consumption rate

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate Yanni Zeng a , Kun Zhao b,∗ a Department of Mathematics, University of Alabama at Birmingham, United States of America b Department of Mathematics, Tulane University, United States of America

Received 8 April 2019; revised 26 August 2019; accepted 28 August 2019

Abstract We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. It is a 2 × 2 reaction-diffusion system describing the interaction of cells and a chemical signal. We study Cauchy problem for the original system and its transformed system, which is one of hyperbolic-parabolic balance laws. Our initial data are generic perturbations of a constant ground state, i.e. the initial mass of perturbation is non-zero. In the case of non-diffusive chemical, we obtain optimal L2 time decay rates for the solution with finite initial data. In the case of diffusive chemical, optimal L2 rates are also obtained with additional assumption on the smallness of the initial amplitude but still allowing large oscillation. © 2019 Elsevier Inc. All rights reserved. MSC: 35B40; 35Q92; 35K57; 35M31 Keywords: Optimal time decay rates; Chemotaxis; Logarithmic sensitivity; Logistic growth; Cauchy problem

* Corresponding author.

E-mail addresses: [email protected] (Y. Zeng), [email protected] (K. Zhao). https://doi.org/10.1016/j.jde.2019.08.050 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction This paper is a continuation of the authors’ recent study ([45]) of the Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate: 

st = εsxx − μus − σ s, ut = Duxx − χ[u(ln s)x ]x + au(1 −

u K ),

x ∈ R, t > 0.

(1.1)

The main purpose is to identify the optimal decay rates of classical solutions (toward constant equilibrium states) to the Cauchy problem of (1.1) and its transformed system, which improves some of the results obtained in [45], via a different analytical approach. In the model (1.1), the unknown functions are s = s(x, t) and u = u(x, t) for the concentration of a chemical signal and the density of a cellular population, respectively. The system parameters are interpreted as follows. • • • • • • •

ε ≥ 0: diffusion coefficient of chemical signal; μ = 0: coefficient of density-dependent production/consumption rate of chemical signal; σ ≥ 0: natural degradation rate of chemical signal; D > 0: diffusion coefficient of cellular population; χ = 0: coefficient of chemotactic sensitivity; a > 0: natural growth rate of cellular population; K > 0: typical carrying capacity of cellular population.

Such a model depicts the movement of a cellular population in response to a chemical signal (chemotaxis), while both entities are naturally diffusing, growing/dying, producing/consuming and degrading in the local environment. Because of its biological background and potential applications in chemotaxis research and mathematical features, the rigorous analysis of (1.1) and its related models has gradually become one of the focal points in applied mathematics in recent years. Here we list some closely related references in connection with this work, in order to put things into perspective. • For the non-growth model, i.e. (1.1) with a = 0, the following results have been established in the literature (see also the references listed therein): – local well-posedness and blowup criteria of large-data classical solutions in Rn [4,15], – global well-posedness and long-time behavior of small-data classical solutions in finite intervals [46], – local stability of traveling wave solutions in R [9,19–22], – global well-posedness of large-data classical solutions in R [6], – global well-posedness of large-data classical solutions in finite intervals [5], – long-time behavior and chemical diffusion limit of large-data classical solutions in finite intervals [17,18,27,30,37], – long-time behavior, chemical diffusion limit and spatial analyticity of large-data classical solutions in R [23,16], – boundary layer formation and characterization of large-data classical solutions in finite intervals [8,17,27],

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– global well-posedness, long-time behavior and chemical diffusion limit of classical solutions under minimal smallness assumptions on initial data in Rn [35] and bounded domains [28]. • For closely related chemotaxis-growth models, the following results are available: – global well-posedness and long-time behavior of large-data classical solutions in multidimensional bounded domains for (1.1) with σ = 0 under certain technical assumptions on the system parameters [12,48], – global existence of large-data generalized solutions in multi-dimensional bounded domains for (1.1) with σ = 0 [11], – global well-posedness and long-time behavior of large-data classical solutions in multidimensional bounded domains for (1.1) with σ = 0 and density-signal governed sensitivity under certain technical assumptions on the system parameters [3], – global well-posedness of large-data classical solutions in multi-dimensional bounded domains for (1.1) with regular sensitivity function and constant rate of production/consumption of the chemical signal [7,24,31,39,40], – global well-posedness of large-data classical solutions in multi-dimensional bounded domains for (1.1) with singular sensitivity functions and constant rate of production/consumption of the chemical signal [1], – global well-posedness of large-data classical solutions to the Cauchy problem of (1.1) in R2 with regular sensitivity function and constant rate of production/consumption of the chemical signal [25]. In addition, we refer the reader(s) to [13,32–34,36,38,41,49,50] and the references therein for recent progress on the existence of large-data weak solutions to other related Keller-Segel type models with logistic growth. • There are works considering time decay rates for chemotaxis models that are more different from (1.1), such as models incorporated with an incompressible flow field. For instance, in [47] a chemotaxis-Navier-Stokes system is considered in a two dimensional bounded domain, and exponential decay rates are obtained. Also see references therein for well-posedness related to similar models. In this paper, we are interested in the Cauchy problem of (1.1), with (s, u)(x, 0) = (s0 , u0 )(x),

x ∈ R,

(1.2)

where s0 > 0 and u0 > 0. In particular, we are interested in Cauchy data being a generic, possibly large amplitude perturbation of a constant ground state (¯s , u). ¯ Here, a generic perturbation means no zero-mass assumption attached, i.e.,   [s0 (x) − s¯ ] dx and [u0 (x) − u] ¯ dx R

R

are allowed to be non-zero. Mathematically, the possible singularity stemming from the logarithmic function in (1.1) might bring significant difficulties to both the analytical and numerical studies of the model.

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A commonly adopted approach for overcoming the technical barrier is to remove the logarithmic function by the inverse Hopf-Cole transformation [14]: v = (ln s)x =

sx . s

(1.3)

Under the new variables v and u, the reaction-diffusion system (1.1) becomes a system of hyperbolic-parabolic balance laws: 

vt + (μu − εv 2 )x = εvxx , ut + χ(uv)x = Duxx + au(1 −

(1.4)

u K ).

In this paper we assume χμ > 0.

(1.5)

This includes two scenarios: χ > 0 and μ > 0, or χ < 0 and μ < 0. The former is interpreted as cells are attracted to and consume the chemical. On the other hand, the latter describes cells depositing the chemical to modify the local environment for succeeding passages [26]. Mathematically, the non-diffusive, non-reactive part of (1.4) is hyperbolic in biologically relevant regimes when χμ > 0, while it may change type when χμ < 0 [45]. Assumption (1.5) allows the adoption of analytical approaches from hyperbolic-parabolic balance laws, such as symmetrization, energy estimation, et al., to investigate the qualitative behavior of the transformed system (1.4). Under (1.5), we use rescaled and dimensionless variables: √

χμK t, t˜ = D

x˜ =

χμK x, D

 χ v˜ = sign(χ) v, μK

u˜ =

u . K

(1.6)

This simplifies (1.4) to  vt + (u − ε2 v 2 )x = ε1 vxx , ut + (uv)x = uxx + ru(1 − u),

x ∈ R, t > 0

(1.7)

after dropping the tilde accent. Here the new parameters are r=

aD > 0, χμK

ε1 =

ε ≥ 0, D

ε2 =

ε . χ

(1.8)

Corresponding to (1.2), the Cauchy data for (1.7) are D    χ χ s0 ( √χμK x) D  , v(x, 0) = v0 (x) ≡ sign(χ) (ln s0 ) ( √ x) = sign(χ) μK μK s0 ( √ D x) χμK χμK

1 D u(x, 0) = u0 ( √ x). K χμK

(1.9)

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For simplicity we write v(x, 0) = v0 (x),

u(x, 0) = u0 (x),

x ∈ R,

(1.10)

keeping in mind that u0 has been rescaled. Noting s0 (x) → s¯ as x → ±∞ and assuming s¯ > 0, from (1.9) we have lim v0 (x) = 0.

x→±∞

Hence, in order for the ground state (v, ¯ u) ¯ = (0, u) ¯ to be an equilibrium state of (1.7), we must have u¯ = 0, 1. Inspired by the classic result on the logistic differential equation, for stability we take u¯ = 1. The Cauchy problem (1.7), (1.8), (1.10) has been considered in [45], where global wellposedness, long-time behavior, vanishing coefficient limit and decay rates of solution have been studied. Here we cite the global existence and asymptotic stability as follows: Theorem 1.1 ([45]). Let r, D > 0, χ = 0 and ε ≥ 0 be constants. Consider the Cauchy problem (1.7), (1.10), with ε1 = ε/D and ε2 = ε/χ . Suppose that the initial data satisfy u0 > 0 and (v0 , u0 − 1) ∈ H 2 (R). Then there exists a unique solution to (1.7), (1.10) for all t > 0, in the following class: (i) For ε > 0, (v, u −1) ∈ C([0, ∞); H 2 (R)) ∩C 1 ([0, ∞); L2 (R)), and (vx , ux ) ∈ L2 ([0, ∞), H 2 (R)). (ii) For ε = 0, v ∈ C([0, ∞); H 2 (R)) ∩ C 1 ([0, ∞); H 1 (R)), u − 1 ∈ C([0, ∞); H 2 (R)) ∩ C 1 ([0, ∞); L2 (R)), vx ∈ L2 ([0, ∞), H 1 (R)), and ux ∈ L2 ([0, ∞), H 2 (R)). Besides, the solution has the properties u(x, t) > 0 for x ∈ R, t > 0, and

(v, u − 1) 2H 2 (t) +

t   ε vx 2H 2 + ux 2H 2 (τ ) dτ ≤ C1 , 0

t

(1.11)

vx 2H 1 (τ ) dτ ≤ C2 (1 + ε), 0

where the constants C1 and C2 are independent of t and ε, and depend on r, D, χ and the initial data. Theorem 1.2 ([45]). Let the conditions of Theorem 1.1 hold. Then the unique global-in-time solution to (1.7), (1.10) has the following property:   lim vx H 1 + u − 1 H 2 + v C 1 (R) + u − 1 C 1 (R) (t) = 0

t→∞

for any ε ≥ 0.

(1.12)

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Theorems 1.1 and 1.2 apply to initial data that satisfy limx→±∞ s0 (x) = s± . In our case that s0 (x) has a ground state s¯ , the corresponding v0 defined in (1.9) has an extra property R v0 (x) dx = 0. Since the equation for v in (1.7) is a conservation law, we have 

 v(x, t) dx =

R

 v(x, 0) dx =

R

v0 (x) dx = 0. R

Therefore, we define x

x

ψ(x, t) ≡

v(y, t) dy,

ψ0 (x) ≡ ψ(x, 0) =

−∞

v0 (y) dy,

(1.13)

−∞

which implies ψ0 (x) = v0 (x).

ψx (x, t) = v(x, t),

(1.14)

In this case some algebraic asymptotic decay rates related to (1.12) have been identified in [45]: Theorem 1.3 ([45]). Let the conditions of Theorem 1.1 hold, and assume that ψ0 ∈ L2 (R). • When ε > 0, let N > 0 be an arbitrarily fixed constant. Then there exists a constant δ > 0, such that if v0 2L2 + u0 2L2 ≤ N and ψ0 2H 1 + u0 − 1 2L2 ≤ δ, the unique global-in-time solution to (1.7), (1.10) satisfies t (t

+ 1) v 2L2 (t) + (t

+ 1)

2

u − 1 2L2 (t) +

(τ + 1) (vx , u − 1) 2L2 (τ ) dτ ≤ C3 , 0

t (t + 1)2 (vx , ux ) 2L2 (t) +

(τ + 1)2 (vxx , ux ) 2L2 (τ ) dτ ≤ C4 ,

(1.15)

0

t (t + 1)

3

(vxx , uxx ) 2L2 (t) +

  (τ + 1)3 ε vxxx 2L2 + uxx 2H 1 (τ ) dτ ≤ C5 ,

0

where the constants C3 , C4 and C5 are independent of t . • When ε = 0, there exists a finite time T0 > 0, such that the global-in-time solution to (1.7), (1.10) enjoys the same decay rates as in (1.15) for t > T0 , and in this case, the temporal integrals in (1.15) are taken from T0 to any t > T0 . We remark that the explicit decay rates recorded in Theorem 1.3 are obtained by using the (time) weighted energy method, which are the best rates possible via such method. However, the parabolic structure of the equations in (1.7) suggests that these rates are not optimal, in the sense that they are slower than the decay rates of the solution to the heat equation. Hence, it is

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desirable to know whether the rates in Theorem 1.3 can be improved or not, which is the primary motivation of this paper. The first goal of this paper is to improve the decay rates in Theorem 1.3. For instance, here we improve the L2 -decay rate of v from (t + 1)−1/2 to (t + 1)−3/4 . Similarly, those for vx and u − 1 are improved from (t + 1)−1 to (t + 1)−5/4 , and the one for ux is from (t + 1)−1 to (t + 1)−7/4 . The optimal L2 rates imply optimal L∞ rates via Sobolev inequality. They are (t + 1)−1 for v and (t + 1)−3/2 for u − 1, respectively. As was mentioned above, the decay rates recorded in Theorem 1.3 are the best ones possible via the (time) weighted energy method. Hence, in order to obtain faster (optimal) rates, one must switch to a different (more accurate) method. Here we resort to spectral analysis and combine it with the energy method, which is a commonly practiced approach in the analysis of large-time asymptotic behavior of solutions to hyperbolic-parabolic balance laws. We realize the goal via first analyzing the spectral properties of the corresponding linearized system, then further iterating the decay rates obtained in Theorem 1.3 for the nonlinear system through the Duhamel’s principle and taking advantage of the decay properties of the linear system. Due to the nonlinear structure of the model, the detailed arguments involved in the proof are much more delicate than the weighted energy estimates. The second goal of this paper is to obtain optimal time-decay rates for the original variables, especially the function s in (1.1). Noting that the equilibrium state of u is K, hence heuristically the first equation in (1.1) suggests that as t → ∞, the variable s will experience exponential decay when μK +σ > 0 or growth when μK +σ < 0. It is worth mentioning that the exponential decay or growth of the chemical concentration function associated with the non-growth model (i.e. (1.1) with a = 0) was verified in the literature (cf. [16]), but the dynamics of the function in the critical case of μK + σ = 0 has not been identified, which was left as an open problem (cf. Remark 1.2 in [16]). In this paper, we are able to fully characterize the motion of the function s associated with (1.1) for all values of μK + σ , and in particular obtain the optimal L2 -decay rate of s − s¯ as (t + 1)−1/4 , and the optimal L∞ rate as (t + 1)−1/2 in the critical case of μK + σ = 0. Therefore, in the critical case, for a generic perturbation of s¯ , the evolution of s(x, t) − s¯ is like a heat kernel, the solution to the diffusion equation with an initial point-mass. The rest of the paper is organized as follows. In Section 2, we state and comment on the main results of this paper. In Section 3 we study the linear system corresponding to the transformed equations. In Section 4 we prove our first result, which concerns the optimal rates for the transformed system. In Section 5, we prove our second result, which is for the original variables. 2. Main results Throughout this paper we use the following notations to abbreviate the norms of Sobolev spaces with respect to x: · k = · H k (R) ,

· = · L2 (R) .

For the readers’ convenience, we restate the Cauchy problem (1.7), (1.10) before we give our first main result:  vt + (u − ε2 v 2 )x = ε1 vxx , x ∈ R, t > 0, ut + (uv)x = uxx + ru(1 − u), (2.1) (v, u)(x, 0) = (v0 , u0 )(x),

x ∈ R,

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where r = aD/(χμK) > 0, ε1 = ε/D ≥ 0 and ε2 = ε/χ are constants. The initial datum v0 satisfies the extra property R v0 (x) dx = 0, hence we define x ψ0 (x) ≡

v0 (y) dy,

ψ0 (x) = v0 (x).

(2.2)

−∞

Theorem 2.1. Suppose that the initial data satisfy ψ0 ∈ H 3 (R) ∩ L1 (R), u0 > 0, and u0 − 1 ∈ H 2 (R) ∩ L1 (R). Then there exists a unique solution to (2.1) for t ≥ 0. The solution satisfies u(x, t) > 0 for all x ∈ R and t ≥ 0, with the following decay property: • When ε = 0, for t ≥ 0 we have 3

5

7

(t + 1) 4 v (t) + (t + 1) 4 ( vx + u − 1 )(t) + (t + 1) 4 ux (t) ≤ C,

(2.3)

where C > 0 is a constant depending only on the system parameters and the initial data. • When ε > 0, for an arbitrarily fixed constant N > 0, there exists a corresponding constant δ > 0, such that if (v0 , u0 ) 2 ≤ N and ψ0 21 + u0 − 1 2 ≤ δ, the solution of (2.1) enjoys 3

(t + 1) 4 v (t) +

1

5 k 9 (t + 1) 4 + 2 ( Dxk+1 v + Dxk (u − 1) )(t) + (t + 1) 4 uxx (t) ≤ C, (2.4) k=0

where C > 0 is a constant depending only on the system parameters and the initial data. Our second result is for the original Keller-Segel-Fisher/KPP chemotaxis model. This is to consider the Cauchy problem (1.1), (1.2): 

st = εsxx − μus − σ s, ut = Duxx − χ[u(ln s)x ]x + au(1 −

u K ),

(s, u)(x, 0) = (s0 , u0 )(x),

x ∈ R, t > 0,

(2.5)

x ∈ R,

where χμ > 0, D, a, K > 0, and ε, σ ≥ 0 are constants. The initial data (s0 , u0 ) are a generic perturbation of a constant ground state (¯s , K), with s¯ > 0. Theorem 2.2. Suppose that the initial data satisfy s0 (x) > 0, s0 − s¯ ∈ H 3 (R) ∩ L1 (R), with a constant s¯ > 0, u0 > 0, and u0 − K ∈ H 2 (R) ∩ L1 (R). Then there exists a unique solution to (2.5) for t ≥ 0. The solution satisfies s(x, t) > 0 and u(x, t) > 0 for all x ∈ R and t ≥ 0. Writing s(x, t) = e−(μK+σ )t s˜ (x, t), the solution has the following decay property:

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• When ε = 0, for t ≥ 0 we have 2 1



1 k 5 k (t + 1) 4 + 2 Dxk (˜s − s¯ ) (t) + (t + 1) 4 + 2 Dxk (u − K) (t) ≤ C, k=0

(2.6)

k=0

where C > 0 is a constant depending only on the system parameters and initial data. • When ε > 0, for an arbitrarily fixed constant N > 0, there exists a corresponding constant δ > 0, such that if (s0 , u0 ) 2 ≤ N and s0 − s¯ 21 + u0 − K 2 ≤ δ, the solution of (2.5) enjoys 3 2



1 k 5 k (t + 1) 4 + 2 Dxk (˜s − s¯ ) (t) + (t + 1) 4 + 2 Dxk (u − K) (t) ≤ C, k=0

(2.7)

k=0

where C > 0 is a constant depending only on the system parameters and initial data. Here δ > 0 is uniform in s¯ ∈ [s ∗ , ∞) for a fixed constant s ∗ > 0. Remark 2.3. We remark that in the critical case of μK + σ = 0 (i.e. s˜ (x, t) = s(x, t)), where μ < 0 hence χ < 0, there is no exponential decay or growth for s(x, t). In this case, s(x, t) algebraically approaches to s¯ , with s − s¯ (t) ≤ C(t + 1)−1/4 and s − s¯ L∞ (t) ≤ C(t + 1)−1/2 for some constant C > 0, under respective assumptions for ε = 0 and ε > 0. Remark 2.4. The rates obtained in Theorems 2.1 and 2.2 are optimal. The transformed variable v(x, t) decays with the same rates as the first derivative of a heat kernel, i.e. (t + 1)−3/4 in L2 and (t + 1)−1 in L∞ (by Sobolev inequality). The rates for the perturbation of the cellular population u(x, t) − 1 (or u(x, t) − K in the original setting) are the same as those of vx , or the second derivative of a heat kernel: (t + 1)−5/4 in L2 and (t + 1)−3/2 in L∞ . The compatibility of vx and u − 1 (or u − K) can be observed from the second equation in (2.1) (or (2.5)). Finally, in the critical case of μK + σ = 0, the perturbation of the concentration of chemical signal s(x, t) − s¯ has the same rates as a heat kernel. Other than the critical case, s(x, t) grows or decays exponentially. Remark 2.5. The optimal decay rates recorded in Theorems 2.1 and 2.2 are the first results of their kind for the chemotaxis model (1.1) and its transformed system (1.7). The novelty of our results lies in the fact that they are valid for potentially large-amplitude perturbations. This is rare among the existing results on the general frameworks for hyperbolic-parabolic balance laws (cf. [43] for the first result in the general system), where the full spectrum of the initial perturbations must be assumed to be small in order to identify optimal time decay rates. In the case of bounded domains, whether for systems similar to (1.1) and (1.7) or for those rather different, the Poincaré inequality is oftentimes utilized to obtain optimal (exponential) decay rates of large-data solutions (cf. [17,18,28,37]) via standard energy method. The application of Poincaré inequality allows one to bypass the estimate of low frequency part of the solution. Identifying optimal decay rates of large-data solutions to the Cauchy problem for hyperbolic-parabolic balance laws is usually much more difficult, due to the non-compactness of the spatial domain. In particular, the Poincaré inequality is not available in the whole space, and the low frequency part is the main difficulty in Cauchy problems. Although the standard Lp -based energy methods are effective in studying some of the fundamental properties of solutions, such as global

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well-posedness and global stability near equilibrium states and algebraic decay rates, they are impractical for dealing with more exquisite situations, such as optimal decay rates. Our studies in this paper and [45] indicate that the combination of energy methods and spectral analysis is an effective approach for handling such delicate problem, and we expect that this will open new doors and offer future opportunities in the rigorous studies of chemotaxis models. In particular, it is well imagined that the combination of the analytical approaches of this paper and [45] can be utilized to obtain similar results for the non-growth model (i.e., (1.1) with a = 0) and other related chemotaxis models (cf. [2]). We leave the investigation in forthcoming papers. Remark 2.6. The transformed system (1.7) itself has deep mathematical interest as it serves as a prototype of general hyperbolic-parabolic balance laws. Those are equations arising from continuum mechanics such as thermal non-equilibrium flows [42,44]. The nature of non-uniform parabolicity coupled with nonlinear flux functions and nonlinear source terms in this type of equations presents significant challenge in mathematical analysis. Besides being an effective vehicle in the study of chemotaxis model (1.1), the analysis of (1.7) helps to shed light on how to advance fundamental research of hyperbolic-parabolic balance laws in related topics by carefully developing a sophisticated approach to meet the specific challenge. This is to start with energy estimate, continued with weighted energy estimate in its full capacity for decay rates. Then a repeated iteration follows based on detailed spectral analysis and Duhamel’s principle. We expect the approach or its varieties to be hopeful to other hyperbolic-parabolic balance laws. Remark 2.7. We further remark that our proof of the optimal decay rates relies heavily on the energy estimates and weighted energy estimates established in [45]. To extend our result to multi-space dimensions, one must overcome the difficulty in obtaining energy estimate on low frequency due to the lack of Poincaré inequality in the whole space. The main issue with ε > 0 is how to find an appropriate entropy function in higher space dimensions, while in the case of ε = 0 the issue is the lack of sufficient dissipation. This raises an interesting and challenging question for future studies in this area. 3. Linear system In this section we carry out spectral analysis and energy estimate for the linear system corresponding to (2.1). They are crucial to the identification of optimal rates for the nonlinear systems. Throughout this section we use C to denote a universal positive constant, depending only on the system parameters. The value of C may vary line be line, according to the context. First we rewrite (2.1) in terms of the perturbation. Let 

 v w1 (x, t) = (x, t), u−1 w2   v0 w01 (x) = (x). w0 (x) = u0 − 1 w02

w(x, t) =

Then (2.1) can be written as

(3.1) (3.2)

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w1t + w2x = ε1 w1xx + (ε2 w12 )x , w2t + w1x = w2xx − rw2 − (w1 w2 )x − rw22 ,

(3.3)

(w1 , w2 )(x, 0) = (w01 , w02 )(x), or in the vector form as wt + Awx = Bwxx + Lw + R, w(x, 0) = w0 (x),

(3.4)

where 

0 1 A= , 1 0 R = R1x + R2x + R3 ,



0 , 1

0 0 L= , 0 −r   0 0 R2 = , R3 = . −w1 w2 −rw22

ε B= 1 0  ε w2 R1 = 2 1 , 0



(3.5)

Consider Fourier transform with respect to x:  w(ξ, ˆ t) =

w(x, t)e−ixξ dx,

R

w(x, t) =

1 2π



(3.6) w(ξ, ˆ t)eixξ dξ.

R

Taking Fourier transform of (3.4) with respect to x, we have ˆ wˆ t = E(iξ )wˆ + R, E(iξ ) = −iξ A − ξ 2 B + L.

(3.7)

The solution of (3.7) is t w(ξ, ˆ t) = e

tE(iξ )

w(ξ, ˆ 0) +

ˆ τ ) dτ. e(t−τ )E(iξ ) R(ξ,

(3.8)

0

To study the solution operator in (3.8), we perform spectral analysis for 

−ε1 ξ 2 E(iξ ) = −iξ

−iξ . −ξ 2 − r

By direct calculation, the eigenvalues of E(iξ ) are 1 λ1,2 (iξ ) = − [r + (ε1 + 1)ξ 2 ] ± 2



1 [r + (ε1 + 1)ξ 2 ]2 − ξ 2 (ε1 r + 1 + ε1 ξ 2 ), 4

(3.9)

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12

and the corresponding eigenprojections are P1,2 (iξ ) =



1 −ξ 2 + (λ1,2 + ε1 ξ 2 )2

−ξ 2 −iξ(λ1,2 + ε1 ξ 2 )

−iξ(λ1,2 + ε1 ξ 2 ) . (λ1,2 + ε1 ξ 2 )2

(3.10)

The eigen-decomposition of E(iξ ) is E(iξ ) = λ1 (iξ )P1 (iξ ) + λ2 (iξ )P2 (iξ ), hence etE(iξ ) = eλ1 (iξ )t P1 (iξ ) + eλ2 (iξ )t P2 (iξ ).

(3.11)

Regarding time decay rates, the leading terms in the solution operator come from small ξ . Therefore, we take Taylor expansions for |ξ |  1 in (3.9) and (3.10). By direct calculation we have: 1 λ1 (iξ ) = −(ε1 + )ξ 2 + O(ξ 4 ), λ2 (iξ ) = −r + O(ξ 2 ), r

  − iξr + O(ξ 3 ) 1 + O(ξ 2 ) O(ξ 2 ) , P2 (iξ ) = P1 (iξ ) = iξ ξ2 3 4 O(ξ ) − r + O(ξ ) − r 2 + O(ξ )

O(ξ ) . 1 + O(ξ 2 )

(3.12)

Next we state a lemma concerning a global decay property of the solution operator. The lemma applies to a general class of systems of hyperbolic-parabolic balance laws that satisfy a set of structural conditions proposed in [42], and (2.1) indeed satisfies that set of conditions, see [43]. A proof for a general system can be found in [10], also see [43]. Such a proof has made the crucial use of existence of a “compensating function”. The existence of a compensating function, on the other hand, is established through an elegant and sophisticated argument in [29]. For (2.1) or (3.3), however, here we construct a compensating function K explicitly as  K=

− 12 0

0 1 2

.

(3.13)

Therefore, we carry out the energy estimate directly to prove the lemma as follows. Lemma 3.1. Let ε1 ≥ 0 and r ≥ 0 be constants. Let E(iξ ) be defined in (3.7), with A, B and L given in (3.5). Then there exist positive constants C and c, depending only on ε1 and r, such that |etE(iξ ) | ≤ Ce

−

cξ 2 t 1+ξ 2

,

ξ ∈ R. t ≥ 0.

Proof. Consider the corresponding linear system of (3.4): wt + Awx = Bwxx + Lw. Taking Fourier transform with respect to x gives us

(3.14)

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13

wˆ t = −iξ Awˆ − ξ 2 B wˆ + Lwˆ = E(iξ )w. ˆ

(3.15)

w(ξ, ˆ t) = etE(iξ ) w(ξ, ˆ 0).

(3.16)

The solution of (3.15) is

Denote the complex conjugate of wˆ by wˆ ∗ . Multiplying (3.15) by (wˆ ∗ )t from the left and taking the real part, we have 1 (|wˆ 1 |2 + |wˆ 2 |2 )t = −ξ 2 (ε1 |wˆ 1 |2 + |wˆ 2 |2 ) − r|wˆ 2 |2 . 2

(3.17)

Next we multiply (3.15) by iξ(wˆ ∗ )t K from the left, with K given in (3.13), and take the real part. This gives us 1 1 iξ(−wˆ 1∗ wˆ 2 + wˆ 1 wˆ 2∗ )t + ξ 2 (|wˆ 1 |2 − |wˆ 2 |2 ) 4 2 1 3 r = − iξ (ε1 + 1)(−wˆ 1∗ wˆ 2 + wˆ 1 wˆ 2∗ ) + iξ(wˆ 1∗ wˆ 2 − wˆ 1 wˆ 2∗ ). 4 2 Adding ξ 2 |wˆ 2 |2 to both sides, we further have 1 1 iξ(wˆ 1 wˆ 2∗ − wˆ 1∗ wˆ 2 )t + ξ 2 (|wˆ 1 |2 + |wˆ 2 |2 ) 4 2 1 r ≤ξ 2 |wˆ 2 |2 + (ε1 + 1)|ξ |3 |wˆ 1 ||wˆ 2 | + |ξ ||wˆ 1 ||wˆ 2 | 2 2 1 1 1 r ≤ξ 2 |wˆ 2 |2 + ξ 2 |wˆ 1 |2 + (ε1 + 1)2 |ξ |4 |wˆ 2 |2 + ξ 2 |wˆ 1 |2 + |wˆ 2 |2 , 8 2 8 2 which is simplified as 1 1 1 r iξ(wˆ 1 wˆ 2∗ − wˆ 1∗ wˆ 2 )t + ξ 2 (|wˆ 1 |2 + |wˆ 2 |2 ) ≤ ξ 2 |wˆ 2 |2 + (ε1 + 1)2 |ξ |4 |wˆ 2 |2 + |wˆ 2 |2 . (3.18) 4 4 2 2 We take a positive quantity α: 

 2 2 α = min 1, , , (ε1 + 1)2 r where we omit 2/r if r = 0. Now we multiply (3.17) by (1 + ξ 2 ), and (3.18) by α. Summing up the results we arrive at (1 + ξ 2 )Mt +

α 2 ξ (|wˆ 1 |2 + |wˆ 2 |2 ) ≤ 0, 4

(3.19)

where 1 iξ α (wˆ 1 wˆ 2∗ − wˆ 1∗ wˆ 2 ). M = M(ξ, t) = (|wˆ 1 |2 + |wˆ 2 |2 ) + 2 4(1 + ξ 2 )

(3.20)

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14

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Y. Zeng, K. Zhao / J. Differential Equations ••• (••••) •••–•••

Note that    iξ α  1 |ξ | ∗ ∗   ( w ˆ |wˆ 1 ||wˆ 2 | ≤ (|wˆ 1 |2 + |wˆ 2 |2 ). w ˆ − w ˆ w ˆ ) 1 2 ≤  4(1 + ξ 2 ) 1 2 8 2(1 + ξ 2 )

(3.21)

Using Euclidean norm for w, ˆ i.e., |w| ˆ 2 = |wˆ 1 |2 + |wˆ 2 |2 , (3.20) and (3.21) imply that M is equiv2 alent to |w| ˆ : 3 2 5 2 |w| ˆ ≤ M ≤ |w| ˆ . 8 8

(3.22)

Applying (3.22) to (3.19) gives us Mt +

2αξ 2 M ≤ 0. 5(1 + ξ 2 )

Solving the inequality we have M(ξ, t) ≤ e

−

2αξ 2 t 5(1+ξ 2 )

M(ξ, 0).

(3.23)

Applying (3.22) to (3.23) we further have 2

3 5 − 2αξ t ˆ 0)|2 . |w(ξ, ˆ t)|2 ≤ e 5(1+ξ 2 ) |w(ξ, 8 8 Together with (3.16), this implies   5 − αξ 2 t    tE(iξ ) w(ξ, ˆ 0) ≤ ˆ 0)|. e 5(1+ξ 2 ) |w(ξ, e 3 √ We thus prove (3.14) with C = 5/3 and c = α/5. 2 Remark 3.2. Lemma 3.1 applies to ε1 ≥ 0 and r ≥ 0, while in this paper we only need r > 0. Lemma 3.3. Let k ≥ 0 be an integer, h1 , h2 ∈ L1 (R), Dxk h1 , Dxk h2 ∈ L2 (R), and  H1 (x) =

h1 (x) , 0

 H2 (x) =

0 . h2 (x)

Let (etE(iξ ) )1,2 denote the first/second row of etE(iξ ) . Then for t ≥ 0, (etE(iξ ) )1 (iξ )k Hˆ 1 (ξ ) ≤ C(t + 1)− 4 − 2 h1 L1 + Ce−ct Dxk h1 ,

(3.24)

(etE(iξ ) )1 (iξ )k Hˆ 2 (ξ ) ≤ C(t + 1)

h2 L1 + Ce−ct Dxk h2 ,

(3.25)

(etE(iξ ) )2 (iξ )k Hˆ 1 (ξ ) ≤ C(t + 1)

h1 L1 + Ce−ct Dxk h1 ,

(3.26)

(etE(iξ ) )2 (iξ )k Hˆ 2 (ξ ) ≤ C(t + 1)

h2 L1 + Ce−ct Dxk h2 ,

(3.27)

1

k

− 34 − k2 − 34 − k2 − 54 − k2

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15

where C and c are positive constants depending only on ε1 ≥ 0 and r > 0. Proof. We prove (3.24) first. For a small constant η > 0, by definition we have (etE(iξ ) )1 (iξ )k Hˆ 1 (ξ ) 2 = I1 + I2 ,  |(etE(iξ ) )1 (iξ )k Hˆ 1 (ξ )|2 dξ, I1 =



|(etE(iξ ) )1 (iξ )k Hˆ 1 (ξ )|2 dξ.

I2 =

|ξ |≤η

(3.28)

|ξ |≥η

Since η > 0 is small, we apply (3.11) and (3.12) to I1 to have 

2 [e− 2r ξ t 2|ξ |k |hˆ 1 (ξ )| + e− 2 t |ξ |k |hˆ 1 (ξ )|]2 dξ 1

I1 ≤

r

|ξ |≤η





e− r ξ t ξ 2k dξ hˆ 1 2L∞ + Ce−rt 1 2

≤C |ξ |≤η

|(iξ )k hˆ 1 (ξ )|2 dξ

(3.29)

|ξ |≤η − 12 −k

≤ C(t + 1)

h1 2L1 + Ce−rt Dxk h1 2 .

For I2 we apply (3.14) to have 

|etE(iξ ) |2 |(iξ )k hˆ 1 (ξ )|2 dξ ≤ C

I2 ≤ C |ξ |≥η

≤ Ce

˜ 2t − 2cη 1+η2

 e

2t 1+ξ 2

˜ − 2cξ

|(iξ )k hˆ 1 (ξ )|2 dξ

|ξ |≥η

(3.30)

Dxk h1 2 ,

where c˜ > 0 is a constant. Substituting (3.29) and (3.30) into (3.28) gives us (3.24), with 

 r cη ˜ 2 c = min , . 2 1 + η2 To prove (3.25), similarly, (etE(iξ ) )1 (iξ )k Hˆ 2 (ξ ) 2 =

 

 +

|ξ |≤η



|(etE(iξ ) )1 (iξ )k Hˆ 2 (ξ )|2 dξ ≡ I3 + I4 .

|ξ |≥η

Here I4 is similar to I2 , while from (3.12) we have  1 2 2 r I3 ≤ [e− 2r ξ t |ξ |k+1 |hˆ 2 (ξ )| + e− 2 t |ξ |k |hˆ 2 (ξ )|]2 dξ r |ξ |≤η



≤C

e

− 1r ξ 2 t 2k+2

ξ

|ξ |≤η



dξ hˆ 2 2L∞ + Ce−rt

|ξ |≤η 3

≤ C(t + 1)− 2 −k h2 2L1 + Ce−rt Dxk h2 2 .

|(iξ )k hˆ 2 (ξ )|2 dξ

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16

This gives us (3.25). The proof of (3.26) and (3.27) is similar.

2

4. Nonlinear transformed system We now start the proof of our main results on the nonlinear systems. We prove Theorem 2.1 on the transformed system in this section, and Theorem 2.2 on the original system in next section. Throughout these two sections, unless specified otherwise, we use C to denote a generic positive constant that depends on the system parameters and the initial data. The value of C may change line by line according to the context. First we rephrase Theorem 1.3 for the case ε = 0, where the constant T0 > 0 is chosen so that for t ≥ T0 , u − 1 L∞ (t) ≤ 1/2, see [45]. Such a constant T0 exists due to Theorem 1.2. Thus T0 depends on the solution, which is uniquely determined by the initial data for a fixed r > 0, with ε1 = ε2 = 0 in (1.7), via Theorem 1.1. That is, T0 depends only on r and the initial data. Let C˜ 3 = max{C3 , sup [(t + 1) v 2 (t) + (t + 1)2 u − 1 2 (t)]} 0≤t≤T0

T0 (t + 1) (vx , u − 1) 2 (t) dt,

+ 0

T0 C˜ 4 = max{C4 , sup [(t + 1) (vx , ux ) (t)]} + (t + 1)2 (vxx , ux ) 2 (t) dt, 2

(4.1)

2

0≤t≤T0

0

C˜ 5 = max{C5 , sup [(t + 1)3 (vxx , uxx ) 2 (t)]} + 0≤t≤T0

T0 (t + 1)3 uxx ) 21 (t) dt, 0

where C3 , C4 and C5 are the same as in Theorem 1.3 for ε = 0. Then C˜ 3 , C˜ 4 and C˜ 5 are welldefined according to Theorem 1.1, and depend on r and the initial data only. Combining (4.1) and (1.15) for the case ε = 0, we restate Theorem 1.3 as follows. Theorem 4.1. Let the hypotheses of Theorem 1.1 hold, and assume that ψ0 ∈ L2 (R). • When ε = 0, the global-in-time solution to (2.1) satisfies the following for t ≥ 0: t (t + 1) v (t) + (t + 1) u − 1 (t) + 2

2

2

(τ + 1) (vx , u − 1) 2 (τ ) dτ ≤ C˜ 3 ,

(4.2)

0

t (t + 1)2 (vx , ux ) 2 (t) +

(τ + 1)2 (vxx , ux ) 2 (τ ) dτ ≤ C˜ 4 ,

(4.3)

0

t (t + 1) (vxx , uxx ) (t) + 3

2

0

(τ + 1)3 uxx 21 (τ ) dτ ≤ C˜ 5 ,

(4.4)

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17

where the positive constants C˜ 3 , C˜ 4 and C˜ 5 depend only on the system parameters and initial data. • When ε > 0, for an arbitrarily fixed constant N > 0, there exists a corresponding constant δ > 0, such that if (v0 , u0 ) 2 ≤ N and ψ0 21 + u0 − 1 2 ≤ δ, the global-in-time solution to (2.1) satisfies the following for t ≥ 0: t (t + 1) v (t) + (t + 1) u − 1 (t) + 2

2

2

(τ + 1) (vx , u − 1) 2 (τ ) dτ ≤ C¯ 3 ,

(4.5)

0

t (t + 1) (vx , ux ) (t) + 2

2

(τ + 1)2 (vxx , ux ) 2 (τ ) dτ ≤ C¯ 4 ,

(4.6)

0

t (t + 1) (vxx , uxx ) (t) + 3

2

(τ + 1)3 [ε vxxx 2 + uxx 21 ](τ ) dτ ≤ C¯ 5 ,

(4.7)

0

where the positive constants C¯ 3 , C¯ 4 and C¯ 5 are independent of t and ε, and depend on r, D, χ and the initial data. To prove Theorem 2.1, we need to improve the time decay rates in (4.2)–(4.7) to the optimal ones in (2.3) and (2.4), under the additional assumption (ψ0 , u0 − 1) ∈ L1 (R). For this we apply Plancherel theorem and (3.8) to have

(k)

(k)

Dxk wj (t) = (iξ )k wˆ j (t) ≤ Ij 1 + Ij 2 , (k) Ij 1

= (iξ ) (e k

tE(iξ )

)j w(ξ, ˆ 0) ,

(k) Ij 2

j = 1, 2,

t =

k ≥ 0,

ˆ τ ) dτ, (iξ )k (e(t−τ )E(iξ ) )j R(ξ,

(4.8)

0

where (etE(iξ ) )j , j = 1, 2, denotes the j th row of etE(iξ ) , and R is given in (3.5). Lemma 4.2. For an integer 0 ≤ k ≤ 2, under the assumptions of Theorem 2.1 we have 3

k

5

k

I11 ≤ C(t + 1)− 4 − 2 ( ψ0 L1 + u0 − 1 L1 ) + Ce−ct [ Dxk+1 ψ0 + Dxk (u0 − 1) ], (4.9) (k)

I21 ≤ C(t + 1)− 4 − 2 ( ψ0 L1 + u0 − 1 L1 ) + Ce−ct [ Dxk+1 ψ0 + Dxk (u0 − 1) ] (4.10) (k)

for t ≥ 0, where C, c > 0 are constants depending only on the system parameters.

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18

Proof. From (4.8), (3.2), (2.2), (3.24) and (3.25), we have (k) 2 (I11 ) =

 |(iξ )k (etE(iξ ) )1 [(vˆ0 , 0)t + (0, wˆ 02 )t ](ξ )|2 dξ R



≤2

|(iξ )k+1 (etE(iξ ) )1 (ψˆ 0 (ξ ), 0)t |2 dξ + 2

R

 |(iξ )k (etE(iξ ) )1 (0, wˆ 02 (ξ ))t |2 dξ R

− 14 − k+1 2

≤ C[(t + 1)

3

k

ψ0 L1 + e−ct Dxk+1 ψ0 ]2 + C[(t + 1)− 4 − 2 w02 L1

+ e−ct Dxk w02 ]2 , where C and c are positive constants depending only on the system parameters. Taking the square root on both sides we obtain (4.9). The proof of (4.10) is similar, using (3.26) and (3.27). 2 Recall Sobolev inequality: For f ∈ H 1 (R), f L∞ ≤

√ 1 1 2 f 2 f  2 .

(4.11)

Lemma 4.3. Let ε ≥ 0 and k = 0, 1. Under the assumptions of Theorem 2.1, we have t

(iξ )k (e(t−τ )E(iξ ) )1 (iξ Rˆ 2 + Rˆ 3 )(ξ, τ ) dτ ≤ C(t + 1)− 4 − 2 3

k

(4.12)

0

for t ≥ 0, where R2 and R3 are defined in (3.5), and C > 0 is a constant depending on the system parameters and the initial data of (2.1). In the special case ε = 0, (4.12) is reduced to 3

k

5

k

I12 ≤ C(t + 1)− 4 − 2 . (k)

(4.13)

In the case ε = 0 we also have I22 ≤ C(t + 1)− 4 − 2 . (k)

(4.14)

(k) Proof. Denote the left-hand side of (4.12) as I˜12 . When ε = 0, ε2 = ε/χ = 0. From (3.5) we have

R = R2x + R3 .

(4.15)

Therefore, (4.8) and (4.15) imply I12 = I˜12 . Equation (4.12) is reduced to (4.13). (k) It remains to prove (4.12) for ε ≥ 0 and (4.14) for ε = 0. Applying (3.25) to I˜12 , k = 0, 1, and noting (3.5), we have (k)

(k)

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Y. Zeng, K. Zhao / J. Differential Equations ••• (••••) •••–••• (k) I˜12 ≤

t (iξ )

k+1

(e

(t−τ )E(iξ )

)1 Rˆ 2 (ξ, τ ) dτ +

0

t

19

(iξ )k (e(t−τ )E(iξ ) )1 Rˆ 3 (ξ, τ ) dτ

0

t ≤C

3

[(t − τ + 1)− 4 −

k+1 2

w1 w2 L1 (τ ) + e−c(t−τ ) Dxk+1 (w1 w2 ) (τ )] dτ

(4.16)

0

t +C

3

k

[(t − τ + 1)− 4 − 2 w22 L1 (τ ) + e−c(t−τ ) Dxk (w22 ) (τ )] dτ,

0

where C, c > 0 are constants depending on the system parameters. From Cauchy-Schwarz inequality, (3.1), (4.2) – (4.7) and (4.11), we have 3

w1 w2 L1 (τ ) ≤ ( w1 w2 )(τ ) = ( v u − 1 )(τ ) ≤ C(τ + 1)− 2 , w22 L1 (τ ) = w2 2 (τ ) ≤ C(τ + 1)−2 , Dx (w1 w2 ) (τ ) ≤ ( w1x w2 + w1 w2x )(τ ) ≤ ( w1x w2 L∞ + w1 L∞ w2x )(τ ) 1

1

1

1

7

≤ C( w1x w2 2 w2x 2 + w1 2 w1x 2 w2x )(τ ) ≤ C(τ + 1)− 4 , 1

1

1

(4.17)

1

Dx2 (w1 w2 ) (τ ) ≤ C( w1xx w2 2 w2x 2 + w1x 2 w1xx 2 w2x 1

1

9

+ w1 2 w1x 2 w2xx )(τ ) ≤ C(τ + 1)− 4 , 3

1

w22 (τ ) ≤ C( w2 2 w2x 2 )(τ ) ≤ C(τ + 1)−2 , 1

3

Dx (w22 ) (τ ) ≤ C( w2 2 w2x 2 )(τ ) ≤ C(τ + 1)−2 , where the constant C > 0 depends on the system parameters and the initial data. Substituting (4.17) into (4.16) gives us

(k) I˜12

t ≤C

5

3

k

3

k

[(t − τ + 1)− 4 − 2 (τ + 1)− 2 + (t − τ + 1)− 4 − 2 (τ + 1)−2 ] dτ

0

t +C

(4.18) − 74 − k2

e−c(t−τ ) [(τ + 1)

− 34 − k2

+ (τ + 1)−2 ] dτ ≤ C(t + 1)

,

k = 0, 1.

0

This is (4.12). (k) Similarly, applying (3.27) to I22 in (4.8) for k = 0, 1 and noting (4.15), we have

(k) I22

t ≤

(iξ )

k+1

0

(e

(t−τ )E(iξ )

)2 Rˆ 2 (ξ, τ ) dτ +

t 0

(iξ )k (e(t−τ )E(iξ ) )2 Rˆ 3 (ξ, τ ) dτ

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20

t ≤C

5

[(t − τ + 1)− 4 −

k+1 2

w1 w2 L1 (τ ) + e−c(t−τ ) Dxk+1 (w1 w2 ) (τ )] dτ

(4.19)

0

t +C

5

k

[(t − τ + 1)− 4 − 2 w22 L1 (τ ) + e−c(t−τ ) Dxk (w22 ) (τ )] dτ,

0

which is parallel to (4.16). However, we replace the first estimate in (4.17) by using (4.13), which has been established. That is, from (4.8), (4.9) and (4.13) we have 3

w1 (τ ) ≤ I11 + I12 ≤ C(τ + 1)− 4 . (0)

(0)

Therefore, the first estimate in (4.17) is improved to 7

w1 w2 L1 (τ ) ≤ ( w1 u − 1 )(τ ) ≤ C(τ + 1)− 4 .

(4.20)

Substituting (4.20) and other estimates in (4.17) into (4.19) we arrive at (k) I22

t ≤C

7

7

k

5

k

[(t − τ + 1)− 4 − 2 (τ + 1)− 4 + (t − τ + 1)− 4 − 2 (τ + 1)−2 ] dτ

0

t +C

7

5

k

k

e−c(t−τ ) [(τ + 1)− 4 − 2 + (τ + 1)−2 ] dτ ≤ C(t + 1)− 4 − 2 ,

k = 0, 1. 2

0

Combining (4.8) and Lemmas 4.2 and 4.3, for ε = 0 we have 3

k

Dxk v (t) = Dxk w1 (t) ≤ C(t + 1)− 4 − 2 , 5

k

Dxk (u − 1) (t) = Dxk w2 (t) ≤ C(t + 1)− 4 − 2 for t ≥ 0 and k = 0, 1, where C > 0 is a constant depending on the system parameters and the initial data of (2.1). This settles the case ε = 0 in Theorem 2.1. Lemma 4.4. Let ε > 0. Under the assumptions of Theorem 2.1, we have 3

v (t) ≤ C(t + 1)− 4 ,

t ≥ 0,

(4.21)

where C > 0 is a constant depending on the system parameters and the initial data of (2.1). Proof. For ε > 0 we define 3

M(t) = sup [(τ + 1) 4 v (τ )], 0≤τ ≤t

which is well-defined according to Theorem 1.1. From (4.22) we have

(4.22)

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Y. Zeng, K. Zhao / J. Differential Equations ••• (••••) •••–•••

21

3

v (t) ≤ M(t)(t + 1)− 4 .

(4.23)

From (4.8) and (3.5), we have (0) I12

t ≤

[ iξ(e(t−τ )E(iξ ) )1 Rˆ 1 (ξ, τ ) + iξ(e(t−τ )E(iξ ) )1 Rˆ 2 (ξ, τ ) + (e(t−τ )E(iξ ) )1 Rˆ 3 (ξ, τ ) ] dτ.

0

Applying (4.12) in Lemma 4.3 we have t

(0) I12

≤

iξ(e(t−τ )E(iξ ) )1 Rˆ 1 (ξ, τ ) dτ + C(t + 1)− 4 . 3

0

Applying (3.24) to the first term, with R1 given in (3.5), we further have (0) I12

t ≤C

− 34

(t − τ + 1)

t ε2 w12 L1 (τ ) dτ

+C

0

3

e−c(t−τ ) ε2 Dx (w12 ) (τ ) dτ + C(t + 1)− 4 .

0

(4.24) By (4.23), (3.1), (4.5), (4.6) and (4.11), we arrive at 3

1

1

3

1

9

w12 L1 (τ ) = w1 2 (τ ) ≤ [M(τ )(τ + 1)− 4 ] 2 [C(τ + 1)− 2 ] 2 = CM 2 (τ )(τ + 1)− 8 , 1

3

7

Dx (w12 ) (τ ) ≤ C( w1 L∞ w1x )(τ ) ≤ C( w1 2 w1x 2 )(τ ) ≤ C(τ + 1)− 4 .

(4.25)

Substituting (4.25) into (4.24) gives us (0) I12

1 2

t

≤ Cε2 M (t)

− 34

(t − τ + 1)

− 98

(τ + 1)

t dτ + Cε2

0

7

e−c(t−τ ) (τ + 1)− 4 dτ (4.26)

0 3

1

3

3

+ C(t + 1)− 4 ≤ Cε2 M 2 (t)(t + 1)− 4 + C(t + 1)− 4 . Combining (4.8), (4.9) and (4.26) we have 3

1

3

v (t) = w1 (t) ≤ I11 + I12 ≤ C(t + 1)− 4 + CM 2 (t)(t + 1)− 4 , (0)

(0)

which implies 1 1 M(t) ≤ C + CM 2 (t) ≤ C + M(t) 2

by (4.22) and Young’s inequality. This is simplified to M(t) ≤ C, or 3

v (t) = w1 (t) ≤ C(t + 1)− 4 by (4.23).

2

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22

Lemma 4.5. Let ε > 0. Under the assumptions of Theorem 2.1, for t ≥ 0 we have 3

k

Dxk v (t) ≤ C(t + 1)− 4 − 2 ,

k = 1, 2,

− 54 − k2

Dxk (u − 1) (t) ≤ C(t + 1)

(4.27)

k = 0, 1, 2,

,

(4.28)

where C > 0 is a constant depending on the system parameters and the initial data of (2.1). Proof. Following the derivation of (4.24) we have (1) I12

t ≤C

− 54

(t − τ + 1)

t ε2 w12 L1 (τ ) dτ

0

+C

5

e−c(t−τ ) ε2 Dx2 (w12 ) (τ ) dτ + C(t + 1)− 4 .

0

(4.29) With (4.21) we have 3

w12 L1 (τ ) = w1 2 (τ ) = v 2 (τ ) ≤ C(τ + 1)− 2 .

(4.30)

With (4.3), (4.4) and (4.11) we also have 3

1

1

1

Dx2 (w12 ) (τ ) ≤ C( vx2 + vvxx )(τ ) ≤ C( vx 2 vxx 2 + v 2 vx 2 vxx )(τ ) 9

≤ C(τ + 1)− 4 .

(4.31)

Substituting (4.30) and (4.31) into (4.29), we arrive at 5

I12 ≤ C(t + 1)− 4 . (1)

(4.32)

Combining (4.8), (4.9) and (4.32) we settle (4.27) for k = 1. For k = 2 we follow the derivation of (4.24) but choose integration by parts differently on different time intervals. That is, applying (4.8), (3.5), (3.24) and (3.25) we have t

(2) I12

2 ≤C

− 74

(t − τ + 1)

t ε2 w12 L1 (τ ) dτ

+C

5

(t − τ + 1)− 4 ε2 Dx (w12 ) L1 (τ ) dτ

t 2

0

t +C

− 94

(t − τ + 1) 0

t +C

t w1 w2 L1 (τ ) dτ + C

7

(t − τ + 1)− 4 rw22 L1 (τ ) dτ

(4.33)

0

e−c(t−τ ) ( ε2 Dx3 (w12 ) + Dx3 (w1 w2 ) + rDx2 (w22 ) )(τ ) dτ.

0

Similar to (4.30) and (4.17), with updated rates for v and vx from (4.21) and (4.27) (for k = 1), respectively, we have

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23

Dx (w12 ) L1 (τ ) ≤ 2( w1 w1x )(τ ) = 2( v vx )(τ ) ≤ C(τ + 1)−2 , 7

w1 w2 L1 (τ ) ≤ ( w1 w2 )(τ ) = ( v u − 1 )(τ ) ≤ C(τ + 1)− 4 , Dx3 (w12 ) (τ ) ≤ C( w1x L∞ w1xx + w1 L∞ w1xxx )(τ ) 1

3

1

1

≤ C( w1x 2 w1xx 2 + w1 2 w1x 2 w1xxx )(τ ) 23

≤ C(τ + 1)− 8 + C(τ + 1)−1 Dx3 v (τ ), Dx3 (w1 w2 ) (τ ) ≤ C( w2 L∞ w1xxx + w2x L∞ w1xx + w1x L∞ w2xx 11

+ w1 L∞ w2xxx )(τ ) ≤ C(τ + 1)− 4 + C(τ + 1)−1 ( Dx3 v + Dx3 u )(τ ), 9

Dx2 (w22 ) (τ ) ≤ C( w2x L∞ w2x + w2 L∞ w2xx )(τ ) ≤ C(τ + 1)− 4 . 3

(4.34) 1

For instance, in the last inequality, ( w2x L∞ w2x )(τ ) ≤ C( ux 2 uxx 2 )(τ ) by (3.1) and 9 (4.11), which is further bounded by C(τ + 1)− 4 according to (4.6) and (4.7). Similarly, 1 1 5 ( w2 L∞ w2xx )(τ ) = ( u − 1 L∞ uxx )(τ ) ≤ C( u − 1 2 ux 2 uxx )(τ ) ≤ C(τ + 1)− 2 according to (4.5) to (4.7), hence is a higher order term and absorbed into the first one. Substituting (4.30), (4.34) and (4.17) into (4.33) give us

(2) I12

− 74

≤ C(t + 1)

t +C

e−c(t−τ ) (τ + 1)−1 ( Dx3 v + Dx3 u )(τ ) dτ

0 − 74

≤ C(t + 1)

t t 1 1 −2c(t−τ ) −5 2 + C[ e (τ + 1) dτ ] [ (τ + 1)3 ( Dx3 v 2 + Dx3 u 2 )(τ ) dτ ] 2 0

− 74

≤ C(t + 1)

0

(4.35)

,

where we have used Cauchy-Schwarz inequality and (4.7). Combining (4.8), (4.9) and (4.35) we obtain (4.27) for k = 2. The proof of (4.28) is similar to that of (4.27). In particular, in the estimate for uxx we use t

2

(2)

I22 ≤ C

9

(t − τ + 1)− 4 ε2 w12 L1 (τ ) dτ + C

C

5

(t − τ + 1)− 4 ε2 Dx2 (w12 ) L1 (τ ) dτ

t 2

0

t

t

9

(t − τ + 1)− 4 ( Dx (w1 w2 ) L1 + rw22 L1 )(τ ) dτ

0

t +C 0

e−c(t−τ ) ( ε2 Dx3 (w12 ) + Dx3 (w1 w2 ) + rDx2 (w22 ) )(τ ) dτ,

(4.36)

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24

which is similar to (4.33). The proof proceeds in the same way as that for vxx , except that we need the following additional, updated estimates: 5

Dx2 (w12 ) L1 (τ ) + Dx (w1 w2 ) L1 (τ ) + w22 L1 (τ ) ≤ C(τ + 1)− 2 ,

(4.37)

noting that we have established the optimal estimates for Dxk v , 0 ≤ k ≤ 2, and Dxk (u − 1) , k = 0, 1, at this point. That is, substituting (4.27) and (4.37) into (4.36), and repeating the calculation in (4.35), we obtain

(2) I22

t

− 94

≤ C(t + 1)

+C

e−c(t−τ ) (τ + 1)−1 ( Dx3 v + Dx3 u )(τ ) dτ

(4.38)

0 9

≤ C(t + 1)− 4 . Combining (4.8), (4.10) and (4.38), we have (4.28) for k = 2.

2

With Lemmas 4.4 and 4.5, we settle the case ε > 0 in Theorem 2.1. 5. Original chemotaxis model In this section we prove Theorem 2.2. First we verify that under the assumptions of Theorem 2.2, the initial data of the transformed system satisfy the hypotheses in Theorem 2.1. We note that by Sobolev embedding theorem, the assumption s0 − s¯ ∈ H 3 (R) in Theorem 2.2 implies s0 ∈ C 2 (R), with s0 (x) → s¯ as x → ±∞. Together with the assumption s¯ > 0 and s0 (x) > 0 for x ∈ R, we conclude that there is a constant s > 0 such that s0 (x) ≥ s for x ∈ R. Recalling (1.13) and (1.9), we have x ψ0 (x) = −∞

|χ| v0 (y) dy = sign(χ) D

√Dx χμK



−∞

 s0 (y) Dx χ dy = ln s0 ( √ ) − ln s¯ . s0 (y) D χμK

(5.1)

By the mean value theorem, |ψ0 (x)| ≤

|χ| Dx |s0 ( √ ) − s¯ |, Ds χμK

(5.2)

which implies ψ0 ∈ L2 (R) since s0 − s¯ ∈ L2 (R). Similarly, we can verify that v0 and its derivatives up to the second order are in L2 (R), or ψ0 ∈ H 3 (R), noting the Sobolev inequality (4.11). Also, under the assumption s0 − s¯ ∈ L1 (R), (5.2) implies ψ0 ∈ L1 (R). Therefore, under the assumptions of Theorem 2.2, we have ψ0 ∈ H 3 (R) ∩ L1 (R), u˜ 0 > 0, and u˜ 0 − 1 ∈ H 2 (R) ∩ L1 (R) for the transformed system, where u˜ 0 is the initial datum u0 in Theorem 2.1: u˜ 0 =

 1  D u0 √ x , K χμK

(5.3)

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25

see (1.9) and (1.10). Applying Theorem 2.1, we have a unique, global solution to (2.1). This is translated to the unique, global solution to (2.5) through the transformation ⎛



μK s(x, t) = s¯ exp ⎝sign(χ) χ

x −∞

⎞ χμK χμK v( y, t) dy − (μK + σ )t ⎠ , D D √

 √χμK χμK  u(x, t) ← Ku x, t . D D

(5.4)

The solution satisfies s(x, t) > 0 and u(x, t) > 0 for all x ∈ R and t ≥ 0. For the case ε = 0, we immediately have the decay estimate (2.3). For the case ε > 0, however, we need the smallness of ψ0 21 + u˜ 0 − 1 2 in Theorem 2.1 to conclude the decay estimate (2.4). The smallness is implied by the smallness of s0 − s¯ 21 + u0 − K 2 in Theorem 2.2 for ε > 0. This is verified as follows. From (5.2) and by similar argument, we have ψ0 21 ≤

C1 s0 − s¯ 21 , s2

v0 2 ≤

u˜ 0 − 1 = C3 u0 − K , 2

2

C2  2 2C2  3  s + 4 s0 s0 , s2 0 s

u˜ 0 2

(5.5)

= C4 u0 2 ,

where Ci > 0, 1 ≤ i ≤ 4, are constants depending on the system parameters. For an arbitrarily fixed constant N > 0, according to Theorem 2.1, there exists a constant δ  > 0, such that if (v0 , u˜ 0 ) 2 ≤ N  ≡

 4C

2 s ∗2

 32C2 1 + C4 N + ∗4 N 2 , s

ψ0 21 + u˜ 0 − 1 2 ≤ δ  ,

(5.6)

the estimate (2.4) holds. Here, Ci are the same as in (5.5), and s ∗ > 0 is a lower bound of s¯ as in Theorem 2.2. Therefore, there exists a constant δ,  s ∗2 s ∗2 δ  δ   δ ≡ min 1, , , > 0. 4 4C1 C3 If s0 − s¯ 21 ≤ δ, by (4.11) we have s0 (x) ≥ s¯ − s0 − s¯ L∞ ≥ s¯ − s0 − s¯ 1 ≥ s¯ −

s∗ s∗ ≥ . 2 2

Thus we take s = s ∗ /2. Now suppose (s0 , u0 ) 2 ≤ N,

s0 − s¯ 21 + u0 − K 2 ≤ δ.

(5.7)

Equations (5.5) and (5.7) imply (5.6), which further implies (2.4). In summary, under the hypotheses of Theorem 2.2, the transformed system has solution estimate (2.3) for ε = 0, and (2.4) for ε > 0. Noting the transformation (5.4), the estimates for u in

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26

(2.3) and (2.4) imply those for u in (2.6) and (2.7), respectively. The rest of this section is, therefore, devoted to the estimates on s˜ in (2.6) and (2.7), keeping in mind that we have established the estimates for u. In particular, 5

k

Dxk (u − K) (t) ≤ C(t + 1)− 4 − 2 ,

k = 0, 1,

(5.8)

hence 3

u − K L∞ (t) ≤ C(t + 1)− 2

(5.9)

by (4.11), where C > 0 is a constant depending on the system parameters and initial data. Writing s(x, t) = e−(μK+σ )t s˜ (x, t),

(5.10)

from the first equation of (2.5) we obtain the equation for s˜ : s˜t = ε s˜xx − μ(u − K)˜s . Let φ(x, t) = s˜ (x, t) − s¯ .

(5.11)

Then φ solves the Cauchy problem φt = εφxx − μ¯s (u − K) − μ(u − K)φ,

(5.12)

φ(x, 0) = φ(0) ≡ s0 (x) − s¯ .

Lemma 5.1. There is a constant C > 0, depending on the system parameters and initial data, such that φ (t) ≤ C

for t ≥ 0.

(5.13)

Proof. Testing (5.12) by φ gives us d dt

 R

1 2 φ (x, t) dx = −ε 2



 φx2 (x, t) dx − μ¯s

R

 [(u − K)φ](x, t) dx − μ

R

[(u − K)φ 2 ](x, t) dx. R

Applying Cauchy-Schwarz inequality we have 1 d φ 2 (t) ≤ |μ| s¯ ( u − K · φ )(t) + |μ| · u − K L∞ (t) φ 2 (t). 2 dt Applying (5.9) we further have 3 d φ 2 (t) ≤ 2 |μ| s¯ ( u − K · φ )(t) + C(t + 1)− 2 φ 2 (t). dt

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27

For an arbitrarily fixed t > 0 and any s ∈ [0, t], by Grönwall’s inequality and (5.8), s   s − 32 2 φ (s) ≤ φ0 + 2 |μ| s¯ ( u − K · φ )(τ ) dτ eC 0 (τ +1) dτ 2

0

t   s − 32 5 ≤ φ0 2 + 2 |μ| s¯ sup φ (τ ) (τ + 1)− 4 dτ eC 0 (τ +1) dτ 0≤τ ≤t

≤ C + C sup φ (τ ) ≤ C + 0≤τ ≤t

0

2 1 sup φ (τ ) . 2 0≤τ ≤t

This implies 

2 2 1 sup φ (τ ) ≤ C + sup φ (τ ) , 2 0≤τ ≤t 0≤τ ≤t 2

which is simplified to (5.13).

By Duhamel’s principle and (5.12), we write x2 1 H (x, t) ≡ √ e− 4εt , 4επt t   φ(x, t) = H (x − y, t)φ0 (y) dy − μ¯s H (x − y, t − τ )(u − K)(y, τ ) dydτ

(5.14)

0 R

R

t  H (x − y, t − τ )[(u − K)φ](y, τ ) dydτ ≡ I1 + I2 + I3 .

−μ 0 R

Lemma 5.2. There is a constant C > 0, depending on the system parameters and initial data, such that 1

φ L∞ (t) ≤ C(t + 1)− 2

for t ≥ 0.

(5.15)

Proof. Consider I2 in (5.14) with t ≥ 2. From the second equation of (2.5) and by integration by parts, we have  t−1  t   I2 = −μ¯s + H (x − y, t − τ )(u − K)(y, τ ) dydτ 0

=

μ¯s a

t−1 0 R

t−1

R

 H (x − y, t − τ ) (u − K)t − Duxx + χ[u(ln s)x ]x

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 a + (u − K)2 (y, τ ) dydτ − μ¯s K μ¯s  = a

t  H (x − y, t − τ )(u − K)(y, τ ) dydτ

t−1 R





H (x − y, 1)(u − K)(y, t − 1) dy − R

H (x − y, t)(u0 − K)(y) dy R

t−1 +

Ht (x − y, t − τ )(u − K)(y, τ ) dydτ 0 R

t−1 −

Hx (x − y, t − τ )[Dux − χu(ln s)x ](y, τ ) dydτ 0 R

t−1 +

 a (u − K)2 (y, τ ) dydτ K

H (x − y, t − τ ) 0 R

t  H (x − y, t − τ )(u − K)(y, τ ) dydτ.

− μ¯s

(5.16)

t−1 R

Therefore, by triangle inequality and Cauchy-Schwarz inequality, 

|I2 | ≤ C u − K

L∞

− 12

(t − 1) + (t + 1)

t−1 u0 − K L1 + [ Ht (t − τ ) u − K (τ ) 0

+ Hx (t − τ )( ux + u L∞ (ln s)x )(τ ) + H L∞ (t − τ ) u − K 2 (τ )] dτ t +

 H (t − τ ) u − K (τ ) dτ ,

(5.17)

t−1

noting t ≥ 2. From (5.4), (2.3) and (2.4), we have s   χμK  3  x (ln s)x (t) =  (t) ≤ C v t ≤ C(t + 1)− 4 . s D

(5.18)

Substituting (5.8), (5.9) and (5.18) into (5.17), for t ≥ 2 we arrive at 

− 12

|I2 | ≤ C (t + 1)

t−1 3 3 1 5 + [(t − τ )− 4 (τ + 1)− 4 + (t − τ )− 2 (τ + 1)− 2 ] dτ 0

t + t−1

− 14

(t − τ )

− 54

(τ + 1)



dτ ≤ C(t + 1)

(5.19) − 12

.

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29

The case of 0 ≤ t ≤ 2 is simpler: From (5.14) and (5.8), t |I2 | ≤ C

t H (t − τ ) u − K (τ ) dτ ≤ C

0

1

5

3

(t − τ )− 4 (τ + 1)− 4 dτ ≤ Ct 4

0 − 12

≤ C(t + 1)

(5.20)

.

Together with (5.19) we have 1

|I2 | ≤ C(t + 1)− 2 ,

t ≥ 0.

(5.21)

By considering t ≷ 1, it is straightforward to bound I1 in (5.14) as  1 1 |I1 | ≤ C(t + 1)− 2 φ0 L1 + φ0 L∞ ≤ C(t + 1)− 2 ,

t ≥ 0,

(5.22)

using (4.11). Finally, with (5.8) and (5.13), I3 in (5.14) is bounded as t |I3 | ≤ C

− 12

(t − τ )

t ( u − K · φ )(τ ) dτ ≤ C

0

1

5

1

(t − τ )− 2 (τ + 1)− 4 dτ ≤ C(t + 1)− 2 (5.23)

0

for t ≥ 0. Combining (5.14) and (5.21)-(5.23) gives us (5.15).

2

Lemma 5.3. There is a constant C > 0, depending on the system parameters and initial data, such that 1

φ (t) ≤ C(t + 1)− 4

for t ≥ 0.

(5.24)

Proof. From (5.16) and Minkowski’s and Young’s inequalities, we have the following parallel to (5.17): For t ≥ 2, t−1   Ht L1 (t − τ ) u − K (τ ) I2 ≤ C H L1 (1) u − K (t − 1) + H (t) u0 − K L1 + 0

 + Hx L1 (t − τ )( ux + u(ln s)x )(τ ) + H (t − τ ) u − K 2L1 (τ ) dτ t +

 H L1 (t − τ ) u − K (τ ) dτ .

t−1

Applying (5.8), (5.9) and (5.18) gives us

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30



− 14

I2 ≤ C (t + 1)

+

t−1 

− 12

(t − τ )

− 34

(τ + 1)

− 14

+ (t − τ )

− 52

(τ + 1)



t dτ +

0 − 14

≤ C(t + 1)

5

(τ + 1)− 4 dτ



t−1

t ≥ 2.

,

For 0 ≤ t ≤ 2, similar to (5.20), we also have t I2 ≤ C

t H L1 (t − τ ) u − K (τ ) dτ ≤ C

0

5

1

(τ + 1)− 4 dτ ≤ C(t + 1)− 4 .

0

Therefore, 1

I2 ≤ C(t + 1)− 4 ,

t ≥ 0.

(5.25)

From (5.14), similarly, for t ≥ 0 we have  I1 ≤

H (t) φ0 L1 if t ≥ 1 H L1 (t) φ0 if 0 ≤ t ≤ 1

1

≤ C(t + 1)− 4 ,

(5.26)

t I3 ≤ C

H (t − τ ) (u − K)φ L1 (τ ) dτ

(5.27)

0

t ≤C

1

5

1

(t − τ )− 4 (τ + 1)− 4 dτ ≤ C(t + 1)− 4 ,

0

where we have applied (5.8) and (5.13). Combining (5.14) and (5.25)-(5.27) gives us 1

φ (t) ≤ I1 + I2 + I3 ≤ C(t + 1)− 4 ,

t ≥ 0.

2

Noting the definition of φ in (5.11), Lemma 5.3 gives us the term k = 0 in the first summation of (2.6) and of (2.7). For the term k = 1, we apply the transformation (5.4) and (5.10) to write  s˜x (x, t) = sign(χ)

 √χμK χμK  μK s˜ (x, t) v x, t . χ D D

This gives us ˜sx (t) ≤ C ˜s L∞ (t) v

 χμK  t . D

Since ˜s L∞ (t) ≤ φ L∞ (t) + s¯ ≤ C

(5.28)

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31

by (5.15), with (2.3) and (2.4) we have 3

˜sx (t) ≤ C(t + 1)− 4 .

(5.29)

Equation (5.29) gives us the term k = 1 in the first summation of (2.6) and of (2.7). The terms for k = 2 in (2.6) and k = 2, 3 in (2.7) are obtained similarly by differentiating (5.28). Acknowledgments The authors would like to thank the anonymous referee for his invaluable comments and suggestions which substantially improved the readability of this paper. The research of Y. Zeng was partially supported by the National Science Foundation under grant DMS-1908195. K. Zhao was partially supported by the Simons Foundation Collaboration Grant for Mathematicians No. 413028. References [1] M. Aida, K. Osaki, T. Tsujikawa, A. Yagi, M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal., Real World Appl. 6 (2005) 323–336. [2] N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. [3] M. Ding, X. Zhao, Global existence, boundedness and asymptotic behavior to a logistic chemotaxis model with density-signal governed sensitivity and signal absorption, arXiv:1806.09914. [4] J. Fan, K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl. 394 (2012) 687–695. [5] M.A. Fontelos, A. Friedman, B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal. 33 (2002) 1330–1355. [6] J. Guo, J. Xiao, H. Zhao, C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed. 29 (2009) 629–641. [7] X. He, S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl. 436 (2016) 970–982. [8] Q. Hou, Z. Wang, K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differ. Equ. 261 (2016) 5035–5070. [9] H. Jin, J. Li, Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equ. 255 (2013) 193–219. [10] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral thesis, Kyoto University, 1983. [11] E. Lankeit, J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity 32 (2019) 1569–1596. [12] E. Lankeit, J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal., Real World Appl. 46 (2019) 421–445. [13] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ. 258 (2015) 1158–1191. [14] H.A. Levine, B.D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (1997) 683–730. [15] D. Li, T. Li, K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci. 21 (2011) 1631–1650. [16] D. Li, R. Pan, K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity 28 (2015) 2181–2210. [17] H. Li, K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differ. Equ. 258 (2015) 302–338. [18] T. Li, R. Pan, K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math. 72 (2012) 417–443.

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