A8) by $\bar{\psi}$ gives

Thus, choosing $T\leq\frac{1}{2}$, we get

Multiplying the second equation of (A8) by $\bar{\psi}_{xx}$, and noting

we get after choosing $T\leq\frac{1}{2}$ that

This also implies

Multiplying the first equation of (A8) by $\bar{\phi}$, noting

we have

Multiplying the first equation of (A8) by $\bar{\phi}_{xx}$, noting

and

we have

Therefore, owing to (A9)–(A10), we can take $T$ small enough to derive

which verifies (ii).

Now we apply the contraction mapping principle to obtain that system (3.10) has a solution. The uniqueness follows from a similar argument as (A11) and the Gronwall's inequality.