On the logarithmic Keller-Segel-Fisher/KPP system

We consider the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. We study the global well-posedness, long-time behavior, vanishing coefficient limit and decay rate of solutions in \begin{document}$ \mathbb{R} $\end{document} . By utilizing energy methods, we show that for any given classical initial datum which is a perturbation around a constant equilibrium state with finite energy (not small), there exists a unique global-in-time solution to the Cauchy problem, and the solution converges to the constant equilibrium state, as time goes to infinity. Under the same initial condition, it is shown that the solution with positive chemical diffusion coefficient converges to the solution with zero chemical diffusion coefficient, as the coefficient goes to zero. Furthermore, for a slightly smaller class of initial data, we identify the algebraic decay rates of the solution to the constant equilibrium state by employing time-weighted energy estimates.

1. Introduction. We consider the following system of partial differential equations: x ∈ R, t > 0, (1.1) where u(x, t) and v(x, t) are unknown functions, and r > 0, ε 1 ≥ 0 and ε 2 ≥ 0 are constants. The purpose of this paper is to study the qualitative behavior (global well-posedness, long time behavior, vanishing coefficient limit, decay rate) of large data solutions to the Cauchy problem of the model under generally prepared initial conditions.
Since the classic Fisher/KPP equation is included in (1.2), by following standard terminologies, we shall term the model as Keller-Segel-Fisher/KPP (KSF) system. The KSF system, (1.2), belongs to a class of nonlinear reaction-diffusion models with chemotactic effects. Biologically, such a model describes the movement of certain biological organism in response to the chemical signal that it releases or consumes in the local environment while both entities are naturally diffusing and reacting (growing, dying, degrading, et al), cf. [2,10,11,14]. One of the characteristic features of (1.2) is the logarithmic (singular) sensitivity function in the first equation, which entails that the chemotactic response of the cellular population to the chemical signal follows the Fechner's law which states that subjective sensation is proportional to the logarithm of the stimulus intensity and has prominent applications in biological modelings (cf. [3,16]). The singular sensitivity function is also the major source where the technical (analytical, numerical) difficulties for studying the qualitative behavior of the model come from.
Biologically, the sign of χ dictates whether the chemotactic movement is attractive (χ > 0) or repulsive (χ < 0). From the mathematical point of view, the sign of the product of χ and µ plays an indispensable role in the qualitative analysis of the model. In this paper, we shall consider the case when χµ > 0, since otherwise the foundation for analytical study will be lost (explained later).
Formally, when χ > 0 and µ > 0, the first equation in (1.2) indicates that while naturally diffusing and growing/dying, the cellular population is additionally driven by the concentration gradient of the chemical signal in the opposite direction of diffusion (due to χ > 0), which suggests that the cellular population may aggregate at certain spatial locations as time evolves. On the other hand, the (exponentially) rapid degradation in the second equation of (1.2) illustrates that the force driving the cellular population to aggregate is diminishing as time goes on. Hence, one may expect that the system will enter an equilibrium state in the long time run, due to the balance between cellular aggregation and chemical degradation. Similarly, when χ < 0 and µ < 0, because of the interaction between chemotactic repulsion and chemical production, the system is also expected to reach a steady state as time goes on. Collectively, when χµ > 0, finite time singularities are not anticipated to develop in the system (1.2), and the synergy of diffusion, chemotactic attraction/repulsion, ON THE LOGARITHMIC KELLER-SEGEL-FISHER/KPP SYSTEM 5367 logistic growth and chemical degradation/production make the dynamics of the model an intriguing problem to pursue.

1.2.
Motivations and literature review. Now we would like to point out the facts that motivate this work, and briefly survey the literature in connection with the current work to put things into perspective.
1.2.1. Connection with Fisher/KPP equation. We mentioned that the coupled system (1.2) contains the classic Fisher/KPP equation [5,17]: which has the dimensionless form: Indeed, when χ = 0, (1.2) is decoupled and the first equation becomes (1.3). Here the positive constants a and K in (1.3) are the relative growth rate and carrying capacity per unit length, respectively. Next we would like to point out one of the major qualitative differences between (1.3) and (1.2). For this purpose, let us consider the Cauchy problem of (1.4) with where u 0 is a perturbation of 1. Using the maximum principle, one can show that if 0 ≤ u 0 (x) ≤ 1, then 0 ≤ u(x, t) ≤ 1. Thus in the original equation (1.3), the population density u stays bounded above by the capacity K. Such a conclusion, however, is not true for the chemotaxis model (1.2) due to the interaction of the cellular population and the chemical signal. In general, we may expect u(x, t) ≥ 0 but we may not have u(x, t) ≤ K.
To illustrate the difference we consider the equation for u in (1.1): where v = c x /c is a working variable for c x , assuming c > 0. The value of v can be any real number depending on the increasing/decreasing of c. In our discussion later, v is a perturbation of zero, thus v x changes sign in R.
Since v x inevitably takes negative values on some intervals, we are not able to apply the maximum principle to conclude u(x, t) ≤ 1.
As an example we consider the Cauchy problem of (1.2) with initial data (u 0 , c 0 ), or equivalently of (1.1) with (u 0 , v 0 ). Let wherec and δ are positive constants, and δ is small. By direct calculation we have (1.6) Figure 1. (1.7) Then it is straightforward to check that Hence (1.5) and (1.7) imply which further implies u(1/ √ 2, 0 + ) > u 0 (1/ √ 2) = 1. To further illustrate the difference between (1.4) and (1.1), we present a numerical simulation to support the above discussion. In the simulation we set ε 1 = 0.05, ε 2 = 0.1 and r = 1 in (1.1). For the initial data we take Here we note that u 0 has two corners, which cause computational difficulties. Utilizing a graph functionality we smooth the corners and replace u 0 by the blue curve in Figure 1. The computation domain is [−25, 25], with Dirichlet boundary condition (1, 0) on both sides. Figure 1 displays the simulation results of u(x, t) for t = 0, 1, 2, 3, over the interval [−10, 10] to highlight the non-trivial parts of the solution. Clearly, u(x, t) is not bounded above by one when t > 0.
Intuitively, due to the interaction of the cellular population and the chemical signal, we do not expect a uniform carrying capacity for everywhere and all time. Hence the parameter K in (1.2) is understood as a typical scale of capacity, and therefore u in the scaled equation (1.1) is about one and not necessarily bounded above by one.
From the analytical point of view, the maximum principle enjoyed by the Fish/KPP equation substantially simplifies the analysis of the qualitative behavior of the model, while the absence of such a property in (1.2) makes the corresponding analysis considerably difficult. Indeed, the question of qualitative behavior of large data solutions to (1.2), such as global well-posedness, long time dynamics, et al, remains largely open. This is the first fact that motivates the current work.

1.2.2.
Connection with balance laws. It was mentioned before that the logarithmic sensitivity function in (1.2) is the major source, from which the technical difficulties for studying the qualitative behavior of the model come. To overcome the possible singularity caused by the logarithmic function, we first pre-process the model by carrying out the following actions including non-dimensionalization, change of variable, rescaling and transformation: • By changing the variable u → u/K, (1.8) • By changing the variable c → e σt c, (1.8) becomes (1.9) • By applying the transformation v = c x /c, (1.9) becomes a system of balance laws (1.10) We remark that the characteristics associated with the flux on the left hand side of (1.10) are from which we see that the principle part of the system (1.10) is hyperbolic in biologically relevant regimes (where u > 0) when χµ > 0, while it may change type when χµ < 0. This is the reason for which we focus on the case when χµ > 0 throughout this paper. • By further applying the re-scalings: to the system (1.10), we obtain a clean version of the model: (1.12) From (1.11) we see that the possible singularity caused by the logarithmic sensitivity function is removed by the nonlinear (Cole-Hopf) transformation. On the other hand, the pre-processing introduces a quadratic (convection-like) nonlinearity into the second equation of the model. Hence, properly balancing diffusion and convection, especially in the regime of large data solutions, brings a significant challenge to the analysis of the transformed model.
In this paper, we consider the Cauchy problem of (1.11), with In particular, we are interested in (u 0 , v 0 ) → (ū,v) as x → ±∞, where (ū,v) is a constant equilibrium state, i.e.ū = 0 or 1. For stability, it is necessary to takē u = 1. For physically interesting scenario we setv = 0. This is to be seen as follows.
Thus from the inverse transform Similar discussion on v for traveling waves can be found in [28].
If we takev = 0 and assume v 0 ∈ L 1 (R), then c 0 (x) ≡ c(x, 0) has finite limits as x → ±∞: In particular, if we further assume that v 0 has zero mass, see (2.2) below, then This impliesc + =c − ≡c. Except for the study of time decay rates, all other results in this paper apply to bothc + =c − andc + =c − . System (1.11) is a special case of the more general system of hyperbolic-parabolic balance laws: where w, f j , r ∈ R n and B jk ∈ R n×n . A set of structural conditions have been proposed in a recent paper [47]. Under those conditions and under the assumption on the smallness of initial perturbations, existence of global-in-time solutions to the Cauchy problem has been established in [47]. L p (p ≥ 2) decay rates have been obtained under the same hypotheses for m ≥ 2 in [48], and for m = 1 in [50]. Asymptotic behavior of solutions has been studied for m ≥ 2 in [49]. System (1.11) fits in the framework of (1.14) with n = 2 and m = 1. It is easy to check that (1.11) satisfies the structural conditions proposed in [47], for both ε = 0 and ε > 0. As a consequence, if (u 0 − 1, v 0 ) is small in H 2 (R), the Cauchy problem (1.11), (1.13) has a unique global-in-time solution. In addition, if (u 0 − 1, v 0 ) is small in H 4 (R) ∩ L 1 (R), then the L 2 decay rates of u − 1 and v are obtained as (t + 1) −3/4 and (t + 1) −1/4 , respectively.
However, we note that all the aforementioned results are obtained under the assumption that the initial perturbations are sufficiently small. Indeed, to the authors' knowledge, the story for large data solutions is completely unknown. This is the second fact that motivates the current work.
1.2.3. Connection with Keller-Segel model. Except the classic Fisher/KPP equation, the coupled system (1.2) is also closely connected with contemporary models in mathematical biosciences. Indeed, when a = 0, (1.2) becomes the following Keller-Segel type chemotaxis model: 15) which was proposed in [32] for describing the movement of chemotactic populations that deposit little-or non-diffusive chemical signals that modify the local environment for succeeding passages, and later found applications in cancer research [20]. By applying the similar pre-processing actions as in the preceding section, one gets the following hyperbolic-parabolic system of conservation laws: Since the model was initiated in the late 1990s, the qualitative behavior of (1.15) has been analyzed to a large extent. The pioneering works of [19,32], where the authors constructed explicit and numerical solutions to (1.15) when ε = 0 exhibiting chemotactic aggregation or collapsing (time asymptotical uniform distribution of cellular population), is followed by a series of recent products [4,6,8,12,13,21,22,23,24,25,26,27,28,29,33,34,40,45], in which the global well-posedness and long time behavior of large data classical solutions, local stability of traveling waves, and zero diffusion limit/boundary layer formation are studied for various types of initial and/or boundary value problems of the transformed model (1.16). We mentioned that the object of this paper is the Cauchy (initial value) problem of (1.2). One of our major interests is the global stability of constant ground states (equilibrium solutions) associated with the model. For (1.15), the results in [22,29] show that any positive constant associated with u is globally asymptotically stable, provided the initial function is perturbed around the prescribed constant state. However, it is not difficult to see that such a result can not be inherited by (1.2), due to the presence of the logistic growth term. Indeed, the logistic growth term indicates that, instead of an arbitrarily prescribed positive ground state, the unique possibly stable constant equilibrium associated with u in (1.2) should be the relative carrying capacity, i.e., K. This is one of the major differences between (1.2) and (1.15). The current work is partially motivated by the question that whether the self-selected constant equilibrium state is globally asymptotically stable or not.
In addition, we observe that in the perturbed system, cf. (3.11), the logistic growth introduces a weak damping mechanism, which may have further influences on the dynamics of the solution. Indeed, by applying the arguments in [47] one can show that, when the initial data are slightly perturbed around the constant equilibrium state, the perturbation associated with (1.2) enjoys faster temporal decay rates than that of (1.16), due to the dissipation induced by the linear damping. Nevertheless, such a phenomenon is not yet clear in the regime of large data solutions, which is the third fact that motivates this work.
1.2.4. Related KSF models. Lastly, we would like to point out that other Keller-Segel-Fisher type models have been studied in the literature. A search in the database shows that those models are studied mostly in bounded domains, in particular, with no-flux boundary condition. They are also studied mostly with constant rate production and degradation of the chemical signal, i.e., the second equation in the model takes the form c t = εc xx + µu − σc.
(1.17) This is to compare with our model with a density-dependent rate. With regular sensitivity and (1.17), global existence of large data classical solutions has been established on bounded domains in all space dimensions under suitable conditions, see [9,30,35,43,42] and references therein. The result of [30] is extended in [1] to apply to systems with singular sensitivity functions such as the logarithmic function. Moreover, Cauchy problem is also considered in R 2 , with regular sensitivity and (1.17), in [31]. There are also recent works dealing with the existence of large data weak solutions to Keller-Segel type models with logistic growth, and we refer the reader to [18,36,37,38,39,41,44,46,51] and the references therein for more information in this direction. However, we notice that with the logarithmic sensitivity function and density-dependent production rate, no result concerning the qualitative behavior of large data solutions is available in the knowledge base. This is the fourth fact that motivates the current work.
1.3. Goals and outcomes. Motivated by the aforementioned facts, we devote this paper to the study of qualitative behavior of classical solutions to the Cauchy problem (1.11)-(1.13). In particular, we aim to remove the restriction of small initial data and obtain similar results as those in [47]. The major outcomes of this paper are listed as follows: • (Global Dynamics) Under the weak assumption of (u 0 −1, v 0 ) in H 2 (R)×H 2 (R) and u 0 ≥ 0 (as to be physically meaningful), we establish the well-posedness of global-in-time solutions of (1.11)-(1.13), for both ε = 0 and ε > 0. Time asymptotic behavior (in particular, global asymptotic stability of the constant equilibrium state (1, 0)) is further studied. See Theorems 2.1-2.2.
• (Zero Diffusion Limit) We mentioned that (1.11) is closely connected with (1.16) which was designed to model the movement of chemotactic populations that deposit little-or non-diffusive chemical signals that modify the local environment for succeeding passages. Hence, it is natural to ask whether the chemically non-diffusive model (i.e., ε = 0) is a good approximation of the diffusive one (i.e., ε > 0) in the process of vanishing diffusion coefficient as ε → 0. Under the same hypotheses as in the preceding point, we also study the zero diffusion limit (as ε → 0) of the solution and the corresponding convergence rate in terms of ε. See Theorem 2.3.
• (Decay Rates) To further study the qualitative behavior of (1.11)-(1.13), we consider the explicit time-decay rates of the solution. For this purpose, we make an extra assumption that v 0 has zero mass. As discussed above, this is equivalent to c 0 has the same positive end-statec as x → ±∞. (Here c 0 −c may have nonzero mass though.) The hypotheses, which are without smallness assumption on the norms of (u 0 − 1, v 0 ), allow us to obtain L 2 rates when ε = 0. The rates are (t + 1) −1 for u − 1 and (t + 1) −3/4 for v.
The better rates here, when comparing with the framework (1.14), are due to the zero mass assumption on v 0 . In the case ε > 0, we do need u 0 − 1, v 0 and x −∞ v 0 (y) dy to be small to obtain the same rates. Comparing to the framework (1.14), however, we do not need the smallness assumption on the derivatives. See Theorem 2.4 for more details. The rest of the paper is organized as follows. In Section 2, we state and comment on the main results of this paper, and briefly explain the ideas used to prove the results. Sections 3-7 are devoted to the proofs of the main results. We then finish the paper with concluding remarks in Section 8.

Statement of results and ideas of proof.
Notation. Throughout this paper, · , · ∞ and · H s denote the norms of the usual Lebesgue measurable function spaces L 2 , L ∞ and Hilbert space H s , respectively. We use ( Unless otherwise specified, c i , C and C i denote generic constants which are independent of the unknown functions. The values of the constants may vary line by line according to the context.

2.1.
Results and comments. The first two theorems are concerned with the global-in-time well-posedness and long time behavior of classical solutions to the Cauchy problem of (1.1) for initial data having potentially large energy.
where r, D > 0, χ = 0 and ε ≥ 0 are fixed constants. Suppose that the initial data satisfy u 0 >0 and Then there exists a unique solution to (2.1) for all t > 0, such that u(x, t)>0 for x ∈ R, t > 0, and where the constants c 1 and c 2 are independent of t and ε, and depend on r, D, χ and the initial data.
The third theorem addresses the relationship between the chemically diffusive (ε > 0) and non-diffusive (ε = 0) solutions, and characterizes the difference between the solutions in terms of ε. Theorem 2.3 (Zero Diffusion Limit and Convergence Rates). Let the conditions of Theorem 2.1 hold. Let (u ε , v ε ) and (u 0 , v 0 ) be the solutions to (2.1) with ε > 0 and ε = 0, respectively, and with the same initial data. Then for any fixed t > 0 we have where the constants c 3 , ..., c 6 are independent of t and ε.
Remark 1. Theorem 2.3 shows that for fixed t > 0, the zeroth and first frequencies of the difference between the diffusive and non-diffusive solutions decay to zero, as ε → 0, at different rates in terms of ε.
Next, we identify the explicit decay rates of the solution. For this purpose, we further assume which, combined with the second equation of (2.1), allows one to define the antiderivative: Then we have the following.
• When ε > 0, let N > 0 be an arbitrarily fixed constant. Then there exists a constant δ > 0, such that if u 0 2 + v 0 2 ≤ N and where the constants c 7 , c 8 , c 9 are independent of t.
• When ε = 0, there exists a finite T 0 > 0, such that the global-in-time solution to (2.1) enjoys the same decay rates as in (2.3) for t > T 0 , and in this case the temporal integrals in (2.3) are taken from T 0 to any t > T 0 .
Remark 2. Theorem 2.4 shows that for the chemically diffusive model (ε > 0) one needs the smallness of the low frequency part of the solution to identify the explicit decay rate, while such a requirement is completely unnecessary when ε = 0. This is so partially because when ε > 0, in the equation of the antiderivative (cf. (6.1)), the quadratic nonlinear term (ψ x ) 2 somehow can not be effectively controlled by the diffusion ψ xx , which is a different scenario from the solution with ε = 0.

2.2.
Ideas. We prove the results by using L p -based energy methods. For the global well-posedness of large data solutions, we first construct an entropy-type estimate which involves the anti-logarithmic function of u and provides the uniform-in-time estimates of v(t) 2 and t 0 √ u(τ ) 2 dτ . The L 2 estimate of the solution is then carefully crafted by exploring the logistic growth and the uniform estimates derived from the entropy estimate. The L 2 estimates of the spatial derivatives of the solution are then obtained in a standard fashion by using the estimates of the low frequency part of the solution. Collectively, these provide the first estimate in the statement of Theorem 2.1.
As a consequence of the preceding uniform-in-time energy estimate, the second estimate in the statement of Theorem 2.1 is obtained by analyzing the lower order dissipation induced by the logistic growth. Such an estimate plays a crucial role in deriving the zero diffusion limit and convergence rates, leading to the result recorded in Theorem 2.3. In addition, the time asymptotic behavior (Theorem 2.2) is proved by applying the uniform-in-time estimate and using the fact that To identify the explicit decay rates of the perturbation, we resort to an equation of the antiderivative of v, from which the uniform temporal integral of v(t) 2 is obtained under the conditions of Theorem 2.4 by utilizing the energy estimates obtained in Theorem 2.1. Then by carrying out time-weighted energy estimates, exploring some fine structure of the logistic growth, and using an iteration scheme, we succeed in obtaining the explicit decay rates recorded in Theorem 2.4.
Lastly, we would like to remark that regarding the explicit decay rate of the Keller-Segel type model (1.16), all of the existing results require the smallness of initial perturbation no matter ε > 0 or ε = 0, see e.g. [22,29]. On the other hand, for (1.1) when ε = 0, one does not need any smallness assumption on the initial perturbation. This is one of the exclusive features that distinguish (1.1) from (1.16).
3.1. Local existence. Equation (3.1) can be written in vector form: Local existence of solution to (3.1), or equivalently to (3.2), is an application of Kawashima's theory [15]. Here we cite Theorem 2.2 from [47]. To simplify our presentation, we restrict the statement of the theorem, which applies to all space dimensions, to one space dimension only: Here n 1 , n 2 ≥ 0 are two constant integers such that n = n 1 + n 2 , and B * ∈ R n2×n2 is non-singular if n 2 > 0. Letw With the specific form of our (3.2), which is supplemented by (3.3), we take η = 1 2 v 2 + u ln u − u [22]. As in [50], by direct calculation we find Note that the constant equilibrium state for Theorem 2.1 isw = (1, 0) t ∈ R 2 . Thus we take O = {(u, v)|u > 0}, which is an open convex set containingw. It is clear that in O, η is strictly convex, η f is symmetric, and η B is symmetric, semi-positive definite.
The positivity of u, obtained via Theorem 3.1, can be justified by two different approaches. First, it is inherited from the invariant set under iterations in Kawashima's local existence theory, see Proposition 2.8 and Theorem 2.9 in [15]. The invariant set is a bounded, open, convex set whose closure is in O, hence u(x, t) > 0 for all x ∈ R, t ∈ [0, T ].
Also, the positivity of u can be justified separately by a straightforward application of the maximum principle for Cauchy problems as follows, at the cost of the strict inequality. For this, we use (u, v) obtained from Theorem 3.1 to construct a parabolic operator L: In what follows, we will concentrate on deriving the a priori estimates of the local solution, in order to extend it to a global one.
3.2. Entropy estimate. By testing the first equation in (3.1) by ln u, we get By testing the second equation in (3.1) by v, we get Note that whereû is between u and 1. Since u ≥ 0, we see that u(u − 1)(ln u − ln 1) ≥ 0. Hence, by integrating (3.9), we obtain where the constant C 1 is independent of t and ε, and we dropped two non-negative terms from the left hand side without affecting the inequality.

L 2
x -estimate of zeroth frequency. By letting u =ũ + 1 in (3.1), we obtain     ũ By testing the first equation in (3.11) withũ and the second with v, then adding the results, we deduce by using integration by parts and the elementary inequality, 5378 YANNI ZENG AND KUN ZHAO ab ≤ 1 2 (a 2 + b 2 ), that d dt where we applied (3.10) in deriving the last inequality. Next, we derive an energy estimate for ũ 2 ∞ . Note that since lim x→±∞ũ (x, t) = 0, we deduce by using the Hölder inequality that (3.13) By feeding (3.13) into (3.12), we obtain d dt which implies d dt (3.15) By integrating (3.15) with respect to t, we obtain where the constant C 2 is independent of t and ε.
As a direct consequence of (3.13), (3.10) and (3.16), we deduce that where the constant C 3 is independent of t and ε.

L 2
x -estimate of 1st order spatial derivatives. By testing the first equation in (3.11) with −ũ xx and the second with −v xx , then adding the results, we deduce d dt which implies d dt (3.19) Next, we estimate I k on the right hand side of (3.19). First of all, we have Second, by using the Sobolev-type inequality: and the Young inequality, we estimate I 2 as

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where we applied the estimate (3.10) for v(t) . For I 3 , we deduce that Similar to (3.22), we can show that (3.23) By feeding the estimates for I k into (3.19), we have d dt (3.24) By applying the Gronwall inequality to (3.24), we deduce where we applied (3.17), (3.16) and (3.10), and the constant C 4 is independent of t and ε. By feeding (3.25) into (3.24), we obtain d dt (3.26)

ON THE LOGARITHMIC KELLER-SEGEL-FISHER/KPP SYSTEM 5381
After integrating (3.26) with respect to t, we conclude where the constant C 5 is independent of t and ε.

L 2
x -estimate of 2nd order spatial derivatives. By taking ∂ x to the equations in (3.11), we have (3.28) By testing the first equation in (3.28) with −ũ xxx and the second with −v xxx , then adding the results, we deduce d dt (3.29) After rearranging terms, we have d dt Next, we are devoted to estimate the J k 's on the right hand side of (3.30). First, by using Cauchy's inequality, we can show that

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where we applied (3.21), (3.16) and (3.25). In a similar fashion, we can show that For the next three terms involvingũ xxx , we can show that For the last two terms, recalling (1.12) we can show that (3.32) By feeding the estimates (3.31)-(3.32) into (3.30), we obtain d dt (3.33) By applying the Gronwall inequality to (3.33), we get where we applied (3.10), (3.16) and (3.27), and the constant C 6 is independent of t and ε. By substituting (3.34) into (3.33), we have d dt (3.35) By integrating (3.35) with respect to time, we obtain where the constant C 7 is independent of t and ε.
3.6. Improved L 2 t H 1 x -estimate of v x . So far we have established the following energy estimate for the solution to (3.1): where the constant C is independent of t and ε. This indicates that the L 2 t H 2 x norm of v x is inversely proportional to ε. In this section, we shall show that the L 2 t H 1 x norm of v x is bounded by some generic constant which is independent of t and is not inversely proportional to ε. In particular, the generic constant is finite when ε = 0. To see this, let us test the second equation of (3.11) with v to get d dt Note that, from the first equation of (3.11), By substituting (3.38) into the integral on the right hand side of (3.37), we have (3.39)

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For the last term on the right hand side of (3.39), we can show that where again we used the second equation of (3.11). By substituting (3.40) into (3.39), we obtain (3.41) By substituting (3.41) into (3.37), we have d dt (3.42) Next, we are devoted to estimate the terms on the right hand side of (3.42). First, we have 1 where the second term on the right hand side can be estimated as where we applied (3.21), (3.10), (3.16) and (3.25). By substituting (3.44) into (3.43), we have (3.45) Next, in a similar fashion we can show that (3.46) By feeding (3.45) and (3.46) into (3.42), we obtain d dt (3.47) After integrating (3.47) with respect to time and applying the previous energy estimates, we get which implies that (3.48)

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In particular, (3.48) yields (3.49) We note that the constant on the right hand side of (3.49) is independent of t and is not inversely proportional to ε. In particular, the constant is finite when ε = 0. Furthermore, by using the same idea and previous estimates, we can establish a similar result for v xx . Since the proof is in the same spirit, we shall omit the technical details here. This completes the proof of Theorem 2.1.

4.
Long time behavior of large data solutions. In this section we prove Theorem 2.2. From (3.18) we see that d dt By applying the arguments in Section 3.3 (cf. (3.20), (3.22)-(3.23)), we can show that d dt (4.1) Upon integrating (4.1) with respect to t, we obtain t 0 d dτ where the constant on the right hand side is independent of t. The preceding estimate implies which, together with (3.16) and (3.49), yields 1 ((0, ∞)).
Hence, it holds that lim In a similar fashion, by using (3.29) and the arguments in Section 3.4, we can show that lim and we omit the details to simplify the presentation. Since ũ(t) H 2 and v(t) H 2 are uniformly bounded with respect to t, according to Morrey's inequality we know that (ũ, v) ∈ C 1 (R) × C 1 (R) and ũ(t) C 1 (R) and v(t) C 1 (R) are uniformly bounded with respect to t. By using (3.21) we can easily show that It then follows from the uniform boundedness of ũ(t) and v(t) and (4. This implies that there is a finite time T > 0, such that |ũ(x, t)| ≤ 1 2 for all x ∈ R and t ≥ T . This in turn shows that 1 +ũ(x, t) ≥ 1 2 for all x ∈ R and t ≥ T . By utilizing such a piece of information in (3.15), we have d dt For any t > T , by integrating (4.5) from T to t, we have in particular, where we applied (3.10) and (3.16). By testing the first equation in (3.11) withũ, we have d dt For any t > T , upon integrating (4.7) from T to t, we get t T d dτ

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where the constant on the right hand side is independent of t. The preceding estimate implies d dt which, together with (4.6), yields ũ(t) 2 ∈ W 1,1 ((T, ∞)).
This completes the proof of Theorem 2.2.

5.
Diffusion limit and convergence rate of large data solutions. This section is devoted to the proof of Theorem 2.3. Let (ũ ε ,ṽ ε ) and (ũ 0 ,ṽ 0 ) be the solutions to (3.11) and with ε > 0 and ε = 0, respectively, and with the same initial data.
By testing the first equation of (5.1) with u and the second equation with v, we have d dt Rṽ εṽε x vdx.

(5.2)
We estimate the right hand side of (5.2) as follows. First, by Cauchy's inequality, we have The third term on the right hand side of (5.2) is estimated as For the last term, we have (5.5) By applying the Gronwall inequality to (5.5), we have From (3.36) we see that the quantity involving exponential on the right hand side of (5.6) is bounded by e Ct for some constant C which is independent of t and ε. Moreover, from the improved estimates in Section 3.5 we know that the remaining integrals on the right hand side of (5.6) are bounded by ε 2 C(1+ε) for some constant which is also independent of t and ε. Hence, (5.6) shows that for any fixed t > 0, the L 2 norm of the difference between the diffusive and non-diffusive solutions converges to zero, as ε → 0, at the rate of O(ε). Next, we investigate the convergence rate of the first order spatial derivatives of the solution.
By testing the first equation in (5.1) with −u xx and the second with −v xx , we have d dt The first term on the right hand side of the above equation can be estimated as and the second term can be estimated as Then under the a priori assumption, it holds that for ∀ 0 < t ≤ t 0 , d dt (6.8) Now let us revisit the entropy estimate (3.9), which in terms ofũ reads Note that by Taylor's theorem, where by definition, After integrating (6.14) with respect to t and using (6.15), we have in particular, Hence, on one hand, in view of (3.21) and (6.16), we see that the a priori assumption (6.7) can be verified by properly choosing the initial data. On the other hand, (6.16) shows that the temporal accumulation of v(t) 2 is uniformly bounded with respect to time, which is the key ingredient for cooking up the subsequent time-weighted energy estimates leading to the algebraic decay rate of the solution.
6.2. Decay rate of spatial derivatives. We recall the estimate (3.24): where the first term on the right hand side can be estimated as where we applied (3.21), (3.16) and (3.25). By feeding (6.23) into (6.22), we obtain (6.24) By multiplying (6.24) by (1 + t), we have By integrating the result with respect to time and applying (6.21), (3.16) and (3.49), we have (6.25) In addition, by applying the arguments in Section 3.5 and using (6.21) and (6.25), we can show that for any t > 0, where the constant C 12 is independent of t.
6.3. Improved decay rate of zeroth order frequency. In this subsection, we improve the decay rate of ũ(t) 2 . From the first equation of (3.11) we havẽ Squaring both sides of the above equation, then integrating over R, we have from which we can show that d dt r ũ 2 + r 2 ũ 2 + ũ t ≤ 5 ũ xx 2 + 20 C 1 C 4 ũ x 2 + (20 C 2 C 4 + 5) v x 2 + 20r 2 C 2 C 4 ũ 2 .

7.
Explicit decay rate of large data solutions. In this last section, we establish the explicit decay rate of the global-in-time solution to (3.11) when ε = 0, and finish the proof of Theorem 2.4. Interestingly, in this case we do not need any smallness assumption on the initial data. However, we will show that the large data solution