Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 48, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS WITH

CHEMO ATTRACTANT AND REPELLENT

AESHA LAGHA, HARUMI HATTORI

Abstract. We study the existence of global solutions and their asymptotic

behavior for a chemotaxis system with chemo-attractant and repellent in three
dimensions. To accomplish this, we use the Fourier transform and energy

method. First, we establish Lq time-decay for the linear homogeneous system

using Fourier transform and finding Green’s matrix. Then, we find Lq time-
decay for the nonlinear system using the Duhamel’s principle and the time-

weighted estimates.

1. Introduction

Chemotaxis refers to the biological process of cells or microscopic organisms mov-
ing toward a favorable chemical concentration gradient. There are many examples
that describe this biological phenomena, such as the oriented movement of certain
cells from higher or lower concentrations of chemicals in the case of immunology,
or the movement of endothelial cells toward the higher concentration of chemo-
attractant that cancer cells produce [1, 4]. A preeminent model for describing the
chemotaxis of cells was proposed by Keller and Segel [11, 12], and is now one of
the most studied and utilized models in mathematical biology. Many mathematical
approaches have since emerged to describe chemotaxis using systems of partial dif-
ferential equations. Detailed introductions to mathematical aspects of Keller-Segel
models are available in [8, 9].

In this article, we use the equations for continuum mechanics to describe the
movement of cells and use the diffusion equations for the chemo-attractant and
repellent. Thus, we consider the initial value problem of the system in R3

∂tn+∇ · (nu) = 0

∂t(nu) +∇ · (nu⊗ u) +∇p(n) = n(χ1(c1)∇c1 − χ2(c2)∇c2)− νnu
∂tc1 = ∆c1 − a11c1 + a12nκ1(c1)

∂tc2 = ∆c2 − a21c2 + a22nκ2(c2)

(1.1)

with the initial data

(n, u, c1, c2)
∣∣
t=0

= (n0, u0, c1,0, c2,0)(x), x ∈ R3. (1.2)

2020 Mathematics Subject Classification. 35Q31, 35Q35, 35Q92, 76N10.
Key words and phrases. Energy method; a priori estimates; Fourier transform;

time-decay rates.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted September 4, 2021. Published July 5, 2022.

1



2 A. LAGHA, H. HATTORI EJDE-2022/48

Here, n(x, t), u(x, t), c1(x, t), c2(x, t) and p(n) for t > 0, x ∈ R3, are the cell con-
centration, velocity of cells, chemo-attractant, chemo-repellent and pressure of the
cells, respectively. A positive constant ν is the coefficient of the damping. Sub-
tracting the first equation from the second equation, we will consider the simplified
chemotaxis fluid equations taking the form

∂tn+∇ · (nu) = 0

∂tu+ u · ∇u+
∇p(n)

n
= χ1(c1)∇c1 − χ2(c2)∇c2 − νu

∂tc1 = ∆c1 − a11c1 + a12nκ1(c1)

∂tc2 = ∆c2 − a21c2 + a22nκ2(c2)

(1.3)

with the initial data

(n, u, c1, c2)
∣∣
t=0

= (n0, u0, c1,0, c2,0)(x), x ∈ R3 (1.4)

satisfying

(n0, u0, c1,0, c2,0)(x)→ (n∞, 0, 0, 0) as |x| → ∞
for a given constant n∞ > 0. Throughout this paper, we assume the following: p(·)
is the smooth functions of n and p′(n) > 0. Also, χ1(·), χ2(·), κ1(·), κ2(·) are the
smooth functions of c1, c2, and κi(0) = 0, χi(0) = 0, i = 1, 2. nκ1(c1) and nκ2(c2)
accelerate the rate of change of c1 and c2, respectively.

The main goals of this article are to show the existence of local and global
solutions in HN (R3) and Lq time-decay rates of solutions for the Cauchy problem
of the above system (1.1)-(1.2). The main result of this paper is stated as follows.
Notations are explained at the end of the section.

Theorem 1.1. Let N ≥ 4. There exists a positive number ε0 such that if

‖(n0 − n∞, u0, c1,0, c2,0)‖N ≤ ε0,
the Cauchy problem (1.3)-(1.4) has a unique solution (n, u, c1, c2)(t) globally in time
which satisfies

(n− n∞, u)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−1(R3)),

(c1, c2)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−2(R3))

and there are constants λ1 > 0, λ2 > 0, λ3 > 0 and C0 > 0 such that

‖(n− n∞, u, c1, c2)‖2N + λ1

∫ t

0

‖∇(n− n∞)‖2N−1

+ λ2

∫ t

0

‖∇(c1, c2)‖2N + λ3

∫ t

0

‖(u, c1, c2)‖2N

≤ C0‖(n0 − n∞, u0, c1,0, c2,0)‖2HN .

(1.5)

Moreover, the global solution [n, u, c1, c2] obtained above satisfies for a sufficiently
large time t > 0

‖n− n∞‖Lq ≤ C(1 + t)
−3
2 + 3

2q , (1.6)

‖u‖Lq ≤ C(1 + t)−2+
3
2q , (1.7)

‖(c1, c2)‖Lq ≤ C(1 + t)−3/2, (1.8)

with 2 ≤ q <∞, where C > 0 is a positive constant independent of time.



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 3

The proof of the existence of global solutions in Theorem 1.1 is based on the
local existence and the a priori estimates. The local existence can be proved by
constructing a sequence of approximation functions based on iteration method dis-
cussed, for example, in Kato [10]. The a priori estimates can be obtained by the
energy method. Moreover, our approach to obtain the time-decay rate in the Lq

norm of solutions in Theorem 1.1 is a combined analysis of Green’s function of the
linear system and the refined energy estimates with the help of Duhamel’s principle.
We obtain Green’s matrix of the linear system by Fourier transform.

Concerning the chemotaxis based on the Keller-Segel model, Wang [17] explored
the interactions between the nonlinear diffusion and logistic source on the solutions
of the attraction-repulsion chemotaxis system in three dimensions. E. Lankeit and
J. Lankeit [13] proved global existence of classical solutions to a chemotaxis system
with singular sensitivity. Liu and Wang [14] established the existence of global
classical solutions and steady states to an attraction-repulsion chemotaxis model in
one dimension based on the energy method. For the long time behavior of solutions,
Tan and Zhou [16] proved the global existence and time-decay estimate of solutions
to the Keller-Segel system in R3 with small initial data.

Modeling the cell movements based on the incompressible or compressible fluid
dynamics has been studied as well. For the incompressible case, Chae, Kang and
Lee [3], and Duan, Lorz, and Markowich [7] showed the global-in-time existence for
the incompressible chemotaxis equations near constant states, if the initial data is
sufficiently small. Rodriguez, Ferreira and Villamizar-Roa [15] showed the global
existence for an attraction-repulsion chemotaxis fluid model with a logistic source.
For the compressible case, D. Ambrosi, F. Bussolino and L. Preziosi [2] discussed the
vasculogenesis using the compressible fluid dynamics for the cells and the diffusion
equation for the attractant. There are many related approaches that use Fourier
transform, and we only mention that Duan [5], and Duan, Liu and Zhu [6] proved
the time-decay rates combining the energy estimates and spectral analysis.

We summarize the notation used in the paper here. C denotes a positive con-
stant, λi, where i = 1, 2, denote positive (generally small) constants, where both
C and λi may take different values in different places. For any integer m ≥ 0, we
use Hm to denote the Sobolev space Hm(R3) and Ḣm the mth-order homogeneous
Sobolev space. Set L2 = H0. For simplicity, the norm of Hm is denoted by ‖ · ‖m
with ‖ · ‖ = ‖ · ‖0. The Lq norms are given by ‖ · ‖Lq . We set ∂α = ∂α1

x1
∂α2
x2
∂α3
x3

for a multi-index α = (α1, α2, α3). The length of α is |.| = α1 + α2 + α3 and we
also set ∂j = ∂xj for j = 1, 2, 3. For an integrable function f : R3 → R, its Fourier

transform is defined by f̂ =
∫
R3 e
−ix·ξf(x)dx, x · ξ =

∑3
i=0 xjξj , x ∈ R3, where

i =
√
−1 is the imaginary unit. Let us denote the space

X(0, T ) =
{

(ρ, u) ∈ C([0, T ];HN (R3)) ∩ C1([0, T ];HN−1(R3)),

(c1, c2) ∈ C([0, T ];HN (R3)) ∩ C1([0, T ];HN−2(R3))
}
.

This article is organized as follows. In section 2, we reformulate the Cauchy problem
under consideration. In section 3, we prove the global existence and uniqueness of
solutions for the reformulated problem using the energy method. In section 4, we
linearize the reformulated problem and obtain the Lq time-decay property for the
linearized equations by the Fourier transform. In section 5, we use the results of
sections 3 and 4 to study the Lq time-decay rates of solutions to the reformulated
nonlinear system and complete the proof of Theorem 1.1.



4 A. LAGHA, H. HATTORI EJDE-2022/48

2. Reformulation of the system (1.3)

Let U(t) = (n, u, c1, c2) be a smooth solution to the Cauchy problem of the
chemotaxis fluid equations (1.3) with initial data U0 = (n0, u0, c1,0, c2,0). Set

n(x, t) = ρ(x, t) + n∞. (2.1)

Then the Cauchy problem (1.3)-(1.4) is reformulated as

∂tρ+ n∞∇ · u = −∇ · (ρu)

∂tu+ u · ∇u+ νu+
p′(ρ+ n∞)

ρ+ n∞
∇ρ = χ1(c1)∇c1 − χ2(c2)∇c2

∂tc1 −∆c1 + (a11 − a12n∞κ′1(0))c1 = a12(ρ+ n∞)κ1(c1)− a12n∞κ′1(0)c1

∂tc2 −∆c2 + (a21 − a22n∞κ′2(0))c2 = a22(ρ+ n∞)κ2(c2)− a22n∞κ′2(0)c2

(2.2)

with the initial data

(ρ, u, c1, c2)
∣∣
t=0

= (ρ0, u0, c1,0, c2,0), (2.3)

where ρ0 = n0 − n∞ and

(ρ0, u0, c1,0, c2,0)→ (0, 0, 0, 0) as |x| → ∞.
We assume that (a11 − n∞a12κ

′
1(0)) > 0 and (a22 − n∞a21κ

′
2(0)) > 0. In the

following, we set N ≥ 4.
Concerning the reformulated Cauchy problem (2.2)-(2.3), one has the following

global existence result.

Proposition 2.1. Suppose that ‖(ρ0, u0, c1,0, c2,0)‖N is sufficiently small. Then the
Cauchy problem (2.2)-(2.3) has a unique solution U(t) = (ρ, u, c1, c2)(t) globally in
time which satisfies U(t) ∈ X(0,∞) and for any t ≥ 0,

EN (U(t)) + λ1

∫ t

0

DN (U(t))ds+ λ2

∫ t

0

DhN (U(t))ds ≤ C0EN (U0), (2.4)

where
EN (U(t)) = ‖(ρ, u, c1, c2)‖2N , (2.5)

is the energy functional, and

DN (U(t)) = ‖∇(c1, c2)‖2N , (2.6)

DhN (U(t)) = ‖∇ρ‖2N−1 + ‖(u, c1, c2)‖2N (2.7)

are the dissipation rates.

The solutions obtained in Proposition 2.1 indeed have the decay rates in time
under additional conditions on the initial data. Given U0 = (ρ0, u0, c1,0, c2,0), set
εN (U0) as

εN (U0) = ‖U0‖N + ‖(ρ0, u0)‖L1 , (2.8)

for N ≥ 4. Then, we have the following two Propositions:

Proposition 2.2. Let U(t) = (ρ, u, c1, c2) be the solution to the Cauchy problem
(2.2) with the initial data U0 = (ρ0, u0, c1,0, c2,0). If εN+1(U0) > 0 is sufficiently
small, then the solution U(t) = (ρ, u, c1, c2) satisfies

‖U(t)‖N ≤ εN+1(U0)(1 + t)−3/4, (2.9)

‖∇U(t)‖N ≤ εN+1(U0)(1 + t)−5/4, (2.10)

for any t ≥ 0.



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 5

Proposition 2.3. Let 2 ≤ q ≤ ∞. Suppose that U(t) = (ρ, u, c1, c2) be the solution
to the Cauchy problem (2.2)-(2.3) obtained in Proposition 2.1. Then the solution
U(t) = (ρ, u, c1, c2) satisfies the following Lq-time decay estimates:

‖ρ‖Lq ≤ C(1 + t)−2+
3
2q , (2.11)

‖u‖Lq ≤ C(1 + t)
−3
2 + 3

2q , (2.12)

‖(c1, c2)‖Lq ≤ C(1 + t)−3/2 (2.13)

for any t ≥ 0, 2 ≤ q ≤ ∞.

The existence of global solutions in Theorem 1.1 is obtained directly from Propo-
sition 2.1 and the derivation of rates in Theorem 1.1 is based on Proposition 2.3.

3. Global solution of the nonlinear system (2.2)

The goal of this section is to prove the existence of global solutions to the Cauchy
problem (2.2), when initial data is a small, smooth perturbation near the steady
state (n∞, 0, 0, 0). The proof is based on uniform a priori estimates combined with
the local existence that will be shown in subsections 3.1 and 3.2.

3.1. Existence of local solutions. The existence of smooth local solutions for
symmetrizable hyperbolic equations (2.2)1 and (2.2)2 can be proved as in [10].
Since (2.2)3 and (2.2)4 are the heat equations, the local solutions obviously exist.

We construct a solution sequence (ρj , uj , cj1, c
j
2)j≥0 by iteratively solving the Cauchy

problem on the following system

∂tρ
j+1 + n∞∇ · uj = −∇ · (ρj+1uj),

∂tu
j+1 + νuj+1 +

p′(n∞)

n∞
∇ρj+1

= −uj · ∇uj+1 + χ1(cj1)∇cj1 − χ1(cj2)∇cj2 − (
p′(ρj + n∞)

ρj + n∞
− p′(n∞)

n∞
)∇ρj+1,

∂tc
j+1
1 −∆cj+1

1 + (a11 − a12n∞κ′1(0))cj+1
1

= a12(ρj + n∞)cj+1
1 − a12n∞κ′1(0)cj+1

1 ,

∂tc
j+1
2 −∆cj+1

2 + (a21 − a22n∞κ′2(0))cj+1
2

= a22(ρj + n∞)cj+1
2 − a22n∞κ′2(0)cj+1

2 ,

(3.1)
with initial data

(ρj+1, uj+1, cj+1
1 , cj+1

2 )
∣∣
t=0

= (ρ0, u0, c1,0, c2,0), (3.2)

for j ≥ 0, where (ρ0, u0, c01, c
0
2) ≡ (0, 0, 0, 0) holds. For simplicity, in what follows,

we write U j = (ρj , uj , cj1, c
j
2) and U0 = (ρ0, u0, c1,0, c2,0).

Lemma 3.1. There are constants T1 > 0, ε0 > 0, B > 0 such that if the initial
data U0 ∈ HN (R3) and ‖U0‖N ≤ ε0, then for each j ≥ 0, U j ∈ C([0, T1] : HN (R3))
is well-defined and

sup
0≤t≤T1

‖U j(t)‖N ≤ B, j ≥ 0. (3.3)



6 A. LAGHA, H. HATTORI EJDE-2022/48

Moreover, (U j)j≥0 is a Cauchy sequence in the Banach space C([0, T1];HN (R3)),
and the limit function U(x, t) of (U j)j≥0 satisfies

sup
0≤t≤T1

‖U(t)‖N ≤ B, (3.4)

and U = (ρ, u, c1, c2) is a solution over [0, T1] to the Cauchy problem (2.2)-(2.3).
Finally, the Cauchy problem (2.2)-(2.3) admits at most one solution U ∈ C([0, T1] :
HN (R3)) satisfying (3.2).

3.2. A Priori Estimates. In this subsection we provide some estimates for the
solutions for any t > 0. We establish the uniform-in-time a priori estimates for
smooth solutions to Cauchy problem (2.2)-(2.3) applying basic energy estimates.

Lemma 3.2 (a priori estimates). Suppose that there exists a solution

U(t) = (ρ, u, c1, c2) ∈ C([0, T ];HN (R3)

to the Cauchy problem (2.2)-(2.3), with

sup
0≤t≤T

‖(ρ, u, c1, c2)(t)‖N ≤ ε (3.5)

for 0 < ε ≤ 1. Then, there are ε0 > 0 , C0 > 0 and λ > 0 such that for any ε ≤ ε0,

EN (U(t)) + λ1

∫ t

0

DN (U(t))ds+ λ2

∫ t

0

DhN (U(t))ds ≤ C0EN (U0) (3.6)

holds for any t ∈ [0, T ].

Proof. First, we find the zero-order estimates. For the estimate of ρ, multiplying ρ
to both sides of the first equation of (2.2) and integrating in x ∈ R3, we obtain∫

R3

ρρtdx+ n∞

∫
R3

ρ∇ · udx = −
∫
R3

ρ∇ · (ρu)dx.

Using integration by parts and the Cauchy-Schwarz inequality, we have

1

2

∫
R3

(
ρ2
)
t
dx+ n∞

∫
R3

ρ∇ · udx ≤ C‖ρ‖2
∫
R3

|u|2 + |∇ρ|2dx. (3.7)

For the estimate of u, multiplying by u on both sides of the second equation of
(2.2) and integrating in x ∈ R3, we obtain∫

R3

uutdx+

∫
R3

u(u · ∇u)dx+ ν

∫
R3

u2dx+
p′(n∞)

n∞

∫
R3

u · ∇ρdx

=

∫
R3

uχ1(c1)∇c1dx−
∫
R3

uχ2(c2)∇c2dx

−
∫
R3

u
(p′(ρ+ n∞)

ρ+ n∞
− p′(n∞)

n∞

)
∇ρdx.

Using integration by parts and the Cauchy-Schwarz inequality, we have

1

2

∫
R3

(
u2
)
t
dx+ ν

∫
R3

|u|2dx− p′(n∞)

n∞

∫
R3

ρ∇ · udx

≤ ‖u‖3
∫
R3

|u|2dx+ C‖c1‖2
∫
R3

|∇c1|2 + |u|2dx+ C‖c2‖2
∫
R3

|∇c2|2 + |u|2dx

+ C‖ρ‖2
∫
R3

|∇ρ|2 + |u|2dx.



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 7

For the estimates of c1, multiplying c1 to both sides of the equation of c1, using
κ1(c1)− κ′1(0)c1 = O(c21), and integrating by parts, we obtain

1

2

∫
R3

(c21)tdx+

∫
R3

|∇c1|2dx+ (a11 − n∞a12κ′1(0))

∫
R3

|c1|2dx

≤ C(‖ρ‖2 + ‖c1‖2)

∫
R3

|c1|2dx.

(3.8)

Similarly, for c2, we have

1

2

∫
R3

(c22)tdx+

∫
R3

|∇c2|2dx+ (a21 − n∞a22κ′2(0))

∫
R3

|c2|2dx

≤ C(‖ρ‖2 + ‖c2‖2)

∫
R3

|c2|2dx.

(3.9)

Then, as long as E1/2N (U) is small so that

(a11 − n∞a12κ′1(0)) > CE1/2N (U),

(a21 − n∞a22κ′2(0)) > CE1/2N (U)

are satisfied, we have

1

2

d

dt

∫
R3

(
|u|2 +

p′(n∞)

n2∞
|ρ|2 + |c1|2 + |c2|2

)
dx

+ ν

∫
R3

|u|2dx+

∫
R3

|∇c1|2dx+

∫
R3

|∇c2|2dx

+ (a11 − n∞a12κ′1(0))

∫
R3

|c1|2dx+ (a21 − n∞a22κ′2(0))

∫
R3

|c2|2dx

≤ C‖ρ‖2
∫
R3

|∇ρ|2dx.

(3.10)

Now, we estimate the higher-order derivatives of (ρ, u, c1, c2). Take α with 1 ≤
|α| ≤ N . Applying ∂α to the second equation of (2.2), multiplying by ∂αu, and
integrating by parts, we have

1

2

∫
R3

(∂αu)
2
t dx+ ν

∫
R3

|∂αu|2dx+

∫
R3

∂αu(
p′(ρ+ n∞)

ρ+ n∞
)∂α∇ρdx

=

∫
R3

∂αu

α∑
β=1

Cβα∂
β(
p′(ρ+ n∞)

ρ+ n∞
)∂α−β∇ρdx

−
∫
R3

∂αu

α∑
β=0

Cβα(∂α−βu · ∇∂βu)dx+

∫
R3

∂αu∂α(χ1(c1)∇c1)dx

−
∫
R3

∂αu∂α(χ2(c2)∇c2)dx.

(3.11)



8 A. LAGHA, H. HATTORI EJDE-2022/48

Using the first equation in (2.2) and the integration by parts, we rewrite the third
term on the left-hand side as follows.

∫
R3

∂αu (
p′(ρ+ n∞)

ρ+ n∞
)∂α∇ρdx

=
1

2

∫
R3

p′(ρ+ n∞)

(ρ+ n∞)2
(∂αρ2)t dx+

∫
R3

p′(ρ+ n∞)

(ρ+ n∞)
∂α(

1

ρ+ n∞
)ρt∂

αρdx

+

∫
R3

1

ρ+ n∞
∂α(∇ρ · u)

p′(ρ+ n∞)

(ρ+ n∞)
∂αρdx

+

∫
R3

∂α(
1

ρ+ n∞
)(∇ρ · u)

p′(ρ+ n∞)

(ρ+ n∞)
∂αρdx

−
∫
R3

∂αu∇(
p′(ρ+ n∞)

ρ+ n∞
)∂αρdx.

(3.12)

Therefore, combining (3.11) and (3.12), we obtain

1

2
‖∂αu‖2 + C1‖∂αρ‖2 + ν

∫ t

0

‖∂αu‖2ds

≤ C‖∂αu0‖+ C‖∂αρ0‖+ C‖ρ‖N
∫ t

0

(‖∂αu‖2 + ‖∂αρ‖2)ds

+ C‖u‖N
∫ t

0

‖∂αρ‖2ds+ C‖u‖N
∫ t

0

‖∂αu‖2ds

+ C‖c1‖N
∫ t

0

(‖∂αu‖2 + ‖∂α∇c1‖2)ds

+ C‖c2‖N
∫ t

0

(‖∂αu‖2 + ‖∂α∇c2‖2)ds.

(3.13)

Similarly, we estimate the higher-order derivatives of c1 and c2 as follows:

1

2
‖∂αc1‖2 +

∫ t

0

‖∇∂αc1‖2ds+ (a11 − n∞a12κ′1(0))

∫ t

0

‖∂αc1‖2ds

≤ C‖∂αc1,0‖+ C‖ρ‖N
∫ t

0

‖∂αc1‖2ds+ C‖c1‖N
∫ t

0

(‖∂αc1‖2 + ‖∂αρ‖2)ds,

(3.14)

and

1

2
‖∂αc2‖2 +

∫ t

0

‖∇∂αc2‖2ds+ (a21 − n∞a22κ′2(0))

∫ t

0

‖∂αc2‖2ds

≤ C‖∂αc2,0‖+ C‖ρ‖N
∫ t

0

(‖∂αc2‖2ds

+ C‖c2‖N
∫ t

0

(‖∂αc2‖2 + ‖∂αρ‖2)ds.

(3.15)



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 9

Then, summing (3.13)-(3.15) over |α| ≤ N , we have

1

2
(‖u‖2N + C1‖ρ‖2N + ‖c1‖2N + ‖c2‖2N ) + ν

∫ t

0

‖u‖2Nds+

∫ t

0

‖∇c1‖2Nds

+

∫ t

0

‖∇c2‖2Nds+ (a11 − n∞a12κ′1(0))

∫ t

0

‖c1‖2Nds

+ (a21 − n∞a22κ′1(0))

∫ t

0

‖c2‖2Nds

≤ C0‖U0‖N + C‖ρ‖N
∫ t

0

(‖u‖2N + ‖ρ‖2N + ‖c1‖2N + ‖c2‖2N )ds

+ C‖u, c1, c2‖N
∫ t

0

‖ρ‖2Nds+ C‖u‖N
∫ t

0

‖u‖2Nds

+ C‖c1‖N
∫ t

0

(‖u‖2N + ‖c1‖2N + ‖∇c1‖2N )ds

+ C‖c2‖N
∫ t

0

(‖u‖2N + ‖c2‖2N + ‖∇c2‖2N )ds.

(3.16)

For the estimates of ρ, let |α| ≤ N − 1. Applying ∂α to (2.2)2, multiplying it by
∂α∇ρ and integrating in x yield∫

R3

∂α∇ρ∂αutdx+ ν

∫
R3

∂α∇ρ∂αudx+
p′(n∞)

n∞

∫
R3

∂α∇ρ∂α∇ρdx

= −
∫
R3

∂α∇ρ∂α(u · ∇u)dx+

∫
R3

∂α∇ρ∂α(χ1(c1)∇c1)dx

−
∫
R3

∂α∇ρ∂α(χ2(c2)∇c2)dx−
∫
R3

∂α∇ρ∂α(
p′(ρ+ n∞)

ρ+ n∞
− p′(n∞)

n∞
)∇ρ)dx,

which can be simplified further by (2.2)∫
R3

(∂α∇ρ∂αu)tdx+
p′(n∞)

n∞

∫
R3

|∂α∇ρ|2dx

= −ν
∫
R3

∂α∇ρ∂αudx−
∫
R3

∂α∇ρ∂α(u · ∇u)dx

+

∫
R3

∂α∇ρ∂α(χ1(c1)∇c1)dx−
∫
R3

∂α∇ρ∂α(χ2(c2)∇c2)dx

−
∫
R3

∂α∇ρ∂α(
p′(ρ+ n∞)

ρ+ n∞
− p′(n∞)

n∞
)∇ρ)dx

−
∫
R3

∂α∇ · u∂α∇ · ((ρ+ n∞)u)dx.

Finally, applying the Cauchy-Schwarz inequality, we obtain

d

dt

∫
R3

(∂α∇ρ∂αu)dx+ λ2‖∂α∇ρ‖2

≤ C(‖∇ · ∂αu‖2 + ‖∂αu‖2) + C(‖(c1, c2)‖‖∂α∇(ρ, c1, c2)‖2)

+ C(‖(ρ, u)‖N‖∇ · ∂α(ρ, u)‖2).



10 A. LAGHA, H. HATTORI EJDE-2022/48

Summing over |α| ≤ N − 1 and integrating with respect to t, we have∑
|α|≤N−1

∫
R3

∂α∇ρ∂αudx+ λ2

∫ t

0

‖∇ρ‖2N−1ds

≤
∑

|α|≤N−1

∫
R3

∂α∇ρ∂αu dx
∣∣
t=0

+ C

∫ t

0

‖u‖2Nds

+ C‖(c1, c2)‖N
∫ t

0

‖∇(ρ, c1, c2)‖2N−1ds

+ C‖(ρ, u)]‖N
∫ t

0

‖∇ · (ρ, u)]‖2N−1ds.

(3.17)

Taking linear combination (3.10) + (3.16) + k(3.17), we see that

‖U‖2N + k
∑

|α|≤N−1

∫
R3

∂α∇ρ∂αu dx+ λ1

∫ t

0

‖∇(c1, c2)‖2Nds

+ λ2

∫ t

0

(‖∇ρ‖2N−1 + ‖(u, c1, c2)‖2N )ds ≤ C0‖U0‖2N

for constant 0 < k � 1. Since the second term on the left hand side is absorbed in
‖U‖2N , we obtain the energy estimate (3.6) in Lemma 3.2. �

Proof of Proposition 2.1. Choose a positive constant ε̄ = min{ε0, ε1}, where ε0 > 0
and ε1 > 0 are given in Lemma 3.1 and Lemma 3.2. Let U0 ∈ HN (R3) satisfy

‖U0‖N ≤
ε̄

2
√
C0 + 1

.

Now, let us define

T = {t ≥ 0 : sup
0≤s≤t

‖U(s)‖HN ≤ ε̄}.

Note that

‖U0‖N ≤
ε̄

2
√
C0 + 1

≤ ε̄

2
< ε̄ ≤ ε0.

Then T > 0 holds true from the local existence result. If T is finite, from definition
of T , we have

sup
0≤s≤t

‖U‖ = ε̄. (3.18)

On the other hand, by Lemma 3.2 we have

sup
0≤s≤t

‖U(s)‖N ≤
√
C0‖U0‖N ≤

ε̄
√
C0

2
√
C0 + 1

≤ ε̄

2
,

which is a contradiction to 3.18. Therefore, T = ∞ holds true. This implies that
local solution U(t) obtained in Lemma 3.1 can be extended to infinity in time.
Thus, we have a global solution (ρ, u, c1, c2)(t) ∈ C([0,∞);HN ). This completes
the proof. �



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 11

4. Linearized homogeneous system

In this section, we study the time-decay property of solutions to the nonlinear
system (2.2). For this purpose, we separate the system into the linear and the
nonlinear parts around the constant state (n∞, 0, 0, 0) and we write the system as
follows: Then U = (ρ, u, c1, c2) satisfies

∂tρ+ n∞∇ · u = g1

∂tu+ νu+
p′(n∞)

n∞
∇ρ = g2

∂tc1 −∆c1 + (a11 − a12n∞κ′1(0))c1 = g3

∂tc2 −∆c2 + (a21 − a22n∞κ′2(0))c2 = g4,

(4.1)

where the nonlinear source terms are

g1 = −∇ · (ρu)

g2 = −u · ∇u+ χ1(c1)∇c1 − χ2(c2)∇c2 − (
p′(ρ+ n∞)

ρ+ n∞
− p′(n∞)

n∞
)∇ρ

g3 = a12ρκ1(c1) + a12n∞(κ1(c1)− κ′1(0)c1)

g4 = a21ρκ2(c2) + a22n∞(κ2(c2)− κ′2(0)c2)

(4.2)

and the initial data are

(ρ, u, c1, c2)
∣∣
t=0

= U0 = (ρ0, u0, c1,0, c2,0). (4.3)

To obtain the time-decay rates of the solution to the system (4.1), we first study
the Cauchy problem for the corresponding linearized homogeneous system

∂tρ+ n∞∇ · u = 0

∂tu+ νu+
p′(n∞)

n∞
∇ρ = 0

∂tc1 −∆c1 + (a12 − a11n∞κ′1(0))c1 = 0

∂tc2 −∆c2 + (a22 − a21n∞κ′2(0))c2 = 0.

(4.4)

For this purpose, we define U1 = (ρ, u) to be the solution of the linearized homo-
geneous equations

∂tρ+ n∞∇ · u = 0

∂tu+ νu+
p′(n∞)

n∞
∇ρ = 0,

(4.5)

with the initial data

U1

∣∣
t=0

= U1,0 = (ρ0, u0), x ∈ R3. (4.6)

4.1. Representation of solutions for (4.5) and (4.6). We first find the explicit
representation of the Fourier transform of the solution U1 = (ρ, u) = etBU1,0 to the
system (4.5) with the initial data (4.6), where etB is the linear solution operator.

First, we take the time derivative for the first equation of (4.5) and using the
second equation to replace ∂tu, we have

∂ttρ+ ν∂tρ− p′(n∞)∆ρ = 0. (4.7)

Initial data are given by

ρ
∣∣
t=0

= ρ0, ∂tρ
∣∣
t=0

= −n∞∇ · u0. (4.8)



12 A. LAGHA, H. HATTORI EJDE-2022/48

Then, taking the Fourier transform of the above equation, we get the second-order
ODE as

∂ttρ̂+ ν∂tρ̂+ p′(n∞)|ξ|2ρ̂ = 0

ρ̂
∣∣
t=0

= ρ̂0, ∂tρ
∣∣
t=0

= −n∞iξ · û0.
(4.9)

It is straightforward to obtain

ρ̂ =
λ1e

λ2t − λ2eλ1t

λ1 − λ2
ρ̂0 − in∞ξ

eλ1t − eλ2t

λ1 − λ2
û0, (4.10)

where λ1,2 = −ν2 ±
1
2

√
ν2 − 4p′(n∞)|ξ|2 are the roots of the characteristic equation

λ2 + νλ + p′(n∞)|ξ|2 = 0. Similarly, taking the time derivative for the second
equation of (4.5) and using the first equation to replace ∂tρ, we have

∂ttu+ ν∂tu− p′(n∞)∇(∇ · u) = 0. (4.11)

Further taking the divergence, one has

∂tt(∇ · u) + ν∂t(∇ · u)− p′(n∞)∆(∇ · u) = 0, (4.12)

∇ · u
∣∣
t=0

= ∇ · u0, (4.13)

∂t∇ · u
∣∣
t=0

= −ν∇ · u0 −
p′(n∞)

n∞
∆ρ0. (4.14)

Here and in the sequel we define ξ̃ = ξ/|ξ| for |ξ| 6= 0. By taking the Fourier
transform of (4.12), (4.13) and (4.14), we get the second-order ODE

∂tt(ξ̃ · û) + ν∂t(ξ̃ · û) + p′(n∞)|ξ|2(ξ̃ · û) = 0

(ξ̃ · û)
∣∣
t=0

= ξ̃ · û0,

∂t(ξ̃ · û)
∣∣
t=0

= −νξ̃ · û0 − i
p′(n∞)

n∞
|ξ|ρ̂0.

(4.15)

Therefore,

ξ̃ · û =
eλ1t − eλ2t

λ1 − λ2

(
− ip

′(n∞)

n∞
|ξ|ρ̂0

)
+

(λ1 + ν)eλ2t − (λ2 + ν)eλ1t

λ1 − λ2
ξ̃ · û0. (4.16)

Now, by taking the curl for the second equation of (4.5), we have

∂t(∇× u) + ν(∇× u) = 0.

Taking the Fourier transform of the above equation, we have

∂t(ξ̃ × û) + νξ̃ × û = 0. (4.17)

Initial data is given as

(ξ̃ × û)
∣∣
t=0

= ξ̃ × û0. (4.18)

Solving the initial value problem (4.17) and (4.18), we have

(ξ̃ × û) = e−νtξ̃ × û0. (4.19)

For t ≥ 0 and ξ ∈ R3 with |ξ| 6= 0, one has the decomposition û = ξ̃ξ̃ · û− ξ̃×(ξ̃× û).
It is straightforward to obtain

û =
eλ1t − eλ2t

λ1 − λ2
(−ip

′(n∞)

n∞
ξρ̂0)

+
(λ1 + ν)eλ2t − (λ2 + ν)eλ1t

λ1 − λ2
ξ̃ξ̃ · û0 − e−ν ξ̃ × (ξ̃ × û0).

(4.20)



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 13

Then

û =
eλ1t − eλ2t

λ1 − λ2
(−ip

′(n∞)

n∞
ξρ̂0)

+
( (λ1 + ν)eλ2t − (λ2 + ν)eλ1t

λ1 − λ2

)ξ ⊗ ξ
|ξ|2

û0 + e−νt(I3 −
ξ ⊗ ξ
|ξ|2

)û0.

(4.21)

Therefore, if we define the Fourier transform Ĝ(t, ξ) of Green’s function G(t, ξ) =
etB to be

Ĝ(t, ξ)

=

[
Ĝ11 Ĝ12

Ĝ21 Ĝ22

]

=

[
λ1e

λ2t−λ2e
λ1t

λ1−λ2
(−in∞ξ) e

λ1t−eλ2t
(λ1−λ2)

eλ1t−eλ2t
λ1−λ2

(−ip
′(n∞)
n∞

ξ) (λ1+ν)e
λ2t−(λ2+ν)e

λ1t

λ1−λ2

ξ⊗ξ
|ξ|2 + e−νt(I3 − ξ⊗ξ

|ξ|2 )

]
,

(4.22)

then (4.10) and (4.21) can be written as[
ρ̂(t, ξ)
û(t, ξ)

]
= Ĝ(t, ξ)

[
ρ̂(0, ξ)
û(0, ξ)

]
.

4.2. Refined L2−Lq time-decay property. In this subsection, we use (4.22) to
obtain some refined L2−Lq time-decay property for U1 = (ρ, u). To do so, we need
to find the time-frequency pointwise estimate on ρ̂ and û in the following Lemma:

Lemma 4.1. Suppose U1 = (ρ, u) is the solution to the linear homogeneous system
(4.5) with the initial data U1

∣∣
t=0

= (ρ0, u0). Then, there are constants ε > 0,

C > 0, λ > 0 such that for all t > 0, |ξ| ≤ ε,

|ρ̂(t, ξ)| ≤ C(e−λ|ξ|
2t + |ξ|2e−νλt)|ρ̂0(ξ)|+ C(|ξ|e−λ|ξ|

2t + |ξ|e−νλt)|û0(ξ)|, (4.23)

|û(t, ξ)| ≤ C|ξ|(e−λ|ξ|
2t + e−νλt)|ρ̂0(ξ)|+ C(|ξ|2e−λ|ξ|

2t + e−νλt)|û0(ξ)|, (4.24)

and for all t > 0, |ξ| ≥ ε,

|ρ̂(t, ξ)| ≤ Ce−λt|ρ̂0(ξ), û0(ξ)|, (4.25)

|û(t, ξ)| ≤ Ce−λt|ρ̂0(ξ), û0(ξ)|. (4.26)

Proof. To obtain the upper bound of ρ̂(t, ξ) and û(t, ξ), we estimate Ĝ11, Ĝ12, Ĝ21,

and Ĝ22 in (4.22). If ν2 − 4p′(n∞)|ξ|2 ≥ 0, then λ1,2 = −ν2 ±
1
2

√
ν2 − 4p′(n∞)|ξ|2

are real. It is straightforward to obtain

λ1 ∼ −O(1)|ξ|2,
λ2 ∼ −ν +O(1)|ξ|2,

as |ξ| → 0.

On other hand, if ν2 − 4p′(n∞)|ξ|2 ≤ 0, then λ1,2 = −ν2 ±
ν
2 i
√

4p′(n∞)
ν2 |ξ|2 − 1

are complex conjugates. Moreover, we have

|λ1,2| ∼ O(1)|ξ|,
λ1 − λ2 ∼ iO(1)|ξ|.



14 A. LAGHA, H. HATTORI EJDE-2022/48

as |ξ| → ∞. Then, there exists ε ≤
√

ν2

4p′(n∞) ≤ R, with 0 < ε� 1� R <∞ such

that one can estimate Ĝ(t, ξ) as follows:

|Ĝ11| ≤ C(e−λ|ξ|
2t + |ξ|2e−νλt)

|Ĝ12| ≤ C|ξ|(e−λ|ξ|
2t + e−νλt)

|Ĝ21| ≤ C|ξ|(e−λ|ξ|
2t + e−νλt)

|Ĝ22| ≤ Ce−νt + C(|ξ|2e−λ|ξ|
2t + e−νλt) ≤ C|ξ|2e−λ|ξ|

2t + Ce−νλt,

as |ξ| ≤ ε, and

|Ĝ11| ≤ Ce−
ν
2 t ≤ Ce−λt

|Ĝ12|+ |Ĝ21| ≤ Ce−
ν
2 t ≤ Ce−λt

|Ĝ22| ≤ Ce−ν + Ce−
ν
2 t ≤ Ce−λt,

as |ξ| > R. When the eigenvalues coalesce, since the real part is negative, we have
te−

ν
2 t in the solution, but this decays exponentially. Then we have te−

ν
2 t ≤ e−λt

and over ε ≤ |ξ| ≤ R we have

|Ĝ11| ≤ C(e−λ|ξ|
2t + |ξ|2e−νλt)

|Ĝ12| ≤ C|ξ|(e−λ|ξ|
2t + e−νλt)

|Ĝ21| ≤ C|ξ|(e−λ|ξ|
2t + e−νλt)

|Ĝ22| ≤ C|ξ|2e−λ|ξ|
2t + Ce−νλt.

Now, we can estimate ρ̂, û as follows

|ρ̂(t, ξ)| = |Ĝ11ρ̂0 + Ĝ12û0|

≤ |Ĝ11||ρ̂0|+ |Ĝ12||û0|

≤ C(e−λ|ξ|
2t + |ξ|2e−νλt)|ρ̂0(ξ)|+ C(|ξ|e−λ|ξ|

2t + |ξ|e−νλt)|û0(ξ)|,

and

|û(t, ξ)| = |Ĝ21ρ̂0 + Ĝ22û0|

≤ |Ĝ21||ρ̂0|+ |Ĝ22||û0|

≤ C|ξ|(e−λ|ξ|
2t + e−νλt)|ρ̂0(ξ)|+ C(|ξ|2e−λ|ξ|

2t + e−νλt)|û0(ξ)|,

for |ξ| ≤ ε. Finally, (4.25) and (4.26) can be proved in the same way as for (4.23)
and (4.24). This completes the proof of Lemma (4.1). �

Theorem 4.2. Let 2 ≤ q ≤ ∞, and let m ≥ 0 be an integer. Assume U1 = eBtU1,0

is the solution to the Cauchy problem (4.5)-(4.6). Then for any t ≥ 0, U1 = (ρ, u)
satisfies:

‖∇mρ(t)‖Lq ≤ C(1 + t)−
3
2 (1−

1
q )−

m
2 ‖ρ0, u0‖L1

+ e−λt‖∇m+[3( 1
2−

1
q )]+(ρ0, u0)‖L2 ,

(4.27)

‖∇mu(t)‖Lq ≤ C(1 + t)−
3
2 (1−

1
q )−

m+1
2 ‖ρ0, u0‖L1

+ e−λt‖∇m+[3( 1
2−

1
q )]+(ρ0, u0)‖L2 ,

(4.28)



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 15

where C = C(m, q) and [3( 1
2 −

1
q )]+ are defined by

[3(
1

2
− 1

q
)]+ =

{
0 if q = 2

[3( 1
2 −

1
q )]− + 1 otherwise.

Here, [·]− denotes the integer part of the argument.

Proof. Take 2 ≤ q ≤ ∞ and let m ≥ 0 be an integer. Set U1 = eBtU1,0. Using the
Hausdorff-Young inequality and (4.23) we prove (4.27) as follows,

‖∇mρ(t)‖Lq(R3
x)
≤ C‖|ξ|mρ̂(ξ, t)‖Lq′ (R3

ξ)

≤ C‖|ξ|mρ̂(ξ, t)‖Lq′ (|ξ|≤ε) + C‖|ξ|mρ̂(ξ, t)‖Lq′ (|ξ|≥ε),
(4.29)

where 1
q + 1

q′ = 1. We estimate the first term on the right-hand side of (4.29) as

‖|ξ|mρ̂(t, ξ)‖q
′

Lq′ (|ξ|≤ε)

≤ C
∫
|ξ|≤ε

|ξ|mq
′
e−q

′λ|ξ|2t + |ξ|(m+2)q′e−q
′νλt)|ρ̂0(ξ)|q

′

+ C(|ξ|mq
′+q′e−q

′λ|ξ|2t + |ξ|mq
′+q′e−q

′νλt)|û0(ξ)|q
′
dξ

≤ C sup
ξ
|ρ̂0|q

′
∫
|ξ|≤ε

|ξ|mq
′
e−q

′λ|ξ|2(1+t)+q′λ|ξ|2 + |ξ|mq
′+2q′e−q

′νλt)dξ

+ C sup
ξ
|û0|q

′
∫
|ξ|≤ε

(|ξ|mq
′+q′e−q

′λ|ξ|2(1+t)+q′λ|ξ|2 + |ξ|mq
′+q′e−q

′νλt)dξ

≤ C(1 + t)−
mq′+3

2 ‖ρ0‖q
′

L1 + C(1 + t)−
mq′+q′+3

2 ‖u0‖q
′

L1 + Ce−q
′νλt‖(ρ0, u0)‖q

′

L1 .

Thus,

‖|ξ|mρ̂(t, ξ)‖Lq′ (|ξ|≤ε) ≤ C(1 + t)
− 3

2q′−
m
2 ‖ρ0‖L1

+ C(1 + t)
− 3

2q′−(
m+1

2 )‖u0‖L1 + Ce−νλt‖(ρ0, u0)‖L1

≤ C(1 + t)−
3
2 [1−

1
q ]−

m
2 ‖(ρ0, u0)‖L1 .

(4.30)

Now, we estimate the second term of (4.29) using the Hölder inequality with suffi-
ciently small ε > 0. Then we obtain

‖|ξ|mρ̂(t, ξ)‖Lq′ (|ξ|≥ε) ≤ C
∫
|ξ|≥ε

|ξ|mq
′
e−q

′λt|(ρ̂0(ξ), û0(ξ))|q
′
dξ

≤ Ce−λt‖|ξ|−(3+ε)‖
2−q′
2q′ ‖|ξ|(3+ε)

2−q′
2q′ +m

(ρ̂0(ξ), û0(ξ))‖L2

≤ Ce−λt‖∇m+3[ 12−
1
q ]−(ρ0, u0)‖L2 .

(4.31)
Substituting (4.30) and (4.31) to (4.29), we have (4.27).

To prove (4.28), it similarly holds that

‖∇mu(t)‖Lq(R3
x)
≤ C‖|ξ|mû(t, ξ)‖Lq′ (R3

ξ)

≤ C‖|ξ|mû(t, ξ)‖Lq′ (|ξ|≤ε) + C‖|ξ|mû(t, ξ)‖Lq′ (|ξ|≥ε).
(4.32)

Using (4.24), we estimate the first term of (4.32) as

‖|ξ|mû(t, ξ)‖q
′

Lq′ (|ξ|≤ε)



16 A. LAGHA, H. HATTORI EJDE-2022/48

≤ C
∫
|ξ|≤ε

(|ξ|mq
′+q′(e−q

′λ|ξ|2(t+1) + e−q
′νλt)|ρ̂0(ξ)|q

′
)dξ

+ C

∫
ξ≤ε

(|ξ|(m+2)q′e−q
′λ|ξ|2(t+1) + e−q

′νλt)|û0(ξ)|q
′
dξ

≤ C(1 + t)−
mq′+q′+3

2 ‖ρ0‖q
′

L1 + (1 + t)−
mq′+2q′+3

2 ‖u0‖q
′

L1

+ Ce−q
′νλt‖(ρ0, u0)‖q

′

L1 .

It follows that

‖|ξ|mû(t, ξ)‖Lq′ (|ξ|≤ε)

≤ C(1 + t)
− 3

2q′−
m+1

2 ‖[ρ0‖L1 + (1 + t)
− 3

2q′−
m+2

2 ‖u0‖L1

+ Ce−νλt‖(ρ0, u0)‖L1

≤ C(1 + t)−
3
2 [1−

1
q ]−

m+1
2 ‖ρ0‖L1 + (1 + t)−

3
2 [1−

1
q ]−

m+2
2 ‖u0‖L1

≤ C(1 + t)−
3
2 [1−

1
q ]−

m+1
2 ‖(ρ0, u0)‖L1 .

(4.33)

Similarly to (4.31), one has

‖|ξ|mû(t, ξ)‖Lq′ (|ξ|≥ε) ≤ Ce
−λt‖∇m+3[ 12−

1
q ]−(ρ0, u0)‖L2 . (4.34)

Thus, substituting (4.34) and (4.33) in (4.32), we obtain (4.28). This completes the
proof. �

Corollary 4.3. Assume that U1 = eBtU1,0 is the solution to the Cauchy problem
(4.6) with initial data U1,0 = (ρ0, u0). Then, U1 = (ρ, u) satisfies the following:

‖ρ(t)‖L2 ≤ C(1 + t)−3/4‖(ρ0, u0)‖L1 + e−λt‖(ρ0, u0)‖L2 , (4.35)

‖u(t)‖L2 ≤ C(1 + t)−
5
4 ‖(ρ0, u0)‖L1 + e−λt‖(ρ0, u0)‖L2 , (4.36)

‖ρ(t)‖L∞ ≤ C(1 + t)−
3
2 ‖(ρ0, u0)‖L1 + e−λt‖∇2(ρ0, u0)‖L2 , (4.37)

‖u(t)‖L∞ ≤ C(1 + t)−2‖(ρ0, u0)‖L1 + e−λt‖∇2(ρ0, u0)‖L2 . (4.38)

5. Time-decay rates for the nonlinear system

In this section, we prove Propositions 2.2 and 2.3. The main idea is to combine
the energy estimates and the spectral analysis. We will apply the linear L2 −
Lq time-decay property of the homogeneous system (4.5) studied in the previous
section to the nonlinear case. We need the mild form of the original nonlinear
Cauchy problem (2.2). Throughout this section, we suppose that U = (ρ, u, c1, c2)
is the solution to the Cauchy problem (2.2) with initial data U0 = (ρ0, u0, c1,0, c2,0)
satisfying (2.3). Then, by Duhamel’s principle, the solution U = (ρ, u, c1, c2) can
be formally written as

U(t) = eBtU0 +

∫ t

0

e(t−s)B(g1, g2, g3, g4)ds, (5.1)

where eBt, t ≥ 0, is called the linear solution operator and the nonlinear source
term takes the form (4.2).



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 17

5.1. Decay rates for the energy functional and high-order energy func-
tional. In this subsection, we prove Proposition 2.2, i.e., the decay rates for the
energy functional ‖U(t)‖2N and for the high-order energy functional ‖∇U(t)‖2N . For
that, we investigate the time-decay rates of solutions in Proposition 2.1 under the
extra condition (2.8).

Proof of Proposition 2.2. Suppose εN+1(U0) is sufficiently small. Then, from Propo-
sition 2.1 the solution U = (ρ, u, c1, c2) satisfies

d

dt
EN (U(t)) + λ1DN (U(t)) + λ2DhN (U(t)) ≤ 0 (5.2)

for any t ≥ 0.
Now, we begin with the time-weighted estimates and iteration for inequality

(5.2). Let l ≥ 0. Multiplying (5.2) by (1 + t)l and integrating over [0, t] give

(1 + t)lEN (U(t)) + λ1

∫ t

0

(1 + s)lDN (U(s))ds+ λ2

∫ t

0

(1 + s)lDhN (U(s))ds

≤ EN (U0) + l

∫ t

0

(1 + s)l−1ENU(s)ds

≤ EN (U0) + Cl

∫ t

0

(1 + s)l−1(DN−1(U(s)) +DhN (U(s)) + ‖ρ(s)‖2)ds,

where we have used

EN (U(t)) ≤ CDN−1(U(t)) + CDhN (U(t)) + ‖ρ(t)‖2.
Using (5.2) again, we have

EN+1(U(t)) + λ1

∫ t

0

DN+1(U(t)) + λ2

∫ t

0

DhN+1(U(t)) ≤ EN+1(U0),

and

(1 + t)l−1EN+1(U(t)) + λ1

∫ t

0

(1 + s)l−1DN+1(U(s))ds

+ λ2

∫ t

0

(1 + s)l−1DhN+1(U(s))ds

≤ EN+1(U0) + C(l − 1)

∫ t

0

(1 + s)l−2EN+1(U(s))ds

≤ EN+1(U0) + C(l − 1)

∫ t

0

(1 + s)l−2(DN (U(s))

+ CDhN+1(U(s)) + ‖ρ(s)‖2)ds.

Then, for 1 < l < 2, iterating the previous estimates, we obtain

(1 + t)lEN (U(t)) + λ1

∫ t

0

(1 + s)lDN (U(s))ds+ λ2

∫ t

0

(1 + s)lDhN (U(s))ds

≤ EN+1(U0) + C

∫ t

0

(1 + s)l−1‖ρ(s)‖2ds.
(5.3)

On the other hand, to estimate the integral term on the right-hand side of the
previous inequality, let us define

EN,∞(U(t)) = sup
0≤s≤T

(1 + t)
3
2 EN (U(t)).



18 A. LAGHA, H. HATTORI EJDE-2022/48

Then, applying the linear estimate on ρ in (4.35) to the mild form (5.1), one has

‖ρ(t)‖ ≤ C(1 + t)−3/4‖ρ0, u0‖L1 + Ce−λt‖(ρ0, u0)‖

+ C

∫ t

0

(1 + t− s)−3/4‖(g1(s), g2(s))‖L1ds

+ C

∫ t

0

e−λ(t−s)‖(g1(s), g2(s))‖ds.

(5.4)

Recall the definitions (4.2) of g1 and g2. It is easy to check that for any 0 ≤ s ≤ t,

‖(g1(s), g2(s))‖L1∩L2 ≤ CEN (U(t)) ≤ C(1 + s)−3/2EN,∞(U(t)).

Substituting this into (5.4) gives

‖ρ(t)‖ ≤ C(1 + t)−3/4(E2N,∞(U(t)) + ‖(ρ0, u0)‖2L1∩L2). (5.5)

Next, we prove the uniform-in-time bound of EN,∞(U(t)) which implies the decay
rates of the energy functional EN (U(t)). In fact, taking l = 3

2 + ε in (5.3), where
ε > 0 is small enough, we see that

(1 + t)
3
2+εEN (U(t)) + λ1

∫ t

0

(1 + s)
3
2+εDN (U(s))ds+ λ2

∫ t

0

(1 + s)
3
2+εDhN (U(s))ds

≤ EN+1(U0) + C

∫ t

0

(1 + s)
1
2+ε‖ρ(s)‖2ds.

For ρ, using (5.5), we obtain∫ t

0

(1 + s)
1
2+ε‖ρ(t)‖2ds ≤ C(1 + t)ε(E2N,∞(U(t)) + ‖(ρ0, u0)‖2L1∩L2).

Therefore,

(1 + t)
3
2+εEN (U(t)) + λ1

∫ t

0

(1 + s)
3
2+εDN (U(s))ds+ λ2

∫ t

0

(1 + s)
3
2+εDhN (U(s))ds

≤ EN+1(U0) + C(1 + t)ε(E2N,∞(U(t)) + ‖(ρ0, u0)‖2L1∩L2),

which implies

(1 + t)
3
2 EN (U(t)) ≤ C(EN+1(U0) + ‖(ρ0, u0)‖2L1 + E2N,∞(U(t))),

and thus

EN,∞(U(t)) ≤ C(ε2N+1(U0) + E2N,∞(U(t))).

Since ε2N+1(U0) > 0 is sufficiently small, it holds that EN,∞(U(t)) ≤ Cε2N+1(U0) for

any t ≥ 0, which gives ‖U(s)‖N ≤ C(EN (U(t)))1/2 ≤ CεN+1(U0)(1 + t)−3/4. This
proves (2.9).

Now, we estimate the high-order energy functional. By comparing the definitions
of EN (U(t)), DN (U(t)) and DhN (U(t)), it follows from (5.2) that

d

dt
‖∇U(t)‖2N + λ‖∇U(t)‖2N ≤ C‖∇ρ(t)‖2,

which implies

‖∇U(t)‖2N ≤ e−λt‖∇U0‖2N + C

∫ t

0

e−λ(t−s)‖∇ρ(s)‖2ds, (5.6)



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 19

for any t ≥ 0. To estimate the time integral term on the right hand side of the
above inequality, one can apply the linear estimate (4.27) to the mild form (5.1) of
the solution U(t) so that

‖∇ρ(t)‖ ≤ C(1 + t)−5/4‖(ρ0, u0)‖L1 + Ce−λt‖∇(ρ0, u0)‖

+ C

∫ t

0

(1 + t− s)−5/4‖(g1(s), g2(s))‖L1ds

+ C

∫ t

0

e−λ(t−s)‖∇(g1(s), g2(s))‖ds.

(5.7)

Recall the definition (4.2) of g1 and g2. It is straightforward to check that for any
0 ≤ s ≤ t

‖(g1(s), g2(s))‖L1∩Ḣ1 ≤ CEN (U(s)) ≤ Cε2N+1(U0)(1 + s)−3/2.

Putting this into (5.7) gives

‖∇ρ(t)‖ ≤ CεN+1(U0)(1 + t)−5/4. (5.8)

Then, by using (5.8) in (5.6), we have

‖∇U(t)‖2N ≤ e−λt‖∇U0‖2N + Cε2N+1(U0)(1 + t)−5/2,

which implies (2.10). The proof of Proposition 2.2 is complete. �

5.2. Decay rates in Lq. In this subsection, we prove Proposition 2.3 for time-
decay rates in Lq corresponding to (1.6)-(1.8) in Theorem 1.1. For N ≥ 4, Propo-
sition 2.2 shows that if εN+1(U0) is small enough,

‖U(s)‖N ≤ CεN+1(U0)(1 + t)−3/4, (5.9)

‖∇U(t)‖N ≤ CεN+1(U0)(1 + t)−5/4. (5.10)

First, we consider the Lq estimates on ρ. We use the L2 − L∞ interpolation
inequality. For the L2 rate, it is easy to see from (5.5) and (5.9) that

‖ρ(t)‖ ≤ CεN+1(U0)(1 + t)−3/4 ≤ C(1 + t)−3/4.

For the L∞ rate, applying the L∞ linear estimate on ρ in (4.37) to the mild form
(5.1), we have

‖ρ(t)‖∞ ≤ C(1 + t)−3/2‖(ρ0, u0)‖L1 + Ce−λt‖∇2(ρ0, u0)‖

+ C

∫ t

0

(1 + t− s)−3/2‖(g1(s), g2(s))‖L1ds

+ C

∫ t

0

e−λ(t−s)‖∇2(g1(s), g2(s))‖ds

≤ C(1 + t)−3/2‖(ρ0, u0)‖L1∩Ḣ2

+ C

∫ t

0

(1 + t− s)−3/2‖(g1(s), g2(s))‖L1∩Ḣ2ds.

(5.11)

Since by (5.9),

‖(g1(s), g2(s))‖L1∩Ḣ2 ≤ C‖∇U(t)‖N‖U(s)‖N ≤ Cε2N+1(U0)(1 + s)−2,

substituting the above inequality into (5.11), we obtain

‖ρ(t)‖L∞ ≤ CεN+1(U0)(1 + t)−3/2.



20 A. LAGHA, H. HATTORI EJDE-2022/48

Then, by the L2 − L∞ interpolation, we have

‖ρ‖Lq ≤ CεN+1(U0)(1 + t)
−3
2 + 3

2q (5.12)

for 2 ≤ q ≤ ∞.
Next, we show the estimates on ‖u(t)‖Lq . For the L2 rate, utilizing the L2

estimate on u in (4.36) to (5.1), we have

‖u(t)‖ ≤ C(1 + t)−5/4‖(ρ0, u0)‖L1 + Ce−λt‖(ρ0, u0)‖

+ C

∫ t

0

(1 + t− s)−5/4‖(g1, g2)‖L1ds

+

∫ t

0

e−λ(t−s)‖(g1(s), g2(s))‖ds.

(5.13)

Since (5.9) implies

‖(g1(s), g2(s))‖L1∩L2 ≤ C‖U(s)‖2N ≤ CεN+1(U0)(1 + t)−3/2,

(5.13) yields a slower decay estimate

‖u(t)‖ ≤ CεN+1(U0)(1 + t)−5/4 ≤ C(1 + t)−5/4. (5.14)

For the L∞ rate, applying the L∞ estimate on u in (4.38) to (5.1), we have

‖u(t)‖L∞ ≤ (1 + t)−2‖ρ0, u0‖L1∩Ḣ2

+ C

∫ t

0

(1 + t− s)−2‖(g1(s), g2(s))‖L1∩Ḣ2ds.
(5.15)

By the estimates (5.9) and (5.10), we obtain

‖(g1(s), g2(s))‖L1∩Ḣ2 ≤ C‖∇U(t)‖N‖U(s)‖N ≤ Cε2N+1(U0)(1 + s)−2,

Therefore, from (5.15) it follows that

‖u(t)‖L∞ ≤ CεN+1(U0)(1 + s)−2,

and consequently, the L2-L∞ interpolation theorem implies

‖u(t)‖Lq ≤ CεN+1(U0)(1 + t)−2+
3
2q (5.16)

for 2 ≤ q ≤ ∞.



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 21

Lastly, we estimate the time-decay rates of (c1, c2). We start with the estimate
on ‖c1(t)‖Lq . For the L2 rate,

‖c1‖L2 ≤ C‖ĉ1‖L2(ξ)

≤ C
[ ∫

ξ

e−2(|ξ|
2+(a11−a12n∞κ′1(0)))t|ĉ0|2dξ

]1/2
+ a11

∫ t

0

[ ∫
ξ

e−2(|ξ|
2+(a11−a12n∞κ′1(0))(t−s)|ĝ3|2dξ

]1/2
ds

≤ e−(a11−a12n∞κ
′
1(0))t

[ ∫
ξ

e−2|ξ|
2(t)|ĉ0|2dξ

]1/2
+ C

∫ t

0

e−(a11−a12n∞κ
′
1(0))(t−s)

[ ∫
ξ

e−2|ξ|
2(t−s+1)|ĝ3|2dξ

]1/2
ds

≤ Ce−(a11−a12n∞κ
′
1(0))t‖ĉ0‖L2

+ C

∫ t

0

e−(a11−a12n∞κ
′
1(0))(t−s)(sup

ξ
e−|ξ|

2(t−s+1))‖g3‖L2ds

(5.17)

By (5.12), we see that

‖g3‖L2 ≤ (‖ρκ1(c1)‖L2 + ‖c1‖L∞‖c1‖2L2) ≤ C‖U(s)‖2N ≤ Cε2N+1(U0)(1 + t)−3/2.

This and (5.17) imply the decay estimate for c1

‖c1‖L2 ≤ CεN+1(U0)(1 + t)−3/2. (5.18)

For L∞ rates, we estimate the low frequency and high frequency separately. From
the Hausdorff-Young inequality and Hölder inequality, we have

‖c1‖L∞
≤ C‖ĉ1‖L1

≤ C
∫
ξ≤ε

e−(|ξ|
2+(a11−a12n∞κ′1(0)))t|ĉ1,0|dξ

+ C

∫ t

0

∫
ξ≤ε

e−(|ξ|
2+(a11−a12n∞κ′1(0)))(t−s)|ĝ3|dξds

+ C

∫
|ξ|≥ε

e−(a11−a12n∞κ
′
1(0))t|ĉ1,0|dξ

+ C

∫ t

0

∫
|ξ|≥ε

e−(a11−a12n∞κ
′
1(0))(t−s)|ĝ3|dξds

≤ Ce−(a11−a12n∞κ
′
1(0))t(1 + t)−3/2‖c0‖L1 + C

∫ t

0

e−(a11−a12n∞κ
′
1(0))(t−s)‖ĝ3(s)‖L1

+ Ce−(a11−a12n∞κ
′
1(0))t

[ ∫
|ξ|≥ε

|ξ|−4dξ
]1/2[ ∫

|ξ|≥ε
|ξ|4|ĉ1,0|2dξ

]1/2
+ C

∫ t

0

e−(a11−a12n∞κ
′
1(0)))(t−s)

[ ∫
|ξ|≥ε

|ξ|−4dξ
]1/2[ ∫

|ξ|≥ε
|ξ|4|ĝ3|2dξ

]1/2
ds

≤ Ce−(a11−a12n∞κ
′
1(0))t(1 + t)−3/2‖c0‖L1

+ C

∫ t

0

e−(a11−a12n∞κ
′
1(0))(t−s)‖g3(s)‖L1ds



22 A. LAGHA, H. HATTORI EJDE-2022/48

+ Ce−(a11−a12n∞κ
′
1(0))t‖∇2c0‖L2

+ C

∫ t

0

e−(a11−a12n∞κ
′
1(0))(t−s)‖∇2g3(s)‖L2ds. (5.19)

By (5.12), we see that

‖g3(s)‖L1∩Ḣ2 ≤ C‖U(s)‖2N ≤ Cε2N+1(U0)(1 + t)−3/2.

Then (5.19) implies the decay estimate

‖c1‖L∞ ≤ CεN+1(U0)(1 + t)−3/2. (5.20)

The similar estimates hold for c2. Therefore, by the L2-L∞ interpolation, we obtain

‖(c1, c2)‖Lq ≤ CεN+1(U0)(1 + t)−3/2 (5.21)

for 2 ≤ q ≤ ∞. This completes the proof of Proposition 2.2 and hence of Theo-
rem 1.1.

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[8] T. Hillen, K. J. Painter; A user’s guide to PDE models for chemotaxis. J. Math. Biol., 58

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Theor. Biol., 26 (1970), 399–415.

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Aesha Lagha

Department of Mathematics, Zawia University, Zawia, Libya
Email address: aelagha46@gmail.com



EJDE-2022/48 CAUCHY PROBLEMS FOR CHEMOTAXIS SYSTEMS 23

Harumi Hattori

Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

Email address: hhattori@wvu.edu


	1. Introduction
	2. Reformulation of the system (??)
	3. Global solution of the nonlinear system (??)
	3.1. Existence of local solutions
	3.2. A Priori Estimates

	4. Linearized homogeneous system
	4.1. Representation of solutions for (??) and (??) 
	4.2. Refined L2-Lq time-decay property

	5. Time-decay rates for the nonlinear system
	5.1. Decay rates for the energy functional and high-order energy functional
	5.2. Decay rates in Lq

	References