Initial and Boundary Value Problem for a System of Balance Laws from Chemotaxis: Global Dynamics and Diffusivity Limit

Zefu Feng, Jiao Xu, Ling Xue and Kun Zhao

1 School of Mathematical Sciences, Chongqing Normal University,Chongqing 400047, China

2 SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China

3 College of Mathematical Sciences, Harbin Engineering University,Harbin 150001, Heilongjiang, China

4 Department of Mathematics, Tulane University, New Orleans,LA 70118, USA

Abstract. In this paper, we study long-time dynamics and diffusion limit of large-data solutions to a system of balance laws arising from a chemotaxis model with logarithmic sensitivity and nonlinear production/degradation rate. Utilizing energy methods, we show that under time-dependent Dirichlet boundary conditions, long-time dynamics of solutions are driven by their boundary data,and there is no restriction on the magnitude of initial energy. Moreover, the zero chemical diffusivity limit is established under zero Dirichlet boundary conditions, which has not been observed in previous studies on related models.

Key words: Balance laws, chemotaxis, initial-boundary value problem, dynamic boundary condition, strong solution, long-time behavior, diffusivity limit.

1 Introduction

1.1 Overview

This paper is oriented around the initial-boundary value problem:

where γ>1 and ε are constant parameters, and p(x), q(x), α(t), and β(t) are given functions. Assuming appropriate conditions for p(x),q(x),α(t)and β(t),we establish global stability of strong solutions to(1.1). Moreover, when α(t)=β(t)≡0,we show that solutions to (1.1) with ε>0 converge to that with ε=0, as ε→0, in certain topology.

1.2 Background

The system of balance laws in problem(1.1)is derived from the following chemotaxis model of Keller-Segel type with logarithmic sensitivity:

Derivation of the system of balance laws in (1.1) from (1.2) can be realized by

first performing the following change of variables and rescaling:

after which (1.2) becomes

Then, taking χ=?1, D=1 and μ<0 in (1.3), one obtains

Note that the case of χ<0 and μ<0 corresponds biologically to the chemotactic process in which the chemotaxis is repulsive and the chemical signal is being produced. Moreover, since the specific values of χ<0 and D>0 do not affect the qualitative behavior of the solution,our choice of the parameter values is to simplify the presentation.

1.3 Literature review

In an effort of studying the chemotactic movement of organisms,such as myxobacteria, that deposit little- or non-diffusive chemical signals modifying succeed passages in local environment, Othmer-Stevens [29] and Levine-Sleeman [16] proposed the model:

which corresponds to (1.2) when ε=0 and ψ(u)=u. Among the first generation of analytical studies of (1.5),we refer the reader to[5–7,18–20,37]for results concerning global well-posedness and large-time behavior of large-data solutions in one space dimension, and to [13,21,22] for nonlinear stability of one-dimensional traveling wave solutions.

Recently, the appended version of (1.5)(by adding diffusion to the second equation): has been investigated in a series of works[8,19,23,24,28,30,33,36],where in addition to global well-posedness, large-time behavior of large-data solutions and nonlinear stability of traveling waves in one space dimension, the vanishing diffusivity limit(as ε→0) and boundary layer formation associated with one-dimensional solutions are studied. There are also similar results in multi-dimensional spaces, and we refer the reader to [9,10,31,32,35] for details.

More recently, the authors of [38] studied another version of (1.2):

where γ>1 is a constant. Applying the same transformation and scaling as above to (1.7), one obtains the system of balance laws in (1.1). In [38], it is shown that the Cauchy problem of the transformed model possesses global (in time) large-data solutions with γ-dependent regularity, and the chemically diffusive (ε>0) and nondiffusive (ε=0) solutions are consistent in the limiting process as ε→0.

One of the major contributions of the results reported for the 1D models (1.5),(1.6) and (1.7) is the global stability of constant equilibrium solutions subject to various initial and/or boundary conditions. The proofs are constructed by utilizing L-based energy methods. One of the key ingredients consists in the implementation of entropy-entropy fluxes associated with the models,which is inspired by analytical techniques in hyperbolic conservation/balance laws.

It is also worth mentioning that very recently the following model:

1.4 Motivation and goal

In this paper, we continue the analytical study of (1.7) by considering the model on a finite interval and studying the initial-boundary value problem (1.1) for the transformed model. A search in the database shows that such a problem has not been investigated in the literature. In real world, physical/biological processes always occur in bounded regions with constraints from boundaries, where initial-boundary value problems appear. Solutions to initial-boundary value problems usually exhibit different behaviors and much richer phenomena comparing with the Cauchy problem.

Our first work targets the long-time dynamics of large-data solutions to (1.1)when α(t) and β(t) are non-trivial functions. One of the motivations of our study comes from the observation that boundary conditions oftentimes vary in time in many in vivo environments such as tumor angiogenesis (c.f. [30]). Since model (1.7)is deeply connected with (1.5) and (1.6) which have found applications in modeling tumor angiogenesis (see e.g., [17]), the problem considered in this paper becomes biologically relevant. Moreover, note that under dynamic boundary conditions, the initial-boundary value problem (1.1) can be reformulated as

where ?p=p?α(t), ?q=q?β(t), and the feedback controls are given by F=α(t)?q+β(t)?p?α(t) and F=2εβ(t)?q+(ψ(?p+α(t))?ψ(?p))?β(t). Hence, the perturbed problem (1.9) can be viewed as a control problem of (1.1) subject to zero Dirichlet boundary conditions with control actions Fand F. As a typical question in control problem, one would ask: For what kind of feedback controls Fand Fdoes the closed-loop system (1.9)stabilize, i.e., its solution (?p,?q)tends to zero in an appropriate topology,as t→∞? We aim to give a definite answer to such a question in this paper.

The second goal of this paper is to investigate the zero chemical diffusivity limit of solutions to (1.1) when p and q satisfy the homogeneous Dirichlet boundary conditions. In all of the previous studies concerning initial-boundary value problems for model(1.6),it has been shown that if p is supplemented with Dirichlet(including time-dependent) boundary conditions, then the transformed variable q=c develops boundary layers at endpoints of the spatial interval as ε→0, see [8,9,19,30]. The inconsistency between chemically diffusive and non-diffusive solutions is caused by the mismatch of boundary conditions for the models with ε>0 and ε=0,respectively.Naturally, one would expect that the singular phenomenon also occurs to solutions to (1.1), no matter what condition α(t) is prescribed with. Indeed, by adopting the energy method in this paper and the idea of effective viscous flux in [30], one can show that as long as the boundary datum of p is not identical to zero, boundary layers develop at endpoints of the spatial interval. On the other hand, however, we discovered that this is not the case when p=0 at x=0 and x=1. Roughly speaking,since(p)=γpp,the zero boundary condition for p generates the same boundary condition for (p)when γ>1. Hence, q=0 at the endpoints when ε=0. This is a strong indication, based on previous studies in [19,36],that the chemically diffusive solutions converge to the non-diffusive one as ε→0. Such a phenomenon is new in this specific research area, and we give it a rigorous demonstration in this paper.

1.5 Statement of results

Now we state the main results of this paper. The first two theorems demonstrate that large-data solutions to (1.1) stabilize as t→∞. The first theorem is concerned with the case when ε>0.

Theorem 1.1. Consider the initial-boundary value problem (1.1) when ε>0. Suppose that the initial data satisfy p>0, (p,q)∈H((0,1)) and are compatible with the boundary conditions. Assume that

? there exist constants α,α,β, such that 0<α ≤α(t)≤α and |β(t)|≤β, for all t≥0,

? (α(t),β(t))∈L(0,∞)∩L(0,∞).

Then for any fixed ε>0 and γ>1 there exists a unique global-in-time strong solution(p,q), such that (p?α(t),q?β(t))∈L(0,∞;H(0,1))∩L(0,∞;H(0,1)) and

When ε=0, we have the following:

Theorem 1.2. Consider the initial-boundary value problem (1.1) when ε=0. Suppose that the initial data satisfy p>0,(p,q)∈H((0,1))and are compatible with the boundary conditions. Assume that there exist constants α,α,such that 0<α≤α(t)≤α for all t≥0, and α(t)∈L(0,∞)∩L(0,∞). Then for any fixed γ ≥2 there exists a unique global-in-time strong solution (p,q), such that (p?α(t))∈L(0,∞;H(0,1))∩L(0,∞;H(0,1)), (q?ˉq)∈L(0,∞;H(0,1))∩L(0,∞;H(0,1)) and

where ˉq denotes the spatial average of q.

We have several remarks regarding Theorem 1.1 and Theorem 1.2.

Remark 1.2. The assumptions for α(t) and β(t) in Theorem 1.1 and Theorem 1.2 imply that the boundary data converge to constants as time goes to infinity. By utilizing the energy methods in this paper, we can show that when there are no variations in the boundary data, i.e., constant Dirichlet boundary conditions, the perturbations converge exponentially rapidly to zero as time goes to infinity. In this case, quantities in the energy estimates involving derivatives of the boundary data vanish,and exponential decay of the perturbations follow from Poincar′e’s inequality.We omit the technical details to simplify the presentation.

The third theorem is concerned with the consistency between the chemically diffusive and non-diffusive solutions under the homogeneous Dirichlet boundary conditions, which has not been observed in previous studies for related models.

Theorem 1.3. Consider the initial-boundary value problem (1.1)with α(t)=β(t)≡0.Suppose that the initial data satisfy p>0, (p,q)∈H((0,1)) and are compatible with the boundary conditions. Then for any fixed γ ≥2 and ε ≥0, there exists a unique global-in-time strong solution (p,q), such that for any 0

(p,q)∈L(0,T;H(0,1))∩L(0,T;H(0,1)) when ε>0, and p ∈L(0,T;H(0,1))∩L(0,T;H(0,1)) and q∈L(0,T;H(0,1))∩L(0,T;H(0,1)) when ε=0. Moreover,let (p,q) and (p,q) denote, respectively, the solution with ε>0 and ε=0, with the same initial data. Then for any t>0 it holds that

where the constant C(t)>0 is independent of ε and remains finite for any finite t>0.

The rest of the paper is devoted to proof of Theorems 1.1–1.3. We utilize entropyand L-based energy methods. Similar to the proof constructed in [38], the point of departure is to implement an entropy-entropy flux pair associated with the system of balance laws in (1.1), which consists of the first order Taylor expansion of a power function of the unknown function p around its boundary datum α. The entropy estimate provides a uniform(with respect to t)bound of the Lnorm of the perturbation around boundary data. However, the first order dissipation associated with p, generated from the entropy flux, may be degenerate (c.f. (2.14)). This is caused by the lack of a strictly positive a priori lower bound of p, which can not be achieved by the entropy estimate. To gain a non-degenerate dissipation mechanism of p, we then perform a series of nonlinear energy estimates. The detailed estimates differ case by case, depending on the value of γ. The proof is carefully crafted by examining some fine properties of power function. On the other hand,because of the time-dependency of the boundary data, the proof constructed in this paper is more involved than the one devised in [38] where the equilibrium solution is constant.We put a considerable amount of effort to make the energy framework developed in[38]fit into the complex situation encountered in this paper. Due to the structural difference between the chemically diffusive(ε>0)and non-diffusive(ε=0)problems,the proof of Theorem 1.1 is different from that of Theorem 1.2. Major difference consists in the estimation of the first order spatial derivative of the solution. For Theorem 1.1,the proof takes advantage of the positivity of chemical diffusivity,while for Theorem 1.2 we extract a non-homogeneous wave type equation for q, which serves as the foundation for deriving the desired energy estimates of the solution.Lastly, Theorem 1.3 is proved by utilizing the energy frameworks developed for Theorems 1.1–1.2 and exploring the zero boundary condition.

2 Proof of Theorem 1.1

This section is denoted to the proof of Theorem 1.1. First of all, using standard arguments (see e.g., [25–27,34]), one can show that under the assumptions of Theorem 1.1 there exists a unique local solution to (1.1) with ε>0, such that(p,q)∈L(0,T;H(0,1))∩L(0,T;H(0,1)) for some T∈(0,∞), and p(x,t)>0 for any (x,t)∈(0,1)×(0,T). The technical details are omitted to simplify the presentation. This section is largely devoted to deriving a priori estimates of the local solution, which not only extend the local solution to a global one, but also play an important role in investigating the global stability of the solution. The following technical lemmas are frequently utilized in the subsequent analysis (c.f. [11,12,38]).

Lemma 2.1. Let a≥?1 and λ≥2. Then it holds that

Lemma 2.2. Let a≥?1 and λ≥2. Then it holds that

Lemma 2.3. Let a≥?1 and λ>1. Then it holds that

Lemma 2.4. Let a≥0 and 0<λ≤1. Then it holds that

Next, we establish the a priori estimates of the local solution utilizing entropyand L-based energy methods.

Notation 2.1. Throughout the rest of the paper, unless otherwise specified, we use C to denote a positive generic constant which is independent of the unknown functions and time. The value of the constant may vary line by line according to the context.

2.1 Entropy estimate

We first derive a uniform estimate based on the entropy-entropy flux pair associated with the initial-boundary value problem (1.1). For this purpose, let us define for any non-negative functions f,g ∈L,

Since γ>1, then it follows from Taylor’s theorem that E(f,g)≥0.

For the first term on the right-hand side of (2.2), using the first equation in (1.1)and noting that α is independent of x, we can show that

Adding (2.7) to (2.5), we obtain

Step 2. In this step,we derive a bound for the Lnorm of p in terms of E(p,α). The idea is to make use of the convexity of the entropy expansion, E(p,α), and compare it with a linear function of p. For this purpose, we set

By direct calculations, we can show that

for p>0, α>0. Moreover, since

Step 3. Utilizing (2.9), we derive from (2.7) that

Using the assumptions in Theorem 1.1 and applying Gr¨onwall’s inequality to(2.10),we have

Combining (2.11) and (2.12) completes the proof of Lemma 2.5.

Next, we derive the key estimate in this paper.

2.2 L2 estimate

Letting ?p=p?α, ?q=q?β, the IBVP (1.1) is reformulated as

which implies

Lemma 2.6. Under the conditions of Theorem 1.1 for any γ>1, ε>0, and t>0,there exists a positive constant C, which is independent of t and ε such that

We divide the proof of Lemma 2.6 into four sub-cases, namely, 1<γ<2,2≤γ≤3,3<γ ≤4, and γ>4, since the technical details differ case by case.

2.2.1 Proof of Lemma 2.6 when 1<γ<2

Step 1. Multiplying the first equation of (2.13) by (?p+α)?αand the second equation by α?q, after integrating by parts, we obtain

Step 3. We have the following observations regarding G(t), H(t) and K(t). First,by virtue of (2.23), we can show that

Using Lemma 2.4 and the Cauchy-Schwarz inequality, we can show that

Using (2.29), we update (2.28) as

Since 0<α≤α≤α, using the Cauchy-Schwarz inequality, we can show that

In view of (2.27), we see that there exists a generic constant C>0, such that

Step 4. We note that the third term on the right-hand side of (2.30)is non-positive.To compensate such a term, we add (2.25) to (2.26) to get

where we used (2.32) and arguments similar to those in (2.31), and

According to (2.30), we have

by using which we update (2.33) as

For the first two terms on the right-hand side of(2.36),by using the same method[38,Lemma 5.1], we can show that Applying Gr¨onwall’s inequality to (2.39) and using (2.12) and the assumptions in Theorem 1.1, we can show that

According to (2.27), the definition of X(t), and (2.34), we conclude that

This completes the proof of Lemma 2.6 when 1<γ<2.

2.2.2 Proof of Lemma 2.6 when 2≤γ ≤3

Step 1. Using (2.14), (2.15), and the assumptions for α, we update (2.10) as

Since 2≤γ ≤3, choosing a=(?p+α)/α in Lemma 2.3 and using Young’s inequality,we deduce that

which implies

where the lower bound of α is used. Using (2.41), we update (2.40) as

For the first term on the right-hand side of (2.43), we use H¨older’s inequality to derive

where we applied (2.19) when deriving the last inequality. Hence, we have

Moreover, since ?p|=0, we can show that

Using (2.45) and (2.46), we update (2.44) as

When 2≤γ ≤3, applying Young’s inequality, we can show that

Using Young’s inequality, we estimate the last two terms on the right-hand side of(2.51) as

Applying Gr¨onwall’s inequality and using the assumptions in Theorems 1.1 and(2.14), we can show that

2.2.3 Proof of Lemma 2.6 when 3<γ ≤4

Note that since 0<α≤α≤α and 3<γ ≤4, then

Using (2.57), we update (2.56) as

where ρ=γ(γ?2)max{1,α}. Next, we derive a similar estimate as (2.50).

Step 2. Similar to (2.43), we can show that

Noting that

we update (2.66) as

which yields

which implies

Integrating (2.70)from 0 to t, using (2.14)and the assumptions in Theorem 1.1, we obtain (2.20). This competes the proof of Lemma 2.6 when 3<γ ≤4.

2.2.4 Proof of Lemma 2.6 when γ>4

Step 1. In this case, (2.56) is still valid. Since γ>4, using Young’s inequality, we have

where

Similar to (2.58), we have

Step 2. Multiplying the first equation of (2.13) by (γ+1)|?p|?p, then integrating by parts with respect to x over (0,1), we obtain

Since γ>4 and ?p+α>0, we estimate the first term on the right-hand side of (2.72)as

where we used (2.19). This implies

Substituting (2.74) into (2.73), we have where δ>0 is a constant to be determined, and we used the inequality: |?p|≤|?p|+1 due to γ>4,and Young’s inequality. For the second term on the right-hand side of (2.72), using (2.19), we can show that

Substituting (2.75) and (2.76) into (2.72), we obtain

Using Young’s inequality, we can show that

Hence, we update (2.77) as

Choosing

we obtain from (2.79) that

Integrating(2.80)with respect to time,using(2.14)and the assumptions in Theorem 1.1, we obtain (2.20). This competes the proof of Lemma 2.6 when γ>4.

2.3 H1 estimate

Now we improve the regularity of the solution.

Lemma 2.7. Under the assumptions in Theorem 1.1, there exists a constant C>0,which is independent on t, such that

Proof. Taking Linner products of the first equation of (2.13) with ??p, and the second with ??q, respectively, then adding the results, we have

Using the Cauchy-Schwarz, Sobolev and Poincar′e inequalities, we can show that

For R, by the Cauchy-Schwarz inequality, we have

When 1<γ ≤4, using Sobolev and Poincar′e inequalities, we can show that

where we also used (2.19) and Young’s inequality. When γ>4, by using (2.74), we have

where we used similar arguments as in (2.75). For R, it is straightforward to show that

and

Applying Gr¨onwall’s inequality to (2.85) and (2.86), and using the assumptions in Theorem 1.1 and estimates established in Subsections 2.1 and 2.2, we see that the estimate (2.81) holds. This completes the proof of Lemma 2.7.

Now we show the asymptotic stability of the solution to complete the proof of Theorem 1.1.

2.4 Global stabilization

Next, testing the two equations in (2.13) by ?p and ?q, respectively, we have

from which we deduce that

3 Proof of Theorem 1.2

In this section,we study the global dynamics of large-data solutions to the chemically non-diffusive problem. Letting ?p=p?α, ?q=q?ˉq,where ˉq denotes the spatial average of q, then we have a new initial-boundary value problem for (?p,?q):

Hence, ?q satisfies Poincar′e inequality.

To prove Theorem 1.2, we first recall that the energy estimates in Lemma 2.5 and Lemma 2.6 are independent of ε. Moreover, it can be readily checked that the arguments in Subsections 2.1–2.2 are valid (indeed simpler) if β(t) is a constant.Hence, by replacing β(t) by the constant ˉq and repeating the arguments in those two sections, one can establish the following lemma.

Lemma 3.1. Under the conditions of Theorem 1.2, for any γ ≥2 and t>0, the solution to (3.1) satisfies the following energy estimates:

where C is a positive constant which is independent of t.

The rest of this section is largely devoted to the H-estimate of (?p,?q). Note that when deriving the H-estimate in Subsection 2.3, we used the Cauchy-Schwarz inequality by taking advantage of the positivity of ε, see (2.83) and (2.84). Apparently, such an approach does not work when ε=0. Hence, we need to develop a different method to deal with such a degeneracy. The major result of this section is recorded in the following lemma.

Lemma 3.2. Under the conditions of Theorem 1.2, for any fixed γ ≥2 and t>0, it holds that

where C is a positive constant which is independent of t.

To overcome the technical barrier brought about by the lack of chemical diffusion,we take ?to the second equation of (3.1), multiply the first equation by γα,then take difference of the resulting equations to get

Multiplying (3.3) by ?qand integrating by parts, we can show that

Next, we carry out energy estimates for J–J. The proof is divided into three subsections for the cases: 2≤γ<3, 3≤γ ≤4, and γ>4.

3.1 Estimates of Jk when 2≤γ<3

Step 1. Note that when 2≤γ<3, it holds that

Again, when 2≤γ<3,by Young’s inequality, we have |?p|≤|?p|+1. Then it holds that

Using (3.5), integration by parts, and the assumptions for α, we estimate Jas

where we also used the second equation of (3.1). For J, using the Cauchy-Schwarz inequality, we can show that

where δ>0 is a constant to be determined later. Similarly, it holds that

Since 2≤γ<3, it holds that (?p+α)≤|?p|+α. Then we estimate Jas

Since 2≤γ<3, then 0≤2(γ?2)<2. It follows from Young’s inequality that

Since ?p|=0, then using Poincar′e’s inequality, we can show that

Using (3.11) and (3.13), we update (3.14) as

Note that by Taylor’s theorem, (?p+α)?α=(γ?1)w?p, where w is between ?p+α and α. Since ?p+α≥0 and α>0, we must have 0≤w≤|?p|+α. Moreover, since 2≤γ<3, it holds that w≤(|?p|+α). Hence, we have

So we update (3.18) as

where we used the lower bound of α. Next, we turn to the estimate of ?p.

Step 2. Taking Linner product of the first equation of (3.1) with ??p, we can show that

where we used similar arguments as those in deriving(3.9). After rearranging terms,we have

Note that the quantity inside the temporal derivative in (3.23)may not be positive.In order to apply Gr¨onwall’s inequality,we make coupling of (3.23)with the entropy estimate (2.8).

Step 3. First, we observe that the quantity inside the temporal derivative in (3.23)satisfies

Next, we recall the entropy estimate (2.8) which, for (3.1), reads

where we applied the Cauchy-Schwarz and Poincar′e’s inequalities to the last term on the right-hand side. Applying Gr¨onwall’s inequality to (3.29) and using Lemma 3.1 and the assumptions for α, we can show that

where the constant C>0 is independent of t. By (3.28) and definition of H(t),(3.30) yields (3.2). This completes the proof of Lemma 3.2 when 2≤γ<3.

3.2 Estimates of Jk when 3≤γ ≤4

Step 1. Similar to (3.6), we can show that

For the Lnorm on the right-hand side of (3.31), using ?p|=0, we can show that

where we used Lemma 3.1 when deriving the last inequality. Since x ∈(0,1) is arbitrary, the estimate (3.32) implies

Note that since 3≤γ ≤4, it follows from Young’s inequality that

The estimates of J, Jand Jare identical to (3.7), (3.8) and (3.9), respectively.Next, we re-estimate Jand J.

Step 2. For J, since 3≤γ ≤4, the function zis convex on [0,∞). This implies

Then we can show that

For the first term on the right-hand side of (3.37), using (3.34), we can show that

Then we update (3.38) as

where we used Young’s inequality. For the second term on the right-hand side of(3.37), we can show that

Using (3.39) and (3.40), we update (3.37) as

For J, similar to (3.16), we can show that

For the first term on the right-hand side of (3.42), using (3.35), we can show that

For the second term on the right-hand side of (3.42), we can show that

By repeating the arguments in Step 2 and Step 3 in Subsection 3.1,we can establish(3.2). This completes the proof of Lemma 3.2 when 3≤γ ≤4.

3.3 Estimates of Jk when γ>4

Similar to Subsection 3.2, we only need to estimate J, Jand J. For J, it follows from (3.31) that Note that the right-hand side of (3.47)is uniformly integrable with respect to time,thanks to Lemma 3.1. For J, recalling (3.37), we have

where we used Young’s inequality: |?p|≤|?p|+1. For the first term on the right-hand side of (3.48), we can show that

where we used Young’s inequality due to γ>4. Using (3.50), we update (3.49) as

where we used Young’s inequality at various places. The estimate of the second term on the right-hand side of (3.48) is identical to (3.40). Hence, we obtain

Following the arguments in Subsection 3.1 and using Lemma 3.1, we can establish(3.2). This completes the proof of Lemma 3.2 when γ>4.

Lemma 3.1 and Lemma 3.2 provide the desired energy estimates for the solution to (3.1), as stated in Theorem 1.2. Long-time behavior of the solution can be established via the same method in Subsection 2.4 for the diffusive problem. We omit the technical details to simplify the presentation. This completes the proof of Theorem 1.2.

4 Proof of Theorem 1.3

In this section, we give a sketched proof of Theorem 1.3. Recall the IBVP:

4.1 Energy estimates

Step 1. For any ε≥0,testing the first equation of(4.1)withpand the second equation with q, and adding the results, we can show that

where C is independent on t and ε. Multiplying the first equation of (4.1) by p, integrating by parts,and applying interpolation and the Cauchy-Schwarz inequalities,we can show that which, together with (4.2), implies

Applying Gr¨onwall’s inequality to (4.3), we infer that

where C(t) is increasing with respect to t, but is independent of ε. Moreover, using similar arguments in previous sections, we can show that

where the constant C(t) is increasing with respect to t, but is independent of ε.

Step 2. Note that since γ≥2 and p|=0, then it holds that (p)|=0. Also note that when ε>0, since q|=0, it holds that q|=0. Hence, the second equation of (4.1) implies q|=0. For any ε ≥0, taking Linner product of the first equation of (4.1) with ?p, differentiating the second one with respect to x, then taking Linner product of the resulting equation with q, and using the boundary conditions, we can show that

where integration by parts is applied to obtain Iwhen ε>0. Using interpolation,Sobolev, and Young inequalities and (4.2), we can show that

and

Step 3. When 2≤γ ≤3, since p≤p+1, we can show that

Substituting (4.7), (4.8),(4.13)and(4.14)into(4.6),we get(4.11), and then(4.12).Step 5. When γ>4, using Young’s inequality, we can show that

Since p≥0 and p|=0, we can show that

Substituting the above estimate into (4.15) and invoking interpolation inequality,we deduce

Lastly, using (3.50), we can show that

Substituting (4.7), (4.8), (4.16) and (4.17) into (4.6), we get

Applying Gr¨onwall’s inequality and invoking (4.2), (4.4) and (4.5), we arrive at(4.12).

It should be stressed that for all γ ≥2, the constants in the energy estimates(4.2), (4.4), (4.5) and (4.12) are independent of ε and remain finite for any finite time. Therefore, the energy estimates stated in Theorem 1.3 are established.

4.2 Diffusion limit

where C(t)>0 is independent of ε and remains finite for any finite time. The argument leading to (4.18) is in the same spirit of [38], and we omit the details to simplify the presentation. This completes the proof of Theorem 1.3.

Acknowledgements

The authors would like to thank the anonymous referees for constructive comments and suggestions which help improve the quality of the paper. Z.-F.Feng was partially supported by China Scholarship Council (No. 201906150159). J. Xu was partially supported by China Scholarship Council (No. 201906150101) and National Natural Science Foundation of China(No.11971176 and No.11871226). L.Xue was partially supported by Fundamental Research Funds for the Central Universities of China(No. 3072020CFT2402). K. Zhao was partially supported by Simons Foundation Collaboration Grant for Mathematicians (No. 413028).