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Article

Global Weak Solution in a p-Laplacian Attraction–Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production

School of Mathematics and Statistics, Linyi University, Linyi 276005, China
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Author to whom correspondence should be addressed.
Axioms 2025, 14(8), 642; https://doi.org/10.3390/axioms14080642
Submission received: 17 July 2025 / Revised: 14 August 2025 / Accepted: 15 August 2025 / Published: 18 August 2025
(This article belongs to the Special Issue Differential Equations and Its Application)

Abstract

This paper is devoted to a p-Laplacian attraction–repulsion chemotaxis system with nonlinear sensitivity and signal production. We obtain the global existence and boundedness of weak solutions by means of energy estimates, the Aubin–Lions lemma, and parabolic and elliptic regularity estimates.
MSC:
35K55; 35K65; 35A07; 35B35

1. Introduction

Below is an introduction to a p-Laplacian chemotaxis system with nonlinear sensitivity:
u t = · ( | u | p 2 u ) χ · ( u α v ) + ξ · ( u β w ) + f ( u ) , ( x , t ) Ω × ( 0 , ) , v t = Δ v σ v + g 1 ( u ) , ( x , t ) Ω × ( 0 , ) , 0 = Δ w δ w + g 2 ( u ) , ( x , t ) Ω × ( 0 , ) , u ν = v ν = w ν = 0 , ( x , t ) Ω × ( 0 , ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x Ω .
Let Ω R n be a smoothly bounded domain with dimension n 2 . Consider positive parameters χ , ξ , α , β , σ , δ and an exponent p 2 . Within this framework, the unknown functions represent the following: u is cell density, v is chemoattractant concentration, and w is chemorepellant concentration. In classical linear diffusion ( p = 2 ) , cell motility is assumed to follow Fick’s law (where the diffusion rate is independent of density). However, in actual biological tissues, cell–cell attraction or repulsion leads to density-dependent diffusion. The p-Laplacian term · ( | u | p 2 u ) can describe this dependency: for p > 2 , it describes suppressed cell motility (slow diffusion), while for 1 < p < 2 , it describes enhanced cell motility (fast diffusion).
The smooth logistic-type source f : R R satisfies for all s > 0
f ( s ) κ μ s m and f ( 0 ) 0
with κ R , μ > 0 and m > 1 . The signal production g i ( i = 1 , 2 ) satisfies for all s > 0
g 1 ( s ) = η s k 1 , g 2 ( s ) = γ s k 2 ,
where η , γ , k 1 , k 2 > 0 .
Considering the effect of attraction without repulsion, problem (1) becomes the following Keller–Segel chemotaxis system:
u t = u χ · ( u v ) , ( x , t ) Ω × ( 0 , ) , v t = v v + g ( u ) , ( x , t ) Ω × ( 0 , ) , u ν = v ν = 0 , ( x , t ) Ω × ( 0 , ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω .
In scenarios where the signal production function g exhibits linearity, specifically satisfying g ( u ) = u , it was shown that the solutions of (4) are global bounded when n = 1 , n = 2 with Ω u 0 d x < 4 π χ , or n 3 with small initial conditions (see [1,2,3,4]), whereas when n = 2 with Ω u 0 d x > 4 π χ , or n 3 , finite-time blow-up may occur for a large class of initial data [5,6]. When the signal production g ( u ) is nonlinear (i.e., 0 < g ( u ) u k ), the boundedness of (4) was studied in [7] if k ( 0 , 2 n ) with n 2 . Considering the solution of (4) with nonlinear production g ( u ) = u ( u + 1 ) k 1 under the condition f ( u ) = u μ u m , it was proved in [8,9] that the weak solution of (4) is bounded globally if 2 ( n + 4 ) n + 6 < m 2 , 0 < k < ( n + 6 ) ( m 1 ) 2 ( n + 2 ) , and n = 2 , 3 . Recently, Zhuang et al. [10] obtained the globally bounded classical solution of (4)= under k < m 1 or k = m 1 with μ > 0 sufficiently large.
Now, recall a more general system
u t = · ( D ( u ) u ) χ · ( u v ) + ξ · ( u w ) + f ( u ) , ( x , t ) Ω × ( 0 , ) , τ 1 v t = v σ v + g 1 ( u ) , ( x , t ) Ω × ( 0 , ) , τ 2 w t = w δ w + g 2 ( u ) , ( x , t ) Ω × ( 0 , ) , u ν = v ν = w ν = 0 , ( x , t ) Ω × ( 0 , ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x Ω ,
where τ i = { 0 , 1 } ( i = 1 , 2 ) . For the case f ( u ) 0 , g 1 ( u ) = η u , g 2 ( u ) = γ u . If τ 1 = 1 , τ 2 = 0 , and D ( u ) satisfies D ( u ) = D 0 u θ , there exists a global weak solution under the case θ > 1 2 n [11]. Later, ref. [12] optimized the condition of θ into θ > 1 4 n + 2 but added ξ γ χ η 0 . If τ 1 = τ 2 = 0 , and considering p-Laplacian diffusion (i.e., D ( u ) = | u | p 2 ), it was shown in [13] that for the system (5) there exists a global bounded weak solution under case (i) ξ γ χ η 0 or case (ii) ξ γ χ η > 0 with p > 3 n n + 1 or 1 < p 3 n n + 1 , with u 0 L ( 3 p ) n p being small. For the case f ( u ) = κ μ u m , g 1 ( u ) = η u k 1 , g 2 ( u ) = γ u k 2 . If τ 1 = 1 and τ 2 = 0 , it was shown that when (i) m > max { 2 k 1 , 2 k 1 n 2 + n + 1 p 1 } or case (ii) k 2 > max { 2 k 1 1 , 2 k 1 n 2 + n + 2 p p 1 } with m > 2 , the problem (5) possesses a global bounded weak solution [14]. If τ 1 = τ 2 = 1 , Jia [15] proved that when max { k 1 , k 2 } < m 1 or max { k 1 , k 2 } = m 1 with large μ > 0 , there exists a global bounded weak solution. More results on p-Laplacian chemotaxis models can be found in [16,17,18,19,20,21].
We now formulate the principal result.
Theorem 1. 
Consider a smoothly bounded domain Ω R n with n 2 and an exponent p 2 . Under the assumptions that (2) and (3) hold for parameters k 1 > 0 , 0 < k 2 1 , and m > 1 , the following conclusion holds.
For arbitrary non-negative initial data u 0 , v 0 , w 0 W 1 , ( Ω ) , the problem (1) possesses a globally bounded weak solution ( u , v , w ) in the function space L ( Ω ) × W 1 , ( Ω ) × W 1 , ( Ω ) , provided either
(i)
m > max 2 k 1 , p α p 1 1
(ii)
k 2 > max 2 k 1 β , p α p 1 β 1 with m > max { 2 , 2 α , 2 β } .
Remark 1.  
When α = β = 1 for (1.1), we optimized the assumption (i) m > max { 2 k 1 , 2 k 1 n 2 + n + 1 p 1 } or (ii) k 2 > max { 2 k 1 1 , 2 k 1 n 2 + n + 2 p p 1 } with m > 2 of reference [14] into (i) m > max { 2 k 1 , 1 p 1 } or (ii) k 2 > max { 2 k 1 1 , 2 p p 1 } with m > 2 .

2. Preliminaries

First, we define the weak solution of (1).
Definition 1.  
Let
0 u L l o c 2 ( [ 0 , T ) ; L 2 ( Ω ) ) ,
0 v , w L l o c 2 ( [ 0 , T ) ; W 1 , 2 ( Ω ) ) .
When the following equation holds, ( u , v , w ) is said to be a weak solution of (1):
0 T Ω u φ t d x d t Ω u 0 ( x ) φ ( x , 0 ) d x = 0 T Ω | u | p 2 u · φ d x d t + χ 0 T Ω u α v · φ d x d t ξ 0 T Ω u β w · φ d x d t + 0 T Ω f ( u ) φ d x d t ,
0 T Ω v φ t d x d t Ω v 0 ( x ) φ ( x , 0 ) d x = 0 T Ω v · φ d x d t σ 0 T Ω v φ d x d t + 0 T Ω g 1 ( u ) φ d x d t ,
and
0 = 0 T Ω w · φ d x d t δ 0 T Ω w φ d x d t + 0 T Ω g 2 ( u ) φ d x d t
for all φ C 0 ( Ω ¯ × [ 0 , T ) ) .
Consider the regularity problem below:
u ε t = · ( ( | u ε | 2 + ε ) p 2 2 u ε ) χ · ( u ε α v ε ) + ξ · ( u ε β w ε ) + f ( u ε ) , v ε t = v ε σ v ε + g 1 ( u ε ) , ( x , t ) Ω × ( 0 , ) , 0 = w ε δ w ε + g 2 ( u ε ) , ( x , t ) Ω × ( 0 , ) , u ε ν = v ε ν = w ε ν = 0 , ( x , t ) Ω × ( 0 , ) , u ε ( x , 0 ) = u 0 ( x ) , v ε ( x , 0 ) = v 0 ( x ) , w ε ( x , 0 ) = w 0 ( x ) , x Ω
for each ε ( 0 , 1 ) . Employing a fixed-point theorem analogous to those in [18,22], the local classical solvability of system (6) can be established.
Lemma 1.  
Suppose p 2 and Ω R n ( n 2 ) is a smooth bounded domain. Given initial data u 0 , v 0 , w 0 W 1 , ( Ω ) , for each ε ( 0 , 1 ) there corresponds a maximal existence time T max , ε ( 0 , ] and a classical solution ( u ε , v ε , w ε ) defined locally in time. This solution lies in the space C ( Ω ¯ × [ 0 , T max , ε ) ) C 2 , 1 ( Ω ¯ × [ 0 , T max , ε ) ) and satisfies Equation (6). When T max , ε < , it holds that
lim t T max , ε u ε ( · , t ) L ( Ω ) .
Lemma 2.  
Assume (2) holds; M > 0 and M : = M ( u 0 , | Ω | ) such that
Ω u ε ( x , t ) d x M f o r t ( 0 , T m a x , ε ) .
Proof. 
Integrating the first functions of (6), we have by using (2) that
d d t Ω u ε d x κ | Ω | μ | Ω | 1 m ( Ω u ε d x ) m , t ( 0 , T m a x , ε )
which implies (7). □
Lemma 3.  
For any ε 1 , ε 2 > 0 and q > 0 , there exist constants C 1 , C 2 > 0 such that the solution z of the problem
z + δ z = γ u k 2 i n Ω , z ν = 0 o n Ω ,
satisfies
Ω z q + 1 d x ε 1 Ω u k 2 ( q + 1 ) d x + C 1 ε 2 Ω u q + 1 d x + C 2
for u L 1 ( Ω ) and 0 < k 2 1 .
Proof. 
The proof closely resembles that of Lemma 2.2 in [23]; therefore, we will omit it. □
Next, we present the Gagliardo–Nirenberg inequality [24,25].
Lemma 4. 
Let s > 0 , 0 < r p , and q 1 . Then there exists a constant C G N > 0 such that
ϕ L p ( Ω ) C G N ( ϕ L q ( Ω ) λ ϕ L r ( Ω ) 1 λ + ϕ L s ( Ω ) ) f o r a n y ϕ W 1 , q ( Ω ) L r ( Ω ) ,
where
λ = n r n p 1 n q + n r ( 0 , 1 ) .

3. Regularity Estimates for (6)

We study the global boundedness of the regularized problem (6).
Lemma 5.  
Given parameters k 1 > 0 , 0 < k 2 1 , and m > 1 satisfying conditions (2) and (3), the following gradient estimate holds whenever either m > 2 k 1 or k 2 > 2 k 1 β with m > max { 2 , 2 α , 2 β } : there exists a positive constant C 1 such that for any ε ( 0 , 1 ) ,
Ω | v ε | 2 d x C 1 f o r a l l t ( 0 , T max ) .
Proof. 
Case (i): m > 2 k 1 .Multiplying ( 6 ) 2 by 2 v ε and integrating by parts, we obtain
d d t Ω | v ε | 2 d x = 2 Ω | v ε | 2 d x 2 σ Ω | v ε | 2 d x 2 η Ω u ε k 1 v ε d x 2 Ω | v ε | 2 d x 2 σ Ω | v ε | 2 d x + 2 Ω | v ε | 2 d x + η 2 2 Ω u ε 2 k 1 d x = 2 σ Ω | v ε | 2 d x + η 2 2 Ω u ε 2 k 1 d x .
Together with (14) and the first equation of (6), ∃ C 2 > 0 such that
d d t ( Ω u ε d x + Ω | v ε | 2 d x ) κ | Ω | μ Ω u ε m d x 2 σ Ω | v ε | 2 d x + η 2 2 Ω u ε 2 k 1 d x 2 σ Ω | v ε | 2 d x + C 2 ,
which used m > 2 k 1 . Combining Lemma 2 and (15), we have
d d t ( Ω u ε d x + Ω | v ε | 2 ) d x + 2 σ ( Ω u ε d x + Ω | v ε | 2 d x ) 2 σ Ω u ε d x + C 2 C 3 .
Case (ii): k 2 > 2 k 1 β with m > max { 2 , 2 α , 2 β } . Multiplying ( 6 ) 1 by 1 + ln u ε , we get
d d t Ω u ε ln u ε d x ( p p 1 ) p Ω | u ε p 1 p | p d x + χ Ω u ε α 1 u ε · v ε d x ξ Ω u ε β 1 u ε · w ε d x + κ | Ω | + κ Ω ln u ε d x μ Ω u ε m d x μ Ω u ε m ln u ε d x χ Ω u ε α 1 u ε · v ε d x ξ Ω u ε β 1 u ε · w ε d x μ 2 Ω u ε m d x + C 4 ,
which used u ε ln u ε e 1 and Ω ln u ε d x Ω u ε d x .
Multiplying ( 6 ) 2 by v ε and integrating by parts, we have
1 2 d d t Ω | v ε | 2 d x = Ω | v ε | 2 d x σ Ω | v ε | 2 d x η Ω u ε k 1 v ε d x Ω | v ε | 2 d x σ Ω | v ε | 2 d x + η 2 Ω u ε 2 k 1 d x + 1 4 Ω | v ε | 2 d x 3 4 Ω | v ε | 2 d x σ Ω | v ε | 2 d x + η 2 Ω u ε 2 k 1 d x .
We estimate that
χ Ω u ε α 1 u ε · v ε d x = χ α Ω u ε α v ε d x χ 2 α 2 Ω u ε 2 α d x + 1 4 Ω | v ε | 2 d x ,
as well as
ξ Ω u ε β 1 u ε · w ε d x = ξ β Ω u ε β w ε d x = δ ξ β Ω u ε β w ε d x γ ξ β Ω u ε k 2 + β d x Ω u ε 2 β d x + δ 2 ξ 2 4 β 2 Ω w ε 2 d x γ ξ β Ω u ε k 2 + β d x Ω u ε 2 β d x + σ Ω u 2 d x γ ξ β Ω u ε k 2 + β d x + C 5 ,
which used Lemma 3, for all t ( 0 , T max ) . Since 2 σ Ω u ε ln u ε d x 2 σ Ω u ε 2 d x , and combining (17)–(20), we have
d d t Ω ( u ε ln u ε + 1 2 | v ε | 2 ) d x + 2 σ Ω u ε ln u ε d x + σ Ω | v ε | 2 d x 3 σ Ω u ε 2 d x + χ 2 α 2 Ω u ε 2 α d x + Ω u ε 2 β d x + η 2 Ω u ε 2 k 1 d x γ ξ β Ω u ε k 2 + β d x μ 2 Ω u ε m d x + C 6 .
It follows from k 2 > 2 k 1 β , m > max { 2 , 2 α , 2 β } and Young’s inequality that
3 σ Ω u ε 2 d x + χ 2 α 2 Ω u ε 2 α d x + Ω u ε 2 β d x + η 2 Ω u ε 2 k 1 d x γ ξ β Ω u ε k 2 + β d x μ 2 Ω u ε m d x C 7 .
Let y ( t ) = Ω ( u ε ln u ε + 1 2 | v ε | 2 ) d x ; together with (21) and (22), this indicates
y ( t ) + 2 σ y ( t ) C 8 .
So we have
Ω u ε ln u ε d x + 1 2 Ω | v ε | 2 d x C 9 .
Then we derive
1 2 Ω | v ε | 2 d x Ω u ε ln u ε d x + C 9 e 1 | Ω | + C 9
for all t ( 0 , T max ) and the desired results are proved. □
Lemma 6.  
Let p 2 and assume that (2) and (3) hold with m > 1 , k 1 > 0 and 0 < k 2 1 . If m > max { 2 k 1 , p α p 1 1 } or k 2 > max { 2 k 1 β , p α p 1 β 1 } with m > max { 2 , 2 α , 2 β } , then for any q > 1 , one can find a constant C 10 > 0 such that for any t ( 0 , T max ) and ε ( 0 , 1 )
Ω u ε q ( x , t ) d x C 10 .
Proof. 
Multiply ( 6 ) 1 by q u ε q 1 and integrate over Ω ; thus
d d t Ω u ε q d x + q ( q 1 ) Ω u ε q 2 ( | u ε | 2 + ε ) p 2 2 | u ε | 2 d x χ q ( q 1 ) Ω u ε q + α 2 u ε · v ε d x + ξ q ( q 1 ) q + β 1 Ω u ε q + β 1 w ε d x + κ q Ω u ε q 1 d x μ q Ω u ε q + m 1 d x
for all t ( 0 , T max ) . Note that for p 2 ,
( | u ε | 2 + ε ) p 2 2 | u ε | 2 | u ε | p .
Together with (27) and (28), it follows that for p 2
d d t Ω u ε q d x + q ( q 1 ) Ω u ε q 2 | u ε | p d x χ q ( q 1 ) Ω u ε q + α 2 u ε · v ε d x + ξ q ( q 1 ) q + β 1 Ω u ε q + β 1 w ε d x + κ q Ω u ε q 1 d x μ q Ω u ε q + m 1 d x χ q ( q 1 ) Ω u ε q + α 2 u ε · v ε d x + ξ q ( q 1 ) q + β 1 Ω u ε q + β 1 w ε d x μ q 2 Ω u ε q + m 1 d x + C 11 .
This is also due to
χ q ( q 1 ) Ω u ε q + α 2 u ε · v ε d x q ( q 1 ) 2 Ω u ε q 2 | u ε | p d x + C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x .
Quoting Lemma 3, we obtain
ξ q ( q 1 ) q + β 1 Ω u ε q + β 1 w ε d x = ξ δ q ( q 1 ) q + β 1 Ω u ε q + β 1 w ε d x ξ γ q ( q 1 ) q + β 1 Ω u ε q + β + k 2 1 d x ξ γ q ( q 1 ) 4 ( q + β 1 ) Ω u ε q + β + k 2 1 d x + C 13 Ω w ε q + β + k 2 1 k 2 d x ξ γ q ( q 1 ) q + β 1 Ω u ε q + β + k 2 1 d x = C 13 Ω w ε q + β + k 2 1 k 2 d x 3 ξ γ q ( q 1 ) 4 ( q + β 1 ) Ω u ε q + β + k 2 1 d x ξ γ q ( q 1 ) 2 ( q + β 1 ) Ω u ε q + β + k 2 1 d x + C 14 .
Together with (29), (30), and (31), we have
d d t Ω u ε q d x + q ( q 1 ) 2 Ω u ε q 2 | u ε | p d x C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x ξ γ q ( q 1 ) 2 ( q + β 1 ) Ω u ε q + β + k 2 1 d x μ q 2 Ω u ε q + m 1 d x + C 11 + C 14 .
based on the fact that
q ( q 1 ) 2 Ω u ε q 2 | u ε | p d x = q ( q 1 ) p p 2 ( p + 1 2 ) p u ε p + q 2 p L p ( Ω ) p .
From (32) and (33), we conclude that
d d t Ω u ε q d x + q ( q 1 ) p p 2 ( p + 1 2 ) p u ε p + q 2 p L p ( Ω ) p C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x ξ γ q ( q 1 ) 2 ( q + β 1 ) Ω u ε q + β + k 2 1 d x μ q 2 Ω u ε q + m 1 d x + C 11 + C 14 .
Analogous to Lemma 3.11 in [26], a positive constant C 15 may be selected satisfying the condition that for all exponents q 2 ,
d d t Ω | v ε | 2 q d x + 2 ( q 1 ) q Ω | | v ε | q | 2 d x C 15 Ω u ε 2 k 1 | v ε | 2 q 2 d x + C 15 .
From the combination of (34) and (35), it follows that
d d t ( Ω u ε q d x + Ω | ε | 2 q d x ) + q ( q 1 ) p p 2 ( p + 1 2 ) p u ε p + q 2 p L p ( Ω ) p + 2 ( q 1 ) q Ω | | v ε | q | 2 d x C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x + C 15 Ω u ε 2 k 1 | v ε | 2 q 2 d x ξ γ q ( q 1 ) 2 ( q + β 1 ) Ω u ε q + β + k 2 1 d x μ q 2 Ω u ε q + m 1 d x + C 16 .
Case (i): m > max { 2 k 1 , p α p 1 1 } . Let q : = ( p 1 ) ( q + m 1 ) ( p 1 ) ( m + 1 ) p α and take q m + 3 2 p α p 1 and m > p α p 1 1 ; we have q 2 . Using Hölder’s inequality, we obtain
C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x C 12 ( Ω u ε q + m 1 d x ) ( p 1 ) ( q 2 ) + p α ( p 1 ) ( q + m 1 ) ( Ω | v ε | p q p 1 d x ) 1 q
for all t ( 0 , T max ) . Similarly,
C 15 Ω u ε 2 k 1 | v ε | 2 q 2 d x C 15 ( Ω u ε q + m 1 d x ) 2 k 1 q + m 1 ( Ω | v ε | ( 2 q 2 ) λ d x ) 1 λ ,
where λ = q + m 1 q + m 2 k 1 1 > 1 . We deduce from the G-N inequality that
( Ω | v ε | p q p 1 d x ) 1 q = | v ε | q L p p 1 ( Ω ) p ( p 1 ) q ( C 17 | v ε | q L 2 ( Ω ) θ 1 | v ε | q L 2 q ( Ω ) 1 θ 1 + C 17 | v ε | q L 2 q ( Ω ) ) p ( p 1 ) q
for all t ( 0 , T max ) , where θ 1 = q 2 p 1 p q 2 + 1 n 1 2 . Since p 2 , selecting q > 2 [ ( p 1 ) ( m + 1 ) p α ] p m + 1 , we have θ 1 ( 0 , 1 ) . We invoke (39) and Lemma 5 with m > 2 k 1 such that
( Ω | v ε | p q p 1 d x ) 1 q C 18 ( Ω | | v ε | q | 2 d x ) p θ 1 2 ( p 1 ) q + C 18
with C 18 > 0 . We estimate again
( Ω | v ε | 2 ( q 1 ) λ d x ) 1 λ = | v ε | q L 2 λ ( q 1 ) q ( Ω ) 2 ( q 1 ) q ( C 19 | v ε | q L 2 ( Ω ) θ 2 | v ε | q L 2 q ( Ω ) 1 θ 2 + C 19 | v ε | q L 2 q ( Ω ) ) 2 ( q 1 ) q
with C 19 > 0 , where θ 2 = q 2 q 2 λ ( q 1 ) q 2 + 1 n 1 2 . Due to m > max { 1 , 2 k 1 } , p 2 and taking q > max { m + 3 2 p α p 1 , ( n 2 ) p α 2 ( p 1 ) n ( m 2 k 1 ) 2 + 2 n 2 } , we get θ 2 ( 0 , 1 ) . Then combining with Lemma 5, we have
( Ω | v ε | 2 ( q 1 ) λ d x ) 1 λ C 20 ( Ω | | v ε | q | 2 d x ) ( q 1 ) θ 2 q + C 20 .
Together with (37), (38), (40), and (42), we obtain
C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x + C 15 Ω u ε 2 k 1 | v ε | 2 q 2 d x C 12 C 18 ( Ω u ε q + m 1 d x ) ( p 1 ) ( q 2 ) + p α ( p 1 ) ( q + m 1 ) ( Ω | | v ε | q | 2 d x ) p θ 1 2 ( p 1 ) q + C 12 C 18 ( Ω u ε q + m 1 d x ) ( p 1 ) ( q 2 ) + p α ( p 1 ) ( q + m 1 ) + C 15 C 20 ( Ω u ε q + m 1 d x ) 2 k 1 q + m 1 ( Ω | | v ε | q | 2 d x ) ( q 1 ) θ 2 q + C 15 C 20 ( Ω u ε q + m 1 d x ) 2 k 1 q + m 1
for p 2 and q > max { m + 3 2 p α p 1 , ( n 2 ) p α 2 ( p 1 ) n ( m 2 k 1 ) 2 + 2 n 2 , 2 [ ( p 1 ) ( m + 1 ) p α ] p m + 1 } .
Next we prove ( p 1 ) ( q 1 ) + 1 ( p 1 ) ( q + m 1 ) + p θ 1 2 ( p 1 ) q < 1 . This is an obvious fact that holds true; otherwise, if ( p 1 ) ( q 1 ) + 1 ( p 1 ) ( q + m 1 ) + p θ 1 2 ( p 1 ) q 1 , this leads to
θ 1 2 ( p 1 ) q p · ( p 1 ) ( m + 1 ) p α ( p 1 ) ( q + m 1 ) . = 2 2 p 1 ,
which used q = ( p 1 ) ( q + m 1 ) ( p 1 ) ( m + 1 ) p α and p 2 . Thus
( p 1 ) ( q 1 ) + 1 ( p 1 ) ( q + m 1 ) + p θ 1 2 ( p 1 ) q < 1 .
Subsequently, we prove 2 k 1 q + m 1 + ( q 1 ) θ 2 q < 1 . Let
2 k 1 q + m 1 + ( q 1 ) θ 2 q = h 1 ( p , q ) h 2 ( p , q ) ,
where
h 1 ( p , q ) = 2 k 1 [ ( p 1 ) ( q + m 1 ) + 2 2 n n ( ( p 1 ) ( m + 1 ) p α ) ] ( q + m 1 ) ( ( p 1 ) ( q 2 ) + p α ) + ( q + m 2 k 1 1 ) ( ( p 1 ) ( q 2 ) + p α ) ,
and
h 2 ( p , q ) = ( q + m 1 ) [ ( p 1 ) ( q + m 1 ) + 2 2 n n ( ( p 1 ) ( m + 1 ) p α ) ] .
Obviously there is
h 2 ( p , q ) h 1 ( p , q ) = ( q + m 2 k 1 1 ) [ ( p 1 ) ( q + m 1 ) + 2 2 n n ( ( p 1 ) ( m + 1 ) p α ) ] + ( q + m 1 ) ( ( p 1 ) ( q 2 ) + p α ) .
Due to q 2 , we have ( p 1 ) ( q + m 1 ) 2 [ ( p 1 ) ( m + 1 ) p α ] . Combining m > max { 2 k 1 , p α p 1 1 } , p 2 , and q > 1 , we have
h 2 ( p , q ) h 1 ( p , q ) 2 n ( q + m 2 k 1 1 ) ( ( p 1 ) ( m + 1 ) p α ) + ( q + m 1 ) ( ( p 1 ) ( q 2 ) + p α ) > 0 .
This means
2 k 1 q + m 1 + ( q 1 ) θ 2 q < 1 .
Together with (43), (45), and (51), we deduce
C 12 Ω u ε ( p 1 ) ( q 2 ) + p α p 1 | v ε | p p 1 d x + C 15 Ω u ε 2 k 1 | v ε | 2 q 2 d x ϵ ( Ω u ε q + m 1 d x + Ω | | v ε | q | 2 d x ) + C 21
for any ϵ > 0 , p 2 , and q > q 0 : = max { 1 , m + 3 2 p α p 1 , n 2 2 ( p 1 ) n ( m 2 k 1 ) 2 + 1 , 2 [ ( p 1 ) m 1 ] p m + 1 } . There exists C 22 > 0 such that
Ω | v ε | 2 q d x = | v ε | q L 2 ( Ω ) 2 C 22 | v ε | q L 2 ( Ω ) 2 θ 3 | v ε | q L 2 q ( Ω ) 2 ( 1 θ 3 ) + C 22 | v ε | q L 2 q ( Ω ) ,
where θ 3 = q 2 1 2 q 2 + 1 n 1 2 ( 0 , 1 ) . From (53) and Lemma 5, we conclude that
Ω | v ε | 2 q d x C 23 | v ε | q L 2 ( Ω ) 2 + C 23 .
Due to
Ω u ε q d x C 24 Ω u ε q + m 1 d x + C 24 .
Combining (36), (52), (54), and (55), we have
d d t ( Ω u ε q d x + Ω | ε | 2 q d x ) + Ω u ε q d x + Ω | ε | 2 q d x C 25
for all q > q 0 > 1 . By Young’s inequality again, we obtain for q > 1 that
Ω u ε q d x C 26 .
Case (ii): k 2 > max { 2 k 1 β , p α p 1 β 1 } with m > max { 2 , 2 α , 2 β } . Following the approach used for Case (i), we reach corresponding outcomes. □
Lemma 7.  
Under the assumption of Lemma 6, there exists C 27 > 0 such that
v ε ( · , t ) W 1 , ( Ω ) C 27 , w ε ( · , t ) W 1 , ( Ω ) C 27 f o r a n y ε ( 0 , 1 ) ,
and
u ε ( · , t ) L ( Ω ) C 27
hold for all t ( 0 , T max ) .
Proof. 
In analogy with Lemma 3.3 of [14], we skip the proof. □

4. Proof of Theorem 1

The existence of global weak solutions to system (1) can be shown by taking limits of ( u ε , v ε , w ε ) . We start this analysis with a fundamental lemma from [13,14].
Lemma 8.  
Under the assumption of Lemma 6, there exist constants C 1 , C 2 > 0 such that for all t > 0
0 t Ω | u ε ( · , t ) | p d x d t C 1 ,
and for all T > 0
t u ε L 1 ( ( 0 , T ) ; ( W 0 2 , p ( Ω ) ) ) C 2 .
Proof. 
Lemma 6’s proof establishes that 0 t Ω u ε q 2 | u ε | p d x d t C 3 . Let q = 2 ; we deduce (60). In analogy with Lemma 4.1 of [14], we deduce (61). □
Lemma 9.  
Let ( u ε , v ε , w ε ) be the classical solution of (6). There exists a non-negative funtion ( u , v , w ) such that as ε 0 ,
u ε u i n L l o c p ( Q T ) ,
u ε u a . e . i n Q T ,
u ε u i n L ( Q T ) ,
u ε u i n L l o c p ( Q T ) ,
( | u ε | 2 + ε ) p 2 2 u ε | u | p 2 u i n L l o c p p 1 ( Q T ) ,
v ε v , u n i f o r m l y ,
v ε v i n W r 2 , 1 ( Q T ) , f o r a n y r > 1 ,
w ε w , i n L l o c 2 ( ( 0 , T ) , W 1 , 2 ( Ω ) ) ,
w ε w , i n L ( Ω × ( 0 , T ) ) ,
w ε w , i n L ( Ω × ( 0 , T ) ) ,
for any T > 0 .
Proof. 
Through Lemmas 7 and 8 and together with the Aubin–Lions lemma, we obtain a non-negative function u that satisfies, as ε 0 ,
u ε u in L loc p ( Q T ) and almost everywhere in Q T .
Combining Lemma 7 and (72), we deduce (60). Together with (60) and (72), we obtain (65).
Collecting (64), (72) and Lemma 7, we obtain (66) by the method of Lemma 4.5 in [19]. We may further conclude from Lemma 7 in conjunction with (3) that for any r > 1 ,
sup t ( 1 , ) t 1 t g 1 ( u ε ) L r r d s C 4 .
Combining this estimate with Lemma 2.4 in [27] yields sup t ( 1 , ) t 1 t ( v ε W 2 , r r + v ε t L r r ) d s C 5 , which implies
v ε W r 2 , 1 ( Ω × ( t 1 , t ) ) C 6 .
with C 6 > 0 . In view of (74), we obtain (68). Since W r 2 , 1 ( Q T ) C 2 n + 2 r , 1 n + 2 2 r ( Q T ) for r > n + 2 2 , we obtain (67). In addition, the estimate (3.46) implies (69)–(71). □
Proof of Theorem 1. 
Combining Lemma 7 and Lemma 1, we know that the system (6) has a globally bounded classical solution ( u ε , v ε , w ε ) . Then by the convergence proof of Lemma 9 and the definition of the weak solution in Definition 1, we know that the system (1) has a globally bounded weak solution ( u , v , w ) , which is the limit of ( u ε , v ε , w ε ) as ε 0 . □
Remark 2.  
The existence of global weak solutions rigorously proves that, under the given parameters, cell density remains finite at all times. This aligns with all empirically observed biological systems and provides a strict guarantee of solvability for complex biological chemotactic processes. Model ( 1.1 ) is a new model that has never been considered in the aforementioned papers. The conclusion obtained in this paper will provide some theoretical support for the theory of partial differential equations, especially for non-Newtonian fluids.

Author Contributions

Conceptualization, Z.J.; methodology, Z.J.; formal analysis, H.R.; investigation, H.R. All authors have read and agreed to the published version of the manuscript.

Funding

Project supported by the National Natural Science Foundation of China (Grant No.12301251, 12271232) and the Scientific Research Foundation of Linyi University, China (Grant No. LYDX2020BS014).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Ren, H.; Jia, Z. Global Weak Solution in a p-Laplacian Attraction–Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production. Axioms 2025, 14, 642. https://doi.org/10.3390/axioms14080642

AMA Style

Ren H, Jia Z. Global Weak Solution in a p-Laplacian Attraction–Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production. Axioms. 2025; 14(8):642. https://doi.org/10.3390/axioms14080642

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Ren, Hengyu, and Zhe Jia. 2025. "Global Weak Solution in a p-Laplacian Attraction–Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production" Axioms 14, no. 8: 642. https://doi.org/10.3390/axioms14080642

APA Style

Ren, H., & Jia, Z. (2025). Global Weak Solution in a p-Laplacian Attraction–Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production. Axioms, 14(8), 642. https://doi.org/10.3390/axioms14080642

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