Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 122, pp. 1–14.

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

EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR A

KELLER-SEGEL SYSTEM WITH GRADIENT DEPENDENT

CHEMOTACTIC SENSITIVITY

JIANLU YAN, YUXIANG LI

Abstract. We consider the Keller-Segel system with gradient dependent chemo-

tactic sensitivity

ut = ∆u−∇ · (u|∇v|p−2∇v), x ∈ Ω, t > 0,

vt = ∆v − v + u, x ∈ Ω, t > 0,

∂u

∂ν
=
∂v

∂ν
= 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω

in a smooth bounded domain Ω ⊂ Rn, n ≥ 2. We shown that for all reasonably

regular initial data u0 ≥ 0 and v0 ≥ 0, the corresponding Neumann initial-

boundary value problem possesses a global weak solution which is uniformly
bounded provided that 1 < p < n/(n− 1).

1. Introduction

In this article, we consider the chemotaxis system with gradient dependent
chemotactic sensitivity

ut = ∆u−∇ · (u|∇v|p−2∇v), x ∈ Ω, t > 0,

vt = ∆v − v + u, x ∈ Ω, t > 0,

∂u

∂ν
=
∂v

∂ν
= 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω,

(1.1)

where Ω ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary and 1 < p <
n/(n− 1).

Keller and Segel [9] introduced a mathematical model to describe chemotactic
aggregation of cellular slime molds. The classical Keller-Segel system is

ut = ∆u−∇(u∇v),

vt = ∆v − v + u,
(1.2)

where u denotes the cell density and v describes the concentration of the chemical
signal secreted by cells. This parabolic-parabolic Keller-Segel system has been
studied extensively in literature, see the review paper [2, 6, 7] for details. Here we

2010 Mathematics Subject Classification. 35K55, 35B40, 35Q92, 92C17.

Key words and phrases. Keller-Segel system; weak solution; chemotactic sensitivity.
c©2020 Texas State University.

Submitted June 4, 2019. Published December 16. 2020.

1



2 J. YAN, Y. LI EJDE-2020/122

point out that the authors in [11] proved that (1.2) has global bounded solutions
under the condition

∫
Ω
u0(x) < 4π in R2 or under the condition

∫
Ω
u0(x) < 8π for

radial solutions on a disk. Winkler[20] proved that finite-time blow-up occurs for
radially symmetric initial data when

∫
Ω
u0 is arbitrary prescribed number.

The chemotactic sensitivity can depend nonlinearly on the cell density. Some
authors studied the system

ut = ∇(D(u)∇u)−∇(S(u)∇v),

vt = ∆v − v + u
(1.3)

in the past decades. Horstmann and Winkler [8] determined the critical blow-up
exponent for (1.3), where D(u) = 1 and the chemotactic sensitivity equals some
nonlinear function of the particle density. In [18], it is proved that if S(u)/D(u)
grows faster than u2/n as u→∞ and D(u) satisfies some technical conditions, then
there exist solutions that blow up in either finite or infinite time. In [14], Tao and
Winkler showed that if S(u)/D(u) ≤ cuα with α < 2/n and D(u) satisfies algebraic
upper and lower growth, then the classical solutions to (1.3) are uniformly bounded.

By the Weber-Fechner law, the classical Keller-Segel system has been modified
to the Keller-Segel system with a singular sensitivity

ut = ∆u− χ∇
(u
v
∇v
)
,

vt = ∆v − v + u.
(1.4)

Winkler [19] proved that if 0 < χ <
√

2/n, (1.4) has a global-in-time classical
solution. Furthermore, relaxing the solution concept, the global existence of weak
solutions is established whenever 0 < χ <

√
(n+ 2)/(3n− 4). In [13], Stinner and

Winkler introduced a generalized solution concept, and then proved that such gen-
eralized solution for any χ > 0. In [10], the authors introduced another generalized
solution concept, which exists for the some range of χ.

Recently, Bellomo and Winkler posed a model where the chemotactic sensitivity
depends on ∇v. In [3] the authors deduced the existence of a unique radial classical
solution to the system

ut = ∇ ·
( u∇u√

u2 + |∇u|2
)
− χ∇ ·

( u∇v√
1 + |∇v|2

)
,

0 = ∆v −M + u,

(1.5)

where M = 1
|Ω|
∫

Ω
u0(x)dx, n ≥ 2 and χ < 1. In [4], it is showed that for some

T > 0, (1.5) possesses a uniquely determined classical solution blowing up at time
T . [22] concerns the null controllability of a control system governed by coupled
degenerate parabolic equations with lower order terms.

Negreanu and Tello [12] proposed the model

ut = ∆u−∇ · (χu|∇v|p−2∇v),

0 = ∆v −M + u,
(1.6)

where M = 1
|Ω|
∫

Ω
u0(x)dx. The authors obtained uniform bounds in L∞(Ω) pro-

vided that 1 < p < n/(n − 1) (n > 1). In the one-dimensional case, they proved
that for any positive constants χ and M , if p ∈ (1, 2), then the model (1.6) has
infinitely many non-constant solutions.



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 3

In this article, we study the global existence and boundedness of (1.1), the
parabolic-parabolic version of (1.6). Now we state the main results of this article.
We assume that the initial data u0 and v0 satisfy

u0 ∈ C0(Ω̄) with u0 ≥ 0 in Ω and u0 6≡ 0,

v0 ∈W 1,∞(Ω) with v0 ≥ 0 in Ω̄.
(1.7)

Our main results read as follows.

Theorem 1.1. Let Ω ⊂ Rn, n ≥ 2 be a bounded domain with smooth boundary.
Then for all u0 and v0 satisfying (1.7), system (1.1) with 1 < p < n/(n − 1)
possesses at least one global weak solution in the sense of Definition 2.1.

Theorem 1.2. Under the assumption of Theorem 1.1, there exists a constant C =
C(u0, p,Ω) > 0, such that

‖u(·, t)‖L∞(Ω) ≤ C for all t > 0.

The rest of this article is organized as follows. In Section 2, we introduce the
conception of the weak solution. Section 3 is devoted to showing the existence of
the weak solution. Finally, we give the proof of the boundedness in Section 4.

2. A weak solution concept and approximate problems

Let us firstly introduce a natural concept of weak solutions to (1.1).

Definition 2.1. Assume that u0 and v0 satisfy (1.7). For all T > 0, a pair (u, v)
of functions

u ∈ L∞(Ω̄× [0, T )), v ∈ L∞(Ω̄× [0, T )) ∩ L2([0, T );W 1,2(Ω)) (2.1)

with

u ≥ 0 a.e. in Ω× (0, T ) and v ≥ 0 a.e. in Ω× (0, T ), (2.2)

and

|∇v|p−2∇v ∈ L2(Ω̄× [0, T )), (2.3)

will be called a weak solution of (1.4) if u has the mass conservation property∫
Ω

u(x, t)dx =

∫
Ω

u0(x) for a.e. t > 0, (2.4)

and the following two identities

−
∫

Ω

u0ϕ(·, 0)−
∫ T

0

∫
Ω

uϕt =

∫ T

0

∫
Ω

u ·∆ϕ+

∫ T

0

∫
Ω

u|∇v|p−2∇v · ∇ϕ (2.5)

and ∫ T

0

∫
Ω

vψt +

∫
Ω

v0ψ(·, 0) =

∫ T

0

∫
Ω

∇v · ∇ψ +

∫ T

0

∫
Ω

vψ −
∫ T

0

∫
Ω

uψ (2.6)

hold for non-negative ϕ, ψ ∈ C∞0 (Ω̄× [0, T )).



4 J. YAN, Y. LI EJDE-2020/122

We intend to construct a solution of (1.1) as the limit of a sequence of solutions
to the approximate problems

uεt = ∆uε −∇ ·
(
uε(|∇vε|2 + ε)

p−2
2 ∇vε

)
, x ∈ Ω, t > 0,

vεt = ∆vε − vε + uε, x ∈ Ω, t > 0,

∂uε
∂ν

=
∂vε
∂ν

= 0, x ∈ ∂Ω, t > 0,

uε(x, 0) = u0(x), vε(x, 0) = v0(x), x ∈ Ω,

(2.7)

where ε ∈ (0, 1) is a positive parameter. We construct a suitable fixed point frame-
work to prove the existence of classical solutions to (2.7).

Lemma 2.2. Assume that (1.7) holds, and let ε ∈ (0, 1). Then there exists Tmax,ε ≤
∞, such that (2.7) possesses a classical solution (uε, vε),

uε ∈ C0(Ω̄× [0, Tmax,ε)) ∩ C2,1(Ω̄× (0, Tmax,ε))

vε ∈ C0(Ω̄× [0, Tmax,ε)) ∩ C2,1(Ω̄× (0, Tmax,ε)) ∩ L∞loc([0, Tmax,ε);W
1,ϑ(Ω))

for each ϑ > n, which satisfies uε > 0 in Ω̄× (0,∞) and∫
Ω

uε(x, t)dx =

∫
Ω

u0(x)dx for all t ∈ (0, Tmax,ε), (2.8)

as well as ∫
Ω

vε(t) =

∫
Ω

u0 +
(∫

Ω

v0 −
∫

Ω

u0

)
e−t for all t ∈ (0, Tmax,ε). (2.9)

Proof. Let us prove the existence of solutions by a standard contraction argument
referring to [8]. For T ∈ (0, 1), we define a Banach space

X := C0(Ω̄× [0, T ])× L∞((0, T );W 1,ϑ(Ω)).

Consider the closed set

S :=
{

(uε, vε) ∈ X : ‖(uε, vε)‖X ≤ R
}

with R = ‖(u0, v0)‖X + 1.

We claim that for T sufficiently small, the map

Ψ(uε, vε)(t) :=

(
Ψ1(uε, vε)(t)
Ψ2(uε, vε)(t)

)
:=

(
et∆u0 −

∫ t
0
e(t−s)∆∇ · (uε(|∇vε|2 + ε)

p−2
2 ∇vε(s))ds

et(∆−1)v0 +
∫ t

0
e(t−s)(∆−1)uε(s)ds

)
is a contraction from S to S. We fix β ∈ ( n2ϑ ,

1
2 ) and δ ∈ (0, 1

2 − β). Then for all
t ∈ [0, T ] we have

‖Ψ1(uε, vε)(t)‖C0(Ω̄)

≤ ‖et∆u0‖C0(Ω̄)

+ C

∫ t

0

‖(−∆ + 1)βe(t−s)∆∇ · (uε(|∇vε|2 + ε)
p−2
2 ∇vε(s))‖Lϑ(Ω)ds

≤ ‖u0‖C0(Ω̄) + C

∫ t

0

(t− s)−β− 1
2−δ‖uε(|∇vε|2 + ε)

p−2
2 ∇vε(s)‖Lϑ(Ω)ds

≤ ‖u0‖C0(Ω̄) + CRpT
1
2−β−δ,

(2.10)



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 5

where we have used the estimate

‖uε(|∇vε|2 + ε)
p−2
2 ∇vε‖Lϑ(Ω) ≤ R‖|∇vε|p−1‖Lϑ(Ω)

≤ R‖∇vε‖p−1
Lϑ(p−1)(Ω)

≤ CR‖∇vε‖p−1
Lϑ(Ω)

.

Let γ ∈ (1/2, 1); for for all t ∈ [0, T ] we have

‖Ψ2(uε, vε)(t)‖w1,q(Ω)

≤ ‖et(∆−1)v0‖W 1,ϑ(Ω) + C

∫ t

0

‖(−∆ + 1)γe(t−s)(∆−1)uε(s)‖Lϑ(Ω)ds

≤ ‖v0‖W 1,ϑ(Ω) + C

∫ t

0

(t− s)γ‖uε(s)‖Lϑ(Ω)ds

≤ ‖v0‖W 1,ϑ(Ω) + CRT 1−γ .

(2.11)

From (2.10) and (2.11), it follows that ΨS ⊂ S if we choose T small. For all
(uε, vε), (ūε, v̄ε) ∈ S, we have

‖Ψ1(uε, vε)(t)−Ψ1(ūε, v̄ε)(t)‖C0(Ω̄)

≤ C
∫ t

0

∥∥∥(−∆ + 1)βe(t−s)∆∇ · (uε(|∇vε|2 + ε)
p−2
2 ∇vε(s)

− ūε(|∇v̄ε|2 + ε)
p−2
2 ∇v̄ε(s))

∥∥∥
Lϑ(Ω)

ds

≤ C
∫ t

0

(t− s)−β− 1
2−δ
∥∥∥uε(|∇vε|2 + ε)

p−2
2 ∇vε(s)

− ūε(|∇v̄ε|2 + ε)
p−2
2 ∇v̄ε(s)

∥∥∥
Lϑ(Ω)

ds

≤ C(R+Rp−1)T
1
2−β−δ‖(uε, vε)− (ūε, v̄ε)‖X

and

‖Ψ2(uε, vε)(t)−Ψ2(ūε, v̄ε)(t)‖W 1,ϑ(Ω)

≤ C
∫ t

0

‖(∆ + 1)γe(t−s)(∆−1)(uε(s)− ūε)‖Lϑ(Ω)ds

≤ C
∫ t

0

(t− s)−γ‖uε(s)− ūε‖Lϑ(Ω)ds

≤ CT 1−γ‖(uε, vε)− (ūε, v̄ε)‖X ,
so Ψ is shown to be a contraction if T is sufficiently small. By the Banach’s fixed
point theorem, we obtain that the existence of (u, v) ∈ X satisfies (u, v) = Ψ(u, v).

Properties (2.8) and (2.9) follow by integrating the PDEs in (2.7) in space. �

3. Existence of the weak solutions

The construction of a global weak solution is based on a limit procedure of
solutions to suitably regularized problems. The Aubin-Lions lemma is very helpful.
We collect some ε-independent a priori estimates of the solutions to (2.7). For the
second equation in (2.7), using the parabolic theory, we obtain the following lemma.

Lemma 3.1 ([19, Lemma 2.4]). Let T > 0 and 1 ≤ θ, µ <∞.



6 J. YAN, Y. LI EJDE-2020/122

(i) If n
2 ( 1

θ −
1
µ ) < 1 then there exists C > 0 such that

‖vε(·, t)‖Lµ(Ω) ≤ C
(

1 + sup
s∈(0,t)

‖uε(·, s)‖Lθ(Ω)

)
(3.1)

for all t ∈ (0, T ) and ε ∈ (0, 1).
(ii) If 1

2 + n
2 ( 1

θ −
1
µ ) < 1 then

‖∇vε(·, t)‖Lµ(Ω) ≤ C
(

1 + sup
s∈(0,t)

‖uε(·, s)‖Lθ(Ω)

)
(3.2)

for all t ∈ (0, T ) and ε ∈ (0, 1) is valid with C > 0.

Proof. For convenience, we give the proof.
(i) We represent vε by

vε(·, t) = et(∆−1)v0 +

∫ t

0

e(t−s)(∆−1)uε(·, s)ds, (3.3)

where (et∆)t≥0 denotes the Neumann heat semigroup. By standard smoothing
estimates, we find that if µ ≥ θ then

‖vε(·, t)‖Lµ(Ω) ≤ C
(
‖v0‖L∞(Ω) +

∫ t

0

(t− s)−
n
2−( 1

θ−
1
µ )‖uε(·, s)‖Lµ(Ω)ds

)
(3.4)

for a constant C > 0. By (3.4) and Hölder’s inequality, we obtain (3.1) for µ < θ.
(ii) Applying ∇ to both sides in (3.3) and invoking corresponding smoothing

properties involving gradient [16], we similarly find that

‖∇vε(·.t)‖Lµ(Ω) ≤ C
(
‖∇v0‖L∞(Ω) +

∫ t

0

(t− s)−
1
2−

n
2−( 1

θ−
1
µ )‖uε(·, s)‖Lµ(Ω)ds

)
with a certain C > 0. So we conclude using the similar method of proving (i). �

With Lemma 3.1 in hand, using the Gagliardo-Nirenberg inequality, we can prove
the boundedness in the L2-norm of uε.

Lemma 3.2. Let 1 < p < n/(n − 1). For all T > 0, there exists C > 0 such that
for any ε ∈ (0, 1), ∫ T

0

∫
Ω

u2
ε ≤ C(T + 1). (3.5)

Proof. We multiply the first equation in (2.7) by uε, and integrate by parts to find
that

1

2

d

dt

∫
Ω

u2
ε = −

∫
Ω

|∇uε|2 +

∫
Ω

uε
(
|∇vε|2 + ε

) p−2
2 ∇vε · ∇uε.

By the Cauchy-Schwarz inequality, we have

d

dt

∫
Ω

u2
ε +

∫
Ω

|∇uε|2 ≤
∫

Ω

u2
ε

(
|∇vε|2 + ε

)p−2 |∇vε|2.



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 7

We can find µ satisfying 2(p− 1) < µ < n/(n− 1). Using Lemma 3.1 and Hölder’s
inequality, we have

d

dt

∫
Ω

u2
ε +

∫
Ω

|∇uε|2 ≤
∫

Ω

u2
ε

(
|∇vε|2 + ε

)p−1

≤
(∫

Ω

u
2µ

µ−2(p−1)
ε

)µ−2p−1
µ

(∫
Ω

(
|∇vε|2 + ε

)µ
2

) 2(p−1)
µ

≤ C
(∫

Ω

u
2µ

µ−2(p−1)
ε

)µ−2p−1
µ

[( ∫
Ω

|∇vε|µ
) 2(p−1)

µ

+ 1
]

≤ C
(∫

Ω

u
2µ

µ−2(p−1)
ε

)µ−2p−1
µ

.

(3.6)

Using the Gagliardo-Nirenberg inequality, we can find a positive constant C > 0
such that

‖uε‖
L

2µ
µ−2(p−1) (Ω)

≤ C‖∇uε‖aL2(Ω)‖uε‖
1−a
L1(Ω) + C‖uε‖L1(Ω), (3.7)

where

a =
1− µ−2(p−1)

2µ
1
2 + 1

n

.

Thanks to 1 < p < n/(n− 1), we have a ∈ (0, 1). We now apply inequality (3.7) to
(3.6), and obtain(∫

Ω

u
2µ

µ−2(p−1)
ε

)µ−2p−1
µ ≤ C

(
‖∇uε‖aL2(Ω)‖uε‖

1−a
L1(Ω) + ‖uε‖L1(Ω)

)2

≤ C(‖∇uε‖2aL2(Ω) + 1).

By Young’s inequality for a positive constant δ ∈ (0, 1), we have

d

dt

∫
Ω

u2
ε +

∫
Ω

|∇uε|2 ≤ C(‖∇uε‖2aL2(Ω) + 1) ≤ δ
∫

Ω

|∇uε|2 + C(δ),

which is equivalent to

d

dt

∫
Ω

u2
ε + (1− δ)

∫
Ω

|∇uε|2 ≤ C.

By the Poincaré-Wirtinger inequality, we obtain∫
Ω

|∇uε|2 ≥ C
∫

Ω

(
uε −

1

|Ω|

∫
Ω

uε

)2

= C
(∫

Ω

u2
ε −

1

|Ω|

∣∣∣ ∫
Ω

uε

∣∣∣2),
which implies

d

dt

∫
Ω

u2
ε +

∫
Ω

u2
ε ≤ C.

Finally using the standard ODE argument, we obtain (3.5). �

Next, we prove the almost everywhere convergence of uεk by referring to the
method in [21].

Lemma 3.3. Let 1 < p < n/(n − 1). For all T > 0, there exists C > 0 such that
for any ε ∈ (0, 1), we have∫ T

0

∫
Ω

|∇ ln(uε + 1)|2 ≤ C(T + 1). (3.8)



8 J. YAN, Y. LI EJDE-2020/122

Proof. We multiply the first equation in (2.7) by 1
uε+1 , and integrate by parts to

obtain

d

dt

∫
Ω

ln(uε + 1)

=

∫
Ω

|∇uε|2

(uε + 1)2
−
∫

Ω

uε
(uε + 1)2

(
∇uε ·

(
|∇vε|2 + ε

) p−2
2 ∇vε

)
=

∫
Ω

|∇ ln(u+ 1)|2 −
∫

Ω

uε
uε + 1

(
∇ ln(uε + 1) ·

(
|∇vε|2 + ε

) p−2
2 ∇vε

)
.

By the Cauchy-Schwarz inequality, we obtain∫
Ω

uε
uε + 1

(
∇ ln(uε + 1) ·

(
|∇vε|2 + ε

) p−2
2 ∇vε

)
≤ 1

2

∫
Ω

|∇ ln(uε + 1)|2 +
1

2

∫
Ω

u2
ε

(uε + 1)2

(
|∇vε|2 + ε

)p−2 |∇vε|2

≤ 1

2

∫
Ω

|∇ ln(uε + 1)|2 +
1

2

∫
Ω

u2
ε

(uε + 1)2

(
|∇vε|2 + ε

)p−1

≤ 1

2

∫
Ω

|∇ ln(uε + 1)|2 +
1

2

∫
Ω

(
|∇vε|2 + ε

)p−1
.

Then, we have

d

dt

∫
Ω

ln(uε + 1) ≥
∫

Ω

|∇ ln(u+ 1)|2 − 1

2

∫
Ω

|∇ ln(uε + 1)|2 − 1

2

∫
Ω

(
|∇vε|2 + ε

)p−1
.

By integrating with respect to time we obtain

1

2

∫ T

0

∫
Ω

|∇ ln(uε + 1)|2

≤
∫

Ω

ln(uε(·, T ) + 1)−
∫

Ω

ln(u0 + 1) +
1

2

∫ T

0

∫
Ω

(|∇vε|2 + ε)p−1

≤
∫

Ω

uε +
1

2

∫ T

0

∫
Ω

(|∇vε|2 + ε)p−1

≤ m+
1

2

∫ T

0

∫
Ω

|∇vε|2(p−1) + C,

where m :=
∫

Ω
u0. From 2(p− 1) < n/(n− 1), we obtain (3.8) by Lemma 3.1. �

Lemma 3.4. Let 1 < p < n/(n − 1). For all T > 0, there exists C > 0 such that
for any ε ∈ (0, 1), ∫ T

0

‖∂t ln(uε + 1)‖(Wn,2(Ω))∗dt ≤ C(T + 1). (3.9)

Proof. Testing the first equation in (2.7) by ψ
uε+1 for fixed t > 0 and arbitrary

ψ ∈ C∞(Ω̄), we obtain∫
Ω

∂t ln(uε + 1) · ψ =

∫
Ω

|∇ ln(uε + 1)|2ψ −
∫

Ω

∇ ln(uε + 1) · ∇ψ

−
∫

Ω

uε
uε + 1

(
∇ ln(uε + 1) ·

(
|∇vε|2 + ε

) p−2
2 ∇vε

)
ψ



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 9

+

∫
Ω

uε
uε + 1

(
|∇vε|2 + ε

) p−2
2 ∇vε · ∇ψ.

By the Cauchy-Schwarz inequality and Young’s inequality, we have∣∣ ∫
Ω

∂t ln(uε + 1) · ψ
∣∣

≤
∫

Ω

|∇ ln(uε + 1)|2‖ψ‖L∞(Ω) +
(∫

Ω

| ln(uε + 1)|2
)1/2

‖∇ψ‖L2(Ω)

+
(1

2

∫
Ω

u2
ε

(uε + 1)2
|∇ ln(uε + 1)|2 +

1

2

∫
Ω

(
|∇vε|2 + ε

)p−2 |∇vε|2
)
‖ψ‖L∞(Ω)

+
(∫

Ω

u2
ε

(uε + 1)2

(
|∇vε|2 + ε

)p−2 |∇vε|2
)1/2

‖∇ψ‖L2(Ω)

≤
(∫

Ω

|∇ ln(uε + 1)|2 +
1

2

∫
Ω

|∇ ln(uε + 1)|2 +
1

2

∫
Ω

(
|∇vε|2 + ε

)p−1
)
‖ψ‖L∞(Ω)

+
((∫

Ω

|∇ ln(uε + 1)|2
)1/2

+
(∫

Ω

(
|∇vε|2 + ε

)p−1
)1/2)

‖∇ψ‖L2(Ω)

≤
(

2

∫
Ω

|∇ ln(uε + 1)|2 +

∫
Ω

(
|∇vε|2 + ε

)p−1
+ 1
)(
‖ψ‖L∞(Ω) + ‖∇ψ‖L2(Ω)

)
.

Since in view of the fact that Wn,2(Ω) ↪→ L∞(Ω) we can fix C > 0 such that

‖∇ψ‖L2(Ω) + ‖ψ‖L∞(Ω) ≤ C‖ψ‖Wn,2(Ω)

for any such ψ, this entails

‖∂t ln(uε(·, t) + 1)‖(Wn,2(Ω))∗

≤ C
(

2

∫
Ω

|∇ ln(uε + 1)|2 +

∫
Ω

(
|∇vε|2 + ε

)p−1
+ 1
)
.

After an integration with respect to time, by Lemmas 3.1 and 3.3, this implies
(3.9). �

On the basis of previous three lemmas, we can extract a subsequence of the
approximate solutions of (2.7). By the compactness arguments, the limit function
can be shown to be a weak solution of (1.1).

Lemma 3.5. Let 1 < p < n/(n−1). There exist non-negative functions u, v defined
on Ω× (0,∞) as well as a sequence (εk)k∈N ⊂ (0, 1), and such that as ε = εk ↘ 0,

uε → u a.e. in Ω× (0, T ), (3.10)

uε ⇀ u in L2(Ω× (0, T )), (3.11)

vε → v in L2((0, T );W 1,2(Ω)), (3.12)

∇vε → ∇v a.e. in Ω× (0, T ), (3.13)

|∇vε|p−2∇vε ⇀ |∇v|p−2∇v in Lp
′
(Ω× (0, T )), (3.14)

where 1
p + 1

p′ = 1.

Proof. By Lemmas 3.3, 3.4 and the Aubin-Lions lemma([15]), we choose a sub-
sequence (εk)k∈N ⊂ (0, 1) such that ln(uε + 1) → ln(u + 1) in L2(Ω × (0, T )) as
ε = εk ↘ 0, k → ∞. Then we have ln(uε + 1) → ln(u + 1) a.e. in Ω × (0, T ) and



10 J. YAN, Y. LI EJDE-2020/122

(3.10) is deduced. By Lemma 3.2 and (3.10), we obtain (3.11). It follows from the
parabolic regularity theory [5, Theorem 3.1] and Lemma 3.2 that

‖vε‖L2((0,T );W 2,2(Ω)) + ‖vεt‖L2(Ω×(0,T )) ≤ C(T + 1).

Choosing an appropriate subsequence again and applying the Aubin-Lions lemma
[15], we obtain (3.12). Then (3.13) results from (3.12). Since∫ T

0

∫
Ω

(
|∇vε|p−2∇vε

)p′ ≤ ∫ T

0

∫
Ω

|∇vε|p
′(p−1)

=

∫ T

0

∫
Ω

|∇vε|p

≤ C
∫ T

0

∫
Ω

|∇vε|2 ≤ C(T + 1),

(3.15)

we obtain (3.14) by (3.13) and (3.15). �

Now we are ready to prove the main result of this section.

Proof of Theorem 1.1. For arbitrary non-negative ϕ ∈ C∞0 (Ω̄× [0, T )), multiplying
the first equation in (2.7) by ϕ, and integrating by parts, we have

−
∫

Ω

u0(x)ϕ(·, 0)−
∫ T

0

∫
Ω

uεϕt

=

∫ T

0

∫
Ω

uε ·∆ϕ+

∫ T

0

∫
Ω

uε(|∇vε|2 + ε)
p−2
2 ∇vε · ∇ϕ

(3.16)

for all ε ∈ (0, 1). Choosing T > 0 large enough such that ϕ ≡ 0 in Ω × (T,∞).
Since uε ⇀ u in L2(Ω× (0, T )) as ε = εk ↘ 0 by (3.11), we have∫ T

0

∫
Ω

uεϕt →
∫ T

0

∫
Ω

uϕt and

∫ T

0

∫
Ω

uε ·∆ϕ→
∫ T

0

∫
Ω

u ·∆ϕ (3.17)

as ε = εk ↘ 0. Moreover, because |∇vε|p−2∇vε ⇀ |∇v|p−2∇v in Lp
′
(Ω × (0, T ))

as ε = εk ↘ 0 by (3.14), we can choose a subsequence which is also written as vεk
such that |∇vε|p−2∇vε → |∇v|p−2∇v in L2(Ω × (0, T )) as ε = εk ↘ 0. Then we
have ∫ T

0

∫
Ω

uε(|∇vε|2 + ε)
p−2
2 ∇vε · ∇ϕ→

∫ T

0

∫
Ω

u|∇v|p−2∇v · ∇ϕ (3.18)

as ε = εk ↘ 0. Then (2.5) follows from (3.16)-(3.18).
Finally, for arbitrary non-negative ψ ∈ C∞0 (Ω̄× [0,∞)), multiplying the second

equation in (2.7) by ψ, and integrating by parts, we have∫
Ω

v0ψ(·, 0) +

∫ T

0

∫
Ω

vεψt =

∫ T

0

∫
Ω

∇vε · ∇ψ +

∫ T

0

∫
Ω

vεψ −
∫ T

0

∫
Ω

uεψ (3.19)

for all ε ∈ (0, 1). Thanks to (3.12), We can find that each of the terms in (3.19)
converges to its expected limits as ε = εk ↘ 0. So (2.6) results from (3.19). �



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 11

4. Boundedness

In this section, our goal is to prove Theorem 1.2. Firstly, by means of a Moser-
Alikakos iteration, we can achieve the following boundedness results.

Lemma 4.1. Let 1 < p < n/(n − 1). For all t > 0, there exists C > 0 such that
for any ε ∈ (0, 1),

‖uε(·, t)‖L∞(Ω) ≤ C. (4.1)

Proof. We multiply the first equation in (2.7) by uq−1
ε (for q > 1), and integrate by

parts to find that

1

q

d

dt

∫
Ω

uqε = −(q − 1)

∫
Ω

uq−2
ε |∇uε|2 + (q − 1)

∫
Ω

uq−1
ε

(
|∇vε|2 + ε

) p−2
2 ∇vε · ∇uε.

By the Cauchy-Schwarz inequality, we have

1

q

d

dt

∫
Ω

uqε +
2(q − 1)

q2

∫
Ω

|∇uq/2ε |2 ≤
q − 1

2

∫
Ω

uqε
(
|∇vε|2 + ε

)p−2 |∇vε|2

≤ q − 1

2

∫
Ω

uqε
(
|∇vε|2 + ε

)p−1

We can find a positive constant µ satisfying 2(p−1) < µ < n/(n−1). Using Lemma
3.1 and Hölder’s inequality, we have

1

q

d

dt

∫
Ω

uqε +
2(q − 1)

q2

∫
Ω

|∇uq/2ε |2

≤ q − 1

2

(∫
Ω

u
q
2

2µ
µ−2(p−2)

ε

)µ−2(p−1)
µ

(∫
Ω

(
|∇vε|2 + ε

)µ
2

) 2(p−1)
µ

≤ C · q − 1

2

(∫
Ω

u
q
2 ·

2µ
µ−2(p−1)

ε

)µ−2p−1
µ

[( ∫
Ω

|∇vε|µ
) 2(p−1)

µ

+ 1
]

≤ C · q − 1

2

(∫
Ω

u
q
2 ·

2µ
µ−2(p−1)

ε

)µ−2p−1
µ

.

(4.2)

By the Gagliardo-Nirenberg inequality, we can find a positive constant C > 0 such
that

‖uq/2ε ‖
L

2µ
µ−2(p−1) (Ω)

≤ C‖∇uq/2ε ‖aL2(Ω)‖u
q/2
ε ‖1−aL1(Ω) + C‖uq/2ε ‖L1(Ω), (4.3)

where

a =
1− µ−2(p−1)

2µ
1
2 + 1

n

.

Since 1 < p < n/(n− 1), we have a ∈ (0, 1). We apply inequality (4.3) to (4.2) and
use Young’s inequality to obtain(∫

Ω

u
q
2 ·

2µ
µ−2(p−1)

ε

)µ−2p−1
µ

= ‖uq/2ε ‖2
L

2µ
µ−2(p−1) (Ω)

≤ C‖∇uq/2ε ‖2aL2(Ω)‖u
q/2
ε ‖

2(1−a)
L1(Ω) + C‖uq/2ε ‖2L1(Ω)

≤ 2

Cq2

∫
Ω

|∇u
a
2
ε |2 + (1− a)[Caq2]

a
1−a

(∫
Ω

uq/2ε

)2

+ C
(∫

Ω

uq/2ε

)2

.



12 J. YAN, Y. LI EJDE-2020/122

Then we have

1

q

d

dt

∫
Ω

uqε +
q − 1

q2

∫
Ω

|∇uq/2ε |2 ≤ C(q − 1)q
2a

1−a

(∫
Ω

uq/2ε

)2

,

which is equivalent to

q

q − 1

d

dt

∫
Ω

uqε +

∫
Ω

|∇uq/2ε |2 ≤ Cq
2

1−a

(∫
Ω

uq/2ε

)2

.

By the Poincaré-Wirtinger inequality, we obtain∫
Ω

|∇uq/2ε |2 ≥ C
∫

Ω

(
uq/2ε − 1

|Ω|

∫
Ω

uq/2ε

)2

= C
(∫

Ω

uqε −
1

|Ω|

∣∣∣ ∫
Ω

uq/2ε

∣∣∣2),
which implies

q

q − 1

d

dt

∫
Ω

uqε + C

∫
Ω

uqε ≤ Cq
2

1−a

(∫
Ω

uq/2ε

)2

≤ Cq
2

1−a

(
sup
t≥0

∫
Ω

uq/2ε

)2

.

By the maximum principle, we have∫
Ω

uqε ≤ max
{∫

Ω

uq(x, 0), Cq
2

1−a

(
sup
t≥0

∫
Ω

uq/2ε

)2}
.

Then let qk := 2k, (k ∈ N), δk := C2
2k

1−a , and a constant K satisfying

K ≥ max
{

1, sup ‖uε(·, t)‖L1(Ω), ‖u(·, 0)‖L∞(Ω)

}
.

Using the Moser-Alikakos iteration [1] and assuming, without loss of generality,
that δk ≥ 1, we have∫

Ω

u2k

ε ≤ max
{
δk

(
sup

∫
Ω

u2k−1

ε

)2

,K2k
}
.

Taking K ≥ 1, it follows that∫
Ω

u2k

ε ≤ δkδ2
k−1δ

22

k−2 · · · δ2k−1

1 K2k ,

then we have ∫
Ω

u2k

ε ≤ C2k−12
2

1−a (−k+2k+1−1)K2k . (4.4)

Finally by taking the 1/2k power of both sides of (4.4) and by passing to the limit
as k →∞ we obtain

sup
t≥0
‖uε(·, t)‖L∞(Ω) ≤ C22 2

1−aK.

�

Next, to obtain the limit function u, we need a regularity estimate for ∂tuε.

Lemma 4.2. Let 1 < p < n/(n − 1). There exists C > 0 such that for any
ε ∈ (0, 1),

‖∂tuε(·, t)‖(W 2,2
0 (Ω))∗ ≤ C for all t > 0. (4.5)

In particular,

‖uε(·, t)− uε(·, s)‖(W 2,2
0 (Ω))∗ ≤ C|t− s| for all t ≥ 0, s ≥ 0. (4.6)



EJDE-2020/122 CHEMOTAXIS SYSTEM WITH GRADIENT DEPENDENT SENSITIVITY 13

Proof. We fix ψ ∈ C∞0 (Ω) and multiply the first equation in (2.7) by ψ. Integrating
by parts we find that∫

Ω

∂tuε · ψ =

∫
Ω

uε ·∆ψ +

∫
Ω

uε(|∇vε|2 + ε)
p−2
2 ∇vε · ∇ψ.

Then by Lemmas 3.1 and 4.1, we obtain the inequality∣∣ ∫
Ω

∂tuε · ψ
∣∣ ≤ ‖uε‖L∞(Ω)

∫
Ω

|∆ψ|+ ‖uε‖L∞(Ω)

∫
Ω

∣∣(|∇vε|2 + ε)
p−2
2 ∇vε · ∇ψ

∣∣
≤ C

∫
Ω

|∆ψ|+ C

∫
Ω

∣∣(|∇vε|p−1 + 1) · ∇ψ
∣∣

≤ C
∫

Ω

|∆ψ|+ C

∫
Ω

|∇ψ|.

This readily establishes (4.5) and thus (4.6). �

Lemma 4.3. Let u be the function asserted in Lemma 3.5. Then

uε
∗
⇀ u in L∞(Ω× (0,∞)), (4.7)

uε → u in C∞loc

(
[0,∞); (W 2,2

0 (Ω))∗
)
, (4.8)

as ε = εk ↘ 0.

Proof. By (4.1) and choosing a subsequence, we can deduce (4.7). Since L∞(Ω) ↪→
(W 2,2

0 (Ω))∗ is compact, by Lemma 4.3 and Aubin-Lions lemma([15]), we can obtain
(4.8) after extracting of an adequate subsequence. �

Finally, we give the proof of Theorem 1.2 by referring to the method in [17].

Proof of Theorem 1.2. From (4.1), it follows that there exists a null set N ⊂ [0,∞)
such that for all t ∈ [0,∞) \ N , we have u(·, t) ∈ L∞(Ω). As [0,∞) \ N is dense
in [0,∞), for an arbitrary t0 ∈ [0,∞) we can find (tk)k∈N ⊂ [0,∞) \ N such
that tk → t0 as k → ∞, and extracting a subsequence if necessary we can also

achieve that u(·, tk)
∗
⇀ ũ in L∞(Ω) as k → ∞ with some ũ ∈ L∞(Ω) satisfying

‖ũ‖L∞(Ω) ≤ C. Since (4.8) asserts that moreover u(·, tk) → u(·, t0) in (W 2,2
0 (Ω))∗

as k →∞, this allows us to identify ũ = u(·, t0) and to conclude that u(·, t) ∈ L∞(Ω)
for all t ∈ [0,∞), with ‖u(·, t)‖L∞(Ω) ≤ C for all t ≥ 0. �

Acknowledgments. This research was supported in part by the China Scholarship
Council (No. 201906090124), by the National Natural Science Foundation of China
(Nos. 11671079, 11701290, 11601127 and 11171063), and by the National Natural
Science Foundation of Jiangsu Provience (No. BK20170896).

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Jianlu Yan
Institute for Applied Mathematics, School of Mathematics, Southeast University, Nan-
jing 211189, China

Email address: 230159430@seu.edu.cn

Yuxiang Li

Institute for Applied Mathematics, School of Mathematics, Southeast University, Nan-

jing 211189, China
Email address: lieyx@seu.edu.cn


	1. Introduction
	2. A weak solution concept and approximate problems
	3. Existence of the weak solutions
	4. Boundedness
	Acknowledgments

	References