Home Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
Article Open Access

Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion

  • Ting Huang , Zhibo Hou and Yongjie Han EMAIL logo
Published/Copyright: August 31, 2021
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

This paper is concerned with a chemotaxis system

u t = Δ u m − ∇ ⋅ ( χ 1 ( w ) u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v n − ∇ ⋅ ( χ 2 ( w ) v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = Δ w − ( α u + β v ) w , x ∈ Ω , t > 0 ,

under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 with smooth boundary, where μ 1 , μ 2 > 0 , a 1 , a 2 > 0 , α , β > 0 , and the chemotactic sensitivity function χ i ∈ C 1 ( [ 0 , ∞ ) ) , χ i ′ ≥ 0 . It is proved that for any large initial data, for any m , n > 1 , the system admits a global weak solution, which is uniformly bounded.

Keywords: chemotaxis system; nonlinear diffusion; boundedness; logistic source
MSC 2010: 92C17; 35K55; 35Q92

1 Introduction

Chemotaxis refers to the effect of chemical substances in the environment on the movement of species. This can lead to strict directional movement or partial orientation and partial tumbling movement. The movement to higher chemical concentrations is called positive chemotaxis, and the movement to lower chemical concentrations is called negative chemotaxis. Chemotaxis is an important means of cellular communication. After the pioneering work of Keller and Segel [1], a number of works concerning on the classical Keller-Segel model and its variations are investigated.

This paper is devoted to making development for the following two species chemotaxis system with nonlinear diffusion and consumption of chemoattractant

(1) u t = Δ u m − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v n − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = Δ w − ( α u + β v ) w , x ∈ Ω , t > 0 , ( ∇ u m − u χ 1 ( w ) ∇ w ) ⋅ υ = ( ∇ v n − v χ 2 ( w ) ∇ w ) ⋅ υ = ∇ w ⋅ υ = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω ,

where Ω ⊂ R 3 is a bounded domain with smooth boundary ∂ Ω , and m , n > 1 , μ 1 , μ 2 > 0 , a 1 , a 2 > 0 , α , β > 0 are positive constants and υ is the outward normal vector to ∂ Ω . The functions u = u ( x , t ) and v = v ( x , t ) denote, respectively, the unknown population density of two species, and w = w ( x , t ) represents the concentration of the chemoattractant. χ i ( w ) ( i = 1 , 2 ) is the sensitivity function of aggregation induced by the concentration changes of chemoattractant, μ 1 u ( 1 − u − a 1 v ) and μ 2 v ( 1 − a 2 u − v ) ( μ i > 0 ) are both the proliferation and death of bacteria according to a generalized logistic law and − ( α u + β v ) w denotes the consumption of chemoattractant.

In order to better understand model (1), we recall some previous contributions in this direction. Consider the following chemotaxis model with consumption of chemoattractant

(2) u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ w ) , x ∈ Ω , t > 0 , w t = Δ w − u w , x ∈ Ω , t > 0 , ( ∇ D ( u ) − χ u ⋅ ∇ w ) ⋅ υ = ∇ w ⋅ υ = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω ,

where Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω , χ > 0 is a parameter referred to as chemosensitivity. When D ( u ) ≡ 1 , in [2], Tao and Winkler proved that problem (2) possesses global bounded smooth solutions in the spatially two-dimensional setting, whereas in the three-dimensional counterpart, at least global weak solutions can be constructed, which eventually become smooth and bounded. When the nonlinear nonnegative function D ( u ) ≥ D 0 ( u + 1 ) m + 1 ( D 0 > 0 ) , it has been shown in [3] that system (2) admits a unique global classical solution that is uniformly bounded when m > 1 2 in the case N = 1 and m > 2 − 2 N in the case N ≥ 2 . Later, Zheng and Wang [4] improved this result to m > 2 − 6 N + 4 when N ≥ 3 .

If the reproduction and death of species themselves are taken into account, some logistic type sources will be added to the first equation of (2). For instance, when D ( u ) ≡ 1 , in the three-dimensional case, Zheng et al. proved that the system (2) with a logistic type source μ u ( 1 − u ) admits a unique global classical solution if the initial data of w are small in [5]. In arbitrary N-dimensional bounded smooth domain, Lankeit and Wang obtained the global bounded classical solutions of (2) for any large initial data in [6] when μ is appropriately large, and they also proved the existence of global weak solutions for any large μ . In a bounded domain Ω ⊂ R 3 , when D ( u ) ≥ C D ( u + 1 ) m − 1 for all u ≥ 0 with some C D > 0 , Zheng [7] studied the issue of boundedness to solutions of (2) without any restriction on the space dimension, if

m > 1 − μ χ ( 1 + λ 0 ‖ w 0 ‖ L ∞ ( Ω ) 2 3 ) , if N ≤ 2 , 1 , if N ≥ 3 ,

where λ 0 is a positive constant which is corresponding to the maximal Sobolev regularity, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded.

Also, multi-species chemotaxis systems have been extensively studied by many authors. When the two species have effect on each other, the system involved Lotka-Volterra competitive kinetics

(3) u t = d 1 Δ u − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = d 2 Δ v − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = d 3 Δ w − ( α u + β v ) w , x ∈ Ω , t > 0 , ∂ u ∂ υ = ∂ v ∂ υ = ∂ w ∂ υ = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω ,

has been proposed to describe the evolution of two competing species that react on a single chemoattractant. Here u , v , and w are represented as model (1), the chemotactic function χ i ( w ) ( i = 1 , 2 ) is smooth. In [8], it is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution. Moreover, the authors there also proved the asymptotic stabilization of solution. When − ( α u + β v ) w in (3) is replaced by − w + α u + β v , this model has been extensively studied by many authors. In the two-dimensional case, Bai and Winkler [9] obtained global existence of solution to the system if χ i ( w ) = χ i are positive constants. Moreover, they also considered asymptotic behavior of solutions to the system: when a 1 , a 2 ∈ ( 0 , 1 ) , u ( ⋅ , t ) → 1 − a 1 1 − a 1 a 2 , v ( ⋅ , t ) → 1 − a 2 1 − a 1 a 2 , w → α ( 1 − a 1 ) + β ( 1 − α 2 ) 1 − a 1 a 2 in L ∞ ( Ω ) as t → ∞ ; when a 1 ≥ 1 > a 2 > 0 , u ( ⋅ , t ) → 0 , v ( ⋅ , t ) → 1 , w → β in L ∞ ( Ω ) as t → ∞ . In the three-dimensional case, Lin and Mu [10] obtained similar results if μ 1 and μ 1 are sufficiently large.

When the two species have no effect on each other, the competitive kinetics terms μ 1 u ( 1 − u − a 1 v ) and μ 2 v ( 1 − a 2 u − v ) will be replaced by μ 1 u ( 1 − u ) and μ 2 v ( 1 − v ) in system (3):

(4) u t = Δ u − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − v ) , x ∈ Ω , t > 0 , w t = d Δ w + h ( u , v , w ) , x ∈ Ω , t > 0 , ∂ u ∂ υ = ∂ v ∂ υ = ∂ w ∂ υ = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ Ω ,

where Ω is a bounded domain in R N with smooth boundary ∂ Ω . Negreanu and Tello [11,12] proved global existence and asymptotic behavior of solutions to the above system when 0 ≤ d < 1 . This result was later improved by Mizukami and Yokota, and they removed the restriction of 0 ≤ d < 1 in [13].

Assuming that the random movement of species is nonlinearly enhanced at large densities, several works addressed a porous medium-type diffusion chemotaxis model (see, e.g. [14,15, 16,17]). Inspired by the aforementioned works, in this paper, we consider the two species chemotaxis system (1) with nonlinear diffusion. The purpose of this paper is to obtain global existence and uniform boundedness of weak solution in a three-dimensional setting.

Throughout this paper, we assume that

(5) u 0 , v 0 , w 0 ∈ C 2 + α ( Ω ¯ ) , u 0 , v 0 , w 0 ≥ 0 , ∂ u 0 ∂ υ ∂ Ω = ∂ v 0 ∂ υ ∂ Ω = ∂ w 0 ∂ υ ∂ Ω = 0 .

The chemotactic sensitivity function χ i ( i = 1 , 2 ) satisfies the following conditions:

(6) χ i ∈ C 1 ( [ 0 , ∞ ) ) , χ i ′ ≥ 0 .

Now, we state the main results of this paper as follows.

Theorem 1

Let Ω ⊂ R 3 be a bounded domain with smooth boundary. Assume that (5) and (6) hold, and m , n > 1 , a 1 , a 2 > 0 , α , β > 0 . Then for any μ i > 0 ( i = 1 , 2 ) , system (1) possesses a nonnegative weak solution ( u , v , w ) with u ∈ A 1 , v ∈ A 2 , w ∈ A 3 , where for any p > 1 ,

A 1 = u ∈ L ∞ ( Ω × ( 0 , ∞ ) ) ; ∇ u m ∈ L ∞ ( ( 0 , ∞ ) ; L 2 ( Ω ) ) ; ( u m + 1 2 ) t , ∇ u m + 1 2 ∈ L loc 2 ( [ 0 , ∞ ) ; L 2 ( Ω ) ) , A 2 = v ∈ L ∞ ( Ω × ( 0 , ∞ ) ) ; ∇ v n ∈ L ∞ ( ( 0 , ∞ ) ; L 2 ( Ω ) ) ; v n + 1 2 t , ∇ v n + 1 2 ∈ L loc 2 ( [ 0 , ∞ ) ; L 2 ( Ω ) ) , A 3 = { w ∈ L ∞ ( ( 0 , ∞ ) ; W 1 , ∞ ( Ω ) ) ; w t , Δ w ∈ L loc p ( [ 0 , ∞ ) ; L p ( Ω ) ) } ,

such that

(7) sup t ∈ ( 0 , ∞ ) ( ‖ u ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ v ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ w ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) ) ≤ C ,

where C depends only on μ 1 , μ 2 , u 0 , v 0 , w 0 .

The rest of this paper is organized as follows. In Section 2, we introduce the conception of the weak solution and summarize some basic definitions and useful lemmas. In Section 3, we shall first establish the existence of global classical solutions to the regularized problems and second show the convergence of the solution of regularized problems and thus obtain the proof of Theorem 1.

2 Some preliminaries

We first give some notations, which will be used throughout this paper.

Notations: Q τ ( t ) = Ω × ( t , t + τ ) , Q T ≔ Q T ( 0 ) = Ω × ( 0 , T ) . W p m , k ( Q T ) = { u ; D α u , D t r u ∈ L p ( Q T ) , ‖ u ‖ W p m , k ( Q T ) < ∞ } , where ∣ α ∣ ≤ m , r ≤ k , m = 0 , 1 , 2 , … , k ∈ { 0 , 1 } , 1 ≤ p < ∞ , and ‖ u ‖ W p m , k ( Q T ) = ∬ Q T ∑ ∣ α ∣ ≤ m ∣ D α u ∣ p + ∑ r ≤ k ∣ D t r u ∣ p d x d t 1 ∕ p .

Next, we introduce the definition of weak solutions.

Definition 1

( u , v , w ) is called a weak solution of (1) on Q T , if u ≥ 0 , v ≥ 0 , w ≥ 0 , with

u ∈ L 2 ( Q T ) , ∇ u m ∈ L 2 ( Q T ) , v ∈ L 2 ( Q T ) , ∇ v n ∈ L 2 ( Q T ) , w ∈ W 2 1 , 0 ( Q T ) ,

for any T > 0 , and

(8) − ∬ Q T u φ t d x d t − ∫ Ω u ( x , 0 ) φ ( x , 0 ) d x + ∬ Q T ( ∇ u m − χ 1 ( w ) u ∇ w ) ∇ φ d x d t = μ 1 ∬ Q T u ( 1 − u − a 1 v ) φ d x d t ,

(9) − ∬ Q T v φ t d x d t − ∫ Ω v ( x , 0 ) φ ( x , 0 ) d x + ∬ Q T ( ∇ v n − χ 2 ( w ) v ∇ w ) ∇ φ d x d t = μ 2 ∬ Q T v ( 1 − a 2 u − v ) φ d x d t ,

(10) − ∬ Q T w φ t d x d t − ∫ Ω w ( x , 0 ) φ ( x , 0 ) d x + ∬ Q T ∇ w ∇ φ d x d t + ∬ Q T ( α u + β v ) w d x d t = 0

hold for any φ ∈ C ∞ ( Q ¯ T ) with ∂ φ ∂ υ ∂ Ω = 0 , φ ( x , T ) = 0 .

Before going further, we give some lemmas, which will be used later. We firstly list the following lemma, a proof of which can be found in [18] (see also [19]).

Lemma 1

Let T > 0 , τ ∈ ( 0 , T ) , a > 0 , b > 0 , and suppose that y : [ 0 , T ) → [ 0 , ∞ ) is absolutely continuous such that

(11) y ′ ( t ) + a y ( t ) ≤ h ( t ) , for t ∈ [ 0 , T ) ,

where h ≥ 0 , h ( t ) ∈ L loc 1 ( [ 0 , T ) ) , and

(12) ∫ t − τ t h ( s ) d s ≤ b , for t ∈ [ τ , T ) .

Then

(13) y ( t ) ≤ max y ( 0 ) + b , b a τ + 2 b , for t ∈ [ 0 , T ) .

By [20,21], we have the following two lemmas.

Lemma 2

Let T > 0 , τ ∈ ( 0 , T ) , σ ≥ 0 , a > 0 , b ≥ 0 , and suppose that f : [ 0 , T ) → [ 0 , ∞ ) is absolutely continuous, and satisfies

(14) f ′ ( t ) + a f 1 + σ ( t ) ≤ h ( t ) , t ∈ R ,

where h ≥ 0 , h ( t ) ∈ L loc 1 ( [ 0 , T ) ) and

∫ t − τ t h ( s ) d s ≤ b , for all t ∈ [ τ , T ) .

Then

(15) sup t ∈ ( 0 , T ) f ( t ) + a sup t ∈ ( τ , T ) ∫ t − τ t f 1 + σ ( s ) d s ≤ b + 2 max f ( 0 ) + b + a τ , b a τ + 1 + 2 b + 2 a τ .

Lemma 3

Assume that p > 1 , u 0 ∈ W 2 , p ( Ω ) , and f ∈ L loc P ( ( 0 , + ∞ ) ; L p ( Ω ) ) satisfying

sup t ∈ ( τ , + ∞ ) ∫ t − τ t ‖ f ‖ L p ( Ω ) p d s ≤ A

with some A > 0 , where τ > 0 is a fixed constant. Then the following system

(16) u t − Δ u + u = f ( x , t ) , ∂ u ∂ υ ∂ Ω = 0 , u ( x , 0 ) = u 0 ( x )

has a unique solution u with u ∈ L loc p ( ( 0 , + ∞ ) ; W 2 , p ( Ω ) ) , u t ∈ L loc p ( ( 0 , + ∞ ) ; L p ( Ω ) ) , and

(17) sup t ∈ ( τ , + ∞ ) ∫ t − τ t ( ‖ u ‖ W 2 , p ( Ω ) p + ‖ u t ‖ L p ( Ω ) p ) d s ≤ A M e p τ e p 2 τ − 1 + M e p 2 τ ‖ u 0 ‖ W 2 , p ( Ω ) p ,

where M is a constant independent of τ .

3 Global existence and boundedness of weak solution

The degeneracy at u = 0 of system (1) results in the failure of the classical parabolic regularity theory. To overcome this difficulty, we shall first consider the following regularized version:

(18) u ε t = Δ ( u ε 2 + ε ) m − 1 2 u ε − ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) + μ 1 u ε ( 1 − u ε − a 1 v ε ) , x ∈ Ω , t > 0 , v ε t = Δ ( v ε 2 + ε ) n − 1 2 v ε − ∇ ⋅ ( v ε χ 2 ( w ε ) ∇ w ε ) + μ 2 v ε ( 1 − a 2 u ε − v ε ) , x ∈ Ω , t > 0 , w ε t = Δ w ε − ( α u ε + β v ε ) w ε , x ∈ Ω , t > 0 , ∂ u ε ∂ υ = ∂ v ε ∂ υ = ∂ w ε ∂ υ = 0 , x ∈ ∂ Ω , t > 0 , u ε ( x , 0 ) = u 0 ( x ) , v ε ( x , 0 ) = v 0 ( x ) , w ε ( x , 0 ) = w 0 ( x ) , x ∈ Ω .

We begin with the local existence of classical solutions to system (18), the proof of which is similar (refer to, e.g., [17,22, 23,24,25], for the details).

Lemma 4

Suppose that (5) and (6) hold. Then there exists T max , ε ∈ ( 0 , + ∞ ] and a unique classical solution ( u ε , v ε , w ε ) ∈ C 2 + α , 1 + α ( Ω ¯ × [ 0 , T max , ε ) ) solving the system (18) in the classical sense with u ε , v ε , w ε ≥ 0 in Ω × ( 0 , T max , ε ) and if T max , ε < ∞ , then

‖ u ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ v ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ w ε ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) → ∞ as t → T max , ε .

In what follows, we shall show that for each ε , the solution ( u ε , v ε , w ε ) is actually global in time, that is, we have

Proposition 1

Suppose that (5) and (6) hold. Then for any ε > 0 , (18) has a global classical solution ( u ε , v ε , w ε ) which is, furthermore, bounded, and

(19) sup t ∈ ( 0 , ∞ ) ∫ Ω ∣ ∇ u ε m ∣ 2 + sup t ∈ ( 0 , ∞ ) u ε m + 1 2 W 2 1 , 1 ( Q 1 ( t ) ) ≤ M 1 ,

(20) sup t ∈ ( 0 , ∞ ) ∫ Ω ∣ ∇ v ε n ∣ 2 + sup t ∈ ( 0 , ∞ ) v ε n + 1 2 W 2 1 , 1 ( Q 1 ( t ) ) ≤ M 2 ,

(21) sup t ∈ ( 0 , ∞ ) ‖ w ε ‖ W p 2 , 1 ( Q 1 ( t ) ) ≤ M 3 , for any p > 1 ,

where M i ( i = 1 , 2 , 3 ) are some constants independent of ε .

Next, we give some estimates of ( u ε , v ε , w ε ) . Take τ = min { 1 , T max , ε } . It is easy to see that τ ≤ 1 .

Lemma 5

Suppose that (5) and (6) hold. Let ( u ε , v ε , w ε ) be the classical solution of (18). Then

(22) ‖ w ε ‖ L ∞ ( Ω ) ≤ ‖ w 0 ‖ L ∞ ( Ω ) , t ∈ ( 0 , T max , ε ) ,

(23) sup t ∈ ( 0 , T max , ε ) ‖ u ε ( ⋅ , t ) ‖ L 1 ( Ω ) + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε 2 d x ≤ C 1 ,

(24) sup t ∈ ( 0 , T max , ε ) ‖ v ε ( ⋅ , t ) ‖ L 1 ( Ω ) + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω v ε 2 d x ≤ C 2 ,

(25) sup t ∈ ( τ , T max , ε ) ∫ t − τ t ‖ w ε ‖ W 2 , 2 ( Ω ) 2 d s ≤ C 3 ,

where C i ( i = 1 , 2 , 3 ) are independent of T max , ε , τ , and ε .

Proof

By comparison principle, it is easy to see that (22) holds. Integrating the first equation of (18) and using Young’s inequality, we obtain

(26) d d t ∫ Ω u ε d x + μ 1 ∫ Ω u ε 2 d x ≤ μ 1 ∫ Ω u ε d x ≤ μ 1 2 ∫ Ω u ε 2 d x + 1 2 μ 1 ∣ Ω ∣ .

We get by (26) and Hölder’s inequality that

d d t ∫ Ω u ε d x + μ 1 2 ∣ Ω ∣ ∫ Ω u ε d x 2 ≤ 1 2 μ 1 ∣ Ω ∣ .

Due to ∫ t − τ t ∣ Ω ∣ d s = ∣ Ω ∣ τ ≤ ∣ Ω ∣ for all t ∈ ( τ , T max , ε ) , here τ ≤ 1 , then by Lemma 2, there exists C 1 > 0 such that

sup t ∈ ( 0 , T max , ε ) ‖ u ε ( ⋅ , t ) ‖ L 1 ( Ω ) + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε 2 d x ≤ C 1 ∣ Ω ∣ , μ 1 , ∫ Ω u 0 d x .

By the same way, there exists C 2 > 0 such that (24) holds. From the third equation of (18), we have

w ε t − Δ w ε + w ε = w ε − ( α u ε + β v ε ) w ε .

By (22), (23), (24), and Minkowski’s inequality, we have

sup t ∈ ( τ , T max , ε ) ∫ t − τ t ‖ w ε − ( α u ε + β v ε ) w ε ‖ L 2 ( Ω ) 2 d s ≤ A ( ‖ w 0 ‖ L ∞ ( Ω ) , α , β , C 1 , C 2 ) .

Using Lemma 3, there exists C 3 > 0 such that

sup t ∈ ( τ , T max , ε ) ∫ t − τ t ‖ w ε ‖ W 2 , 2 ( Ω ) 2 d s ≤ A M e 2 τ e τ − 1 + M e τ ‖ w 0 ‖ W 2 , 2 ( Ω ) 2 ≤ A M e 2 + M e 2 ‖ w 0 ‖ W 2 , 2 ( Ω ) 2 = C 3 ,

(25) is proved. The proof is complete.□

Lemma 6

Suppose that (5) and (6) hold. Let ( u ε , v ε , w ε ) be the classical solution of (18). Then for any k = 1 , 2 , 3 , … ,

(27) sup t ∈ ( 0 , T max , ε ) ‖ u ε ( ⋅ , t ) ‖ L 2 m k − 1 ( Ω ) 2 m k − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε m + 2 m k − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m k d x d s ≤ C k ,

(28) sup t ∈ ( 0 , T max , ε ) ‖ v ε ( ⋅ , t ) ‖ L 2 n k − 1 ( Ω ) 2 n k − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω v ε n + 2 n k − 4 ∣ ∇ v ε ∣ 2 d x + ∫ Ω v ε 2 n k d x d s ≤ C k ′ ,

(29) sup t ∈ ( τ , T max , ε ) ∫ t − τ t ( ‖ w ε ‖ W 2 , 2 m k ( Ω ) 2 m k + ‖ w ε t ‖ L 2 m k ( Ω ) 2 m k ) d s ≤ C k ˜ ,

(30) sup t ∈ ( τ , T max , ε ) ∫ t − τ t ( ‖ w ε ‖ W 2 , 2 n k ( Ω ) 2 n k + ‖ w ε t ‖ L 2 n k ( Ω ) 2 n k ) d s ≤ C k ′ ˜ ,

where C k , C k ′ , C k ˜ , C k ′ ˜ depend on k and all of them are independent of T max , ε , τ and ε .

Proof

Multiplying the first equation of (18) by u ε r for any r > 0 and integrating over Ω by parts,

(31) 1 r + 1 d d t ∫ Ω u ε r + 1 d x + r ∫ Ω u ε m + r − 2 ∣ ∇ u ε ∣ 2 d x + μ 1 ∫ Ω u ε r + 2 d x ≤ r ∫ Ω χ 1 ( w ε ) u ε r ∇ u ε ∇ w ε d x + μ 1 ∫ Ω u ε r + 1 d x ≤ r 2 ∫ Ω u ε m + r − 2 ∣ ∇ u ε ∣ 2 d x + μ 1 2 ∫ Ω u ε r + 2 + r 2 ∫ Ω χ 1 2 ( w ε ) u ε r + 2 − m ∣ ∇ w ε ∣ 2 d x + C 1 , t ∈ ( 0 , T max , ε ) .

It is easy to see that

(32) 1 r + 1 d d t ∫ Ω u ε r + 1 d x + r 2 ∫ Ω u ε m + r − 2 ∣ ∇ u ε ∣ 2 d x + μ 1 2 ∫ Ω u ε r + 2 d x ≤ r 2 ∫ Ω χ 1 2 ( w ε ) u ε r + 2 − m ∣ ∇ w ε ∣ 2 d x + C 1 , t ∈ ( 0 , T max , ε ) .

By Gagliardo-Nirenberg inequality and (22), for any k = 0 , 1 , 2 , … , we have

(33) ∫ t − τ t ‖ ∇ w ε ‖ L 4 m k ( Ω ) 4 m k d s ≤ C 2 ∫ t − τ t ( ‖ Δ w ε ‖ L 2 m k ( Ω ) 4 m k α ‖ w ε ‖ L ∞ ( Ω ) 4 m k ( 1 − α ) + ‖ w ε ‖ L ∞ ( Ω ) 4 m k ) d s ≤ C 3 1 + ∫ t − τ t ‖ Δ w ε ‖ L 2 m k ( Ω ) 2 m k d s , t ∈ ( τ , T max , ε ) ,

where α = 1 2 , C 3 depends on m , k , Ω , and ‖ w 0 ‖ L ∞ ( Ω ) . Inserting r = 2 ( m − 1 ) in (32), by (6) and (22), we infer from Young’s inequality that in ( 0 , T max , ε )

(34) 1 2 m − 1 d d t ∫ Ω u ε 2 m − 1 d x + ( m − 1 ) ∫ Ω u ε 3 m − 4 ∣ ∇ u ε ∣ 2 d x + μ 1 2 ∫ Ω u ε 2 m d x ≤ ( m − 1 ) ∫ Ω χ 1 2 ( w ε ) u ε m ∣ ∇ w ε ∣ 2 d x + C 1 ≤ μ 1 4 ∫ Ω u ε 2 m d x + C 4 ∫ Ω ∣ ∇ w ε ∣ 4 d x + C 1 ,

it implies

d d t ∫ Ω u ε 2 m − 1 d x + ∫ Ω u ε 3 m − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m d x + ∫ Ω u ε 2 m − 1 d x ≤ C 5 ∫ Ω ∣ ∇ w ε ∣ 4 d x + C 5 ,

then, taking k = 0 in (33) and using (25), we infer from Lemma 1 that

sup t ∈ ( 0 , T max , ε ) ‖ u ε ‖ L 2 m − 1 ( Ω ) 2 m − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε 3 m − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m d x d s ≤ C .

By the same way, we have

sup t ∈ ( 0 , T max , ε ) ‖ v ε ‖ L 2 n − 1 ( Ω ) 2 n − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω v ε 3 n − 4 ∣ ∇ v ε ∣ 2 d x + ∫ Ω v ε 2 n d x d s ≤ C ′ .

Next, we use recursive method to prove (27) and (28). Assume that i ∈ N +

(35) sup t ∈ ( 0 , T max , ε ) ‖ u ε ‖ L 2 m i − 1 ( Ω ) 2 m i − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε m + 2 m i − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m i d x d s ≤ C i ,

(36) sup t ∈ ( 0 , T max , ε ) ‖ v ε ‖ L 2 n i − 1 ( Ω ) 2 n i − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω v ε n + 2 n i − 4 ∣ ∇ v ε ∣ 2 d x + ∫ Ω v ε 2 n i d x d s ≤ C i ′ .

Due to u , v ≥ 0 , and the third equation of (18), using Lemma 3, (22), we derive

(37) sup t ∈ ( τ , T max , ε ) ∫ t − τ t ( ‖ w ε ‖ W 2 , 2 m i ( Ω ) 2 m i + ‖ w ε t ‖ L 2 m i ( Ω ) 2 m i ) d s ≤ C ˜ i ,

(38) sup t ∈ ( τ , T max , ε ) ∫ t − τ t ( ‖ w ε ‖ W 2 , 2 n i ( Ω ) 2 n i + ‖ w ε t ‖ L 2 n i ( Ω ) 2 n i ) d s ≤ C ˜ i ′ .

Taking r = 2 m i + 1 − 2 in (32), we can obtain in t ∈ ( 0 , T max , ε )

(39) 1 2 m i + 1 − 1 d d t ∫ Ω u ε 2 m i + 1 − 1 d x + ( m i + 1 − 1 ) ∫ Ω u ε m + 2 m i + 1 − 4 ∣ ∇ u ε ∣ 2 d x + μ 1 2 ∫ Ω u ε 2 m i + 1 d x ≤ ( m i + 1 − 1 ) ∫ Ω χ 1 2 ( w ε ) u ε 2 m i + 1 − m ∣ ∇ w ε ∣ 2 d x + C 1 ≤ C 6 ∫ Ω ∣ ∇ w ε ∣ 4 m i d x + μ 1 4 ∫ Ω u ε 2 m i + 1 + C 1 ,

it implies

d d t ∫ Ω u ε 2 m i + 1 − 1 d x + ∫ Ω u ε m + 2 m i + 1 − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m i + 1 d x + ∫ Ω u ε 2 m i + 1 − 1 d x ≤ C 7 ∫ Ω ∣ ∇ w ε ∣ 4 m i d x + C 7 .

Combining (33) and (37) with Lemma 1, a direct calculation shows

sup t ∈ ( 0 , T max , ε ) ‖ u ε ( ⋅ , t ) ‖ L 2 m i + 1 − 1 ( Ω ) 2 m i + 1 − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω u ε m + 2 m i + 1 − 4 ∣ ∇ u ε ∣ 2 d x + ∫ Ω u ε 2 m i + 1 d x d s ≤ C i + 1 .

Arguing similarly as above, we see that

(40) sup t ∈ ( 0 , T max , ε ) ‖ v ε ( ⋅ , t ) ‖ L 2 n i + 1 − 1 ( Ω ) 2 n i + 1 − 1 + sup t ∈ ( τ , T max , ε ) ∫ t − τ t ∫ Ω v ε n + 2 n i + 1 − 4 ∣ ∇ v ε ∣ 2 d x + ∫ Ω v ε 2 n i + 1 d x d s ≤ C i + 1 ′ .

The proof is complete.□

Lemma 7

Suppose that (5) and (6) hold. Let ( u ε , v ε , w ε ) be the classical solution of (18). Then

(41) sup t ∈ ( 0 , T max , ε ) ( ‖ u ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ v ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ w ε ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) ) ≤ C ,

where C is independent of T max , ε and ε .

Proof. We take k appropriately large such that 2 m k ≥ 10 in Lemma 6. Then, by t -anisotropic embedding theorem [26], there exists C 0 such that

(42) ‖ w ε ‖ C 3 2 , 3 4 ( Q τ ( t ) ) ≤ C 0 sup t ∈ ( τ , T max , ε ) ‖ w ε ‖ W 10 2 , 1 ( Q τ ( t ) ) ,

where Q τ ( t ) = Ω × ( t − τ , t ) , it means

(43) sup t ∈ ( 0 , T max , ε ) ‖ w ε ‖ W 1 , ∞ ( Ω ) ≤ C 0 ′ .

Multiplying the first equation of (18) by r u ε r − 1 with r ≥ 3 m , integrating it over Ω , and using (6), (22), (43),

d d t ∫ Ω u ε r d x + r ( r − 1 ) ∫ Ω u ε m + r − 3 ∣ ∇ u ε ∣ 2 d x + r μ 1 ∫ Ω u ε r + 1 d x + ∫ Ω u ε r d x ≤ r ( r − 1 ) ∫ Ω χ 1 ( w ε ) u ε r − 1 ∇ u ε ∇ w ε d x + r μ 1 ∫ Ω u ε r d x + ∫ Ω u ε r d x ≤ r ( r − 1 ) 2 ∫ Ω ( u ε m + r − 3 ∣ ∇ u ε ∣ 2 + χ 1 2 ( w ε ) u ε r + 1 − m ∣ ∇ w ε ∣ 2 ) d x + ( r μ 1 + 1 ) ∫ Ω u ε r d x ≤ r ( r − 1 ) 2 ∫ Ω u ε m + r − 3 ∣ ∇ u ε ∣ 2 + C 1 r 2 ∫ Ω u ε r + 1 − m d x + r μ 1 2 ∫ Ω u ε r + 1 d x + C 1 , t ∈ ( 0 , T max , ε ) .

It is easy to see that

(44) d d t ∫ Ω u ε r d x + ∫ Ω ∇ u ε r + m − 1 2 2 d x + r μ 1 2 ∫ Ω u ε r + 1 d x + ∫ Ω u ε r d x ≤ C 1 r 2 ∫ Ω u ε r + 1 − m d x + C 1 , t ∈ ( 0 , T max , ε ) .

Using the Gagliardo-Nirenberg interpolation inequality and Young’s inequality, we can derive in ( 0 , T max , ε )

C 1 r 2 ‖ u ε ‖ L r + 1 − m ( Ω ) r + 1 − m = C 1 r 2 u ε r + m − 1 2 L 2 ( r + 1 − m ) r + m − 1 ( Ω ) 2 ( r + 1 − m ) r + m − 1 ≤ C 2 r 2 ∇ u ε r + m − 1 2 L 2 ( Ω ) 2 s ( r + 1 − m ) r + m − 1 u ε r + m − 1 2 L r m + r − 1 ( Ω ) 2 ( 1 − s ) ( r + 1 − m ) r + m − 1 + C 3 r 2 u ε r + m − 1 2 L r m + r − 1 ( Ω ) 2 ( r + 1 − m ) r + m − 1 ≤ 1 2 ∇ u ε r + m − 1 2 L 2 ( Ω ) 2 + C 4 r 6 ( m − 1 ) + 5 r 6 ( m − 1 ) + r ∥ u ε ∥ L r 2 ( Ω ) r ( 2 m + r − 2 ) 6 m + r − 6 + C 3 r 2 ‖ u ε ‖ L r 2 ( Ω ) r + 1 − m ≤ 1 2 ∇ u ε r + m − 1 2 L 2 ( Ω ) 2 + C 4 r 5 ‖ u ε ‖ L r 2 ( Ω ) r ( 2 m + r − 2 ) 6 m + r − 6 + C 3 r 2 ‖ u ε ‖ L r 2 ( Ω ) r + 1 − m ,

where s = m + r − 1 r + 1 − m ⋅ − 3 r − 6 + 6 m − 5 r + 6 − 6 m ∈ ( 0 , 1 ) , taking the above inequality into (44), in ( 0 , T max , ε )

(45) d d t ∫ Ω u ε r d x + ∫ Ω u ε r d x ≤ C 4 r 5 ‖ u ε ‖ L r 2 ( Ω ) r ( 2 m + r − 2 ) 6 m + r − 6 + C 3 r 2 ‖ u ε ‖ L r 2 ( Ω ) r + 1 − m + C 1 .

By (27), we can take k * > 0 appropriately large such that 2 m k * − 1 > 3 m , then

sup t ∈ ( 0 , T max , ε ) ‖ u ε ‖ L 2 m k * − 1 ( Ω ) ≤ C 5 .

Taking r j = 2 r j − 1 = 2 j r 0 , r 0 = 2 m k * − 1 , M j = max { 1 , sup t ∈ ( 0 , T max , ε ) ‖ u ε ‖ L r j ( Ω ) } , then by (45), we have

M j ≤ ( C 3 + C 4 + C 1 ) 1 r j r j 5 r j M j − 1 = ( C 3 + C 4 + C 1 ) 1 2 j r 0 r 0 5 2 j r 0 2 5 j 2 j r 0 M j − 1 ≤ ( C 3 + C 4 + C 1 ) ∑ k = 1 j 1 2 k r 0 r 0 ∑ k = 1 j 5 2 k r 0 2 ∑ k = 1 j 5 k 2 k r 0 M 0 ≤ C ,

where C is independent of j . Letting j → ∞ , sup t ∈ ( 0 , T max , ε ) ‖ u ε ‖ L ∞ ( Ω ) ≤ C is obtained. By the same way, sup t ∈ ( 0 , T max , ε ) ‖ v ε ‖ L ∞ ( Ω ) ≤ C is obtained. Then (41) is complete.

Proof of Proposition 1

By Lemmas 4 and 7, for any ε > 0 , T max , ε = + ∞ , the system (18) has a global classical solution which is, furthermore, bounded. Namely, there exists C that is independent of ε , T max , ε such that

(46) sup t ∈ ( 0 , ∞ ) ( ‖ u ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ v ε ( ⋅ , t ) ‖ L ∞ ( Ω ) + ‖ w ε ( ⋅ , t ) ‖ W 1 , ∞ ( Ω ) ) ≤ C .

Meanwhile, due to T max , ε = + ∞ , it means that τ = 1 .

Multiplying the first equation of (18) by u ε and integrating it over Ω , we have

1 2 d d t ∫ Ω u ε 2 d x = ∫ Ω u ε u ε t = ∫ Ω u ε Δ ( ( u ε 2 + ε ) m − 1 2 u ε ) − ∫ Ω u ε ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) = − ∫ Ω ∇ u ε ∇ ( u ε 2 + ε ) m − 1 2 u ε + ∫ Ω χ 1 ( w ε ) u ε ∇ w ε ∇ u ε + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) = − ∫ Ω ∇ u ε ∇ ( u ε 2 + ε ) m − 1 2 u ε + 1 2 ∫ Ω χ 1 ( w ε ) ∇ w ε ∇ u ε 2 + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) = − ∫ Ω ∇ u ε ∇ ( u ε 2 + ε ) m − 1 2 u ε − 1 2 ∫ Ω χ 1 ′ ( w ε ) u ε 2 ∣ ∇ w ε ∣ 2 − 1 2 ∫ Ω χ 1 ( w ε ) u ε 2 Δ w ε + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) ≤ − ∫ Ω ∇ u ε ∇ ( u ε 2 + ε ) m − 1 2 u ε − 1 2 ∫ Ω χ 1 ( w ε ) u ε 2 Δ w ε + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε )

≤ − ( m − 1 ) ∫ Ω ( u ε 2 + ε ) m − 3 2 u ε 2 ∣ ∇ u ε ∣ 2 − ∫ Ω ( u ε 2 + ε ) m − 1 2 ∣ ∇ u ε ∣ 2 + 1 4 χ 1 2 ( ‖ w 0 ‖ L ∞ ( Ω ) ) ∫ Ω u ε 4 + 1 4 ∫ Ω ∣ Δ w ε ∣ 2 + μ 1 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) , t ∈ ( 0 , ∞ ) ,

by (46), we have

d d t ∫ Ω u ε 2 + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∣ ∇ u ε ∣ 2 ≤ C 1 ( 1 + ∫ Ω ∣ Δ w ε ∣ 2 ) , t ∈ ( 0 , + ∞ ) .

By (25) and Lemma 1, it implies

(47) sup t ∈ ( 1 , ∞ ) ∫ t − 1 t ∫ Ω ( u ε 2 + ε ) m − 1 2 ∣ ∇ u ε ∣ 2 ≤ C 2 ,

where C 2 is independent of ε . Multiplying the first equation of (18) by ∂ ( u ε 2 + ε ) m − 1 2 u ε ∂ t , and integrating it over Ω ,

∫ Ω u ε t ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t = ∫ Ω ( m − 1 ) ( u ε 2 + ε ) m − 3 2 u ε 2 + ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 = ∫ Ω Δ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t − ∫ Ω ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t + μ 1 ∫ Ω u ε ( 1 − u ε − a 1 v ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t = − ∫ Ω ∇ ⋅ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ ∇ ⋅ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t − ∫ Ω ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t + μ 1 ∫ Ω u ε ( 1 − u ε − a 1 v ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t , t ∈ ( 0 , + ∞ ) ,

then

∫ Ω ∇ ⋅ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ ∇ ⋅ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 = 1 2 d d t ∫ Ω ∇ ( ( u ε 2 + ε ) m − 1 2 u ε ) 2 + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 ≤ − ∫ Ω ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t + μ 1 ∫ Ω u ε ( 1 − u ε − a 1 v ε ) ∂ ( ( u ε 2 + ε ) m − 1 2 u ε ) ∂ t ≤ m 2 ∫ Ω ∣ ∇ ⋅ ( u ε χ 1 ( w ε ) ∇ w ε ) ∣ 2 ( u ε 2 + ε ) m − 1 2 + 1 2 ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 + μ 1 2 m 2 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) 2 ( u ε 2 + ε ) m − 1 2 ≤ C 3 ∫ Ω χ 1 2 ( w ε ) ∣ ∇ u ε ∣ 2 ∣ ∇ w ε ∣ 2 ( u ε 2 + ε ) m − 1 2 + ∫ Ω ∣ χ 1 ′ ( w ε ) ∣ 2 ∣ ∇ w ε ∣ 4 u ε 2 ( u ε 2 + ε ) m − 1 2 + ∫ Ω χ 1 2 ( w ε ) u ε 2 ∣ Δ w ε ∣ 2 ( u ε 2 + ε ) m − 1 2 + 1 2 ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 + μ 1 2 m 2 ∫ Ω u ε 2 ( 1 − u ε − a 1 v ε ) 2 ( u ε 2 + ε ) m − 1 2 ≤ C 4 1 + ∫ Ω ∣ Δ w ε ∣ 2 + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∣ ∇ u ε ∣ 2 + 1 2 ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 , t ∈ ( 0 , + ∞ ) .

It implies

d d t ∫ Ω ∇ ( ( u ε 2 + ε ) m − 1 2 u ε ) 2 + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 + ∫ Ω ∇ ( ( u ε 2 + ε ) m − 1 2 u ε ) 2 ≤ C 5 1 + ∫ Ω ∣ Δ w ε ∣ 2 + ∫ Ω ( u ε 2 + ε ) m − 1 2 ∣ ∇ u ε ∣ 2 , t ∈ ( 0 , + ∞ ) .

By (47), we infer from Lemma 1 that

sup t ∈ ( 0 , + ∞ ) ∫ Ω ∇ ( ( u ε 2 + ε ) m − 1 2 u ε ) 2 + sup t ∈ ( 1 , + ∞ ) ∫ t − 1 t ∫ Ω ( u ε 2 + ε ) m − 1 2 ∂ u ε ∂ t 2 ≤ C ,

which implies (19). By the same way, we can obtain (20). The estimate of (21) can be easily obtained from (46) and Lemma 3.□

Next, we can use this to show Theorem 1.

Proof of Theorem 1

According to Proposition 1, for any T > 0 , we have subsequence of ( u ε , v ε , w ε ) (for simplicity, we still denote subsequence by ( u ε , v ε , w ε )) and ( u , v , w ) such that

u ε ⇀ ∗ u and v ε ⇀ ∗ v , in L ∞ ( Q T ) , u ε m + 1 2 ⇀ u m + 1 2 and v ε n + 1 2 ⇀ v n + 1 2 , in W 2 1 , 1 ( Q T ) , w ε ⇀ w , in W p 2 , 1 ( Q T ) , for any p ∈ ( 1 , ∞ ) .

By the Aubin-Lions theorem, (19) and (20), we have

u ε m + 1 2 → u m + 1 2 and v ε n + 1 2 → v n + 1 2 , in C ( [ 0 , T ] ; L 2 ( Ω ) ) ,

then u ε → u , v ε → v a.e. in Q T . Using (46) and Lebesgue’s dominated convergence theorem,

u ε → u and v ε → v , in C ( [ 0 , T ] ; L p ( Ω ) ) , for any p ∈ ( 1 , ∞ ) .

By (21) and W p 2 , 1 ( Q T ) ↪ C 2 − 5 p , 1 − 5 2 p ( Q T ) for any p > 5 2 , then

w ε → w , in C 2 − 5 p , 1 − 5 2 p ( Q T ) .

And (7) holds. Due to u ε → u in L p ( Q T ) , ∇ w ε ⇀ ∇ w in L p ( Q T ) for any p > 1 , (6), we have

χ 1 ( w ε ) u ε ∇ w ε ⇀ χ 1 ( w ) u ∇ w , in L p ( Q T ) , for any p ∈ ( 1 , ∞ ) .

By the same way, we also have

χ 2 ( w ε ) v ε ∇ w ε ⇀ χ 2 ( w ) v ∇ w in L p ( Q T ) , for any p ∈ ( 1 , ∞ ) .

We fix φ ∈ C ∞ ( Q ¯ T ) with ∂ φ ∂ υ ∂ Ω = 0 and φ ( x , T ) = 0 , multiplying the equation in (18) by φ and integrating by parts, we obtain

− ∬ Q T u ε φ t d x d t − ∫ Ω u 0 φ ( x , 0 ) d x − ∬ Q T ( u ε 2 + ε ) m − 1 2 u ε Δ φ d x d t = ∬ Q T χ 1 ( w ε ) u ε ∇ w ε ∇ φ d x d t + ∬ Q T μ 1 u ε ( 1 − u ε − a 1 v ε ) φ d x d t , − ∬ Q T v ε φ t d x d t − ∫ Ω v 0 φ ( x , 0 ) d x − ∬ Q T ( v ε 2 + ε ) n − 1 2 v ε Δ φ d x d t = ∬ Q T χ 2 ( w ε ) v ε ∇ w ε ∇ φ d x d t + ∬ Q T μ 2 v ε ( 1 − a 2 u ε − v ε ) φ d x d t ,

− ∬ Q T w ε φ t d x d t − ∫ Ω w 0 φ ( x , 0 ) d x + ∬ Q T ∇ w ε ∇ φ d x d t + ∬ Q T ( α u ε + β v ε ) w ε φ d x d t = 0 .

Letting ε → 0 , we can get

− ∬ Q T u φ t d x d t − ∫ Ω u 0 φ ( x , 0 ) d x − ∬ Q T u m Δ φ d x d t = ∬ Q T χ 1 ( w ) u ∇ w ∇ φ d x d t + ∬ Q T μ 1 u ( 1 − u − a 1 v ) φ d x d t , − ∬ Q T v φ t d x d t − ∫ Ω v 0 φ ( x , 0 ) d x − ∬ Q T v n Δ φ d x d t = ∬ Q T χ 2 ( w ) v ∇ w ∇ φ d x d t + ∬ Q T μ 2 v ( 1 − a 2 u − v ) φ d x d t , − ∬ Q T w φ t d x d t − ∫ Ω w 0 φ ( x , 0 ) d x + ∬ Q T ∇ w ∇ φ d x d t + ∬ Q T ( α u + β v ) w φ d x d t = 0 .

By ∇ u m ∈ L 2 ( Q T ) , ∇ v n ∈ L 2 ( Q T ) we obtain

− ∬ Q T u φ t d x d t − ∫ Ω u 0 φ ( x , 0 ) d x + ∬ Q T ( ∇ u m − χ 1 ( w ) u ∇ w ) ∇ φ d x d t = ∬ Q T μ 1 u ( 1 − u − a 1 v ) φ d x d t , − ∬ Q T v φ t d x d t − ∫ Ω v 0 φ ( x , 0 ) d x + ∬ Q T ( ∇ v n − χ 2 ( w ) v ∇ w ) ∇ φ d x d t = ∬ Q T μ 2 v ( 1 − a 2 u − v ) φ d x d t ,

which means ( u , v , w ) with u ∈ A 1 , v ∈ A 2 , w ∈ A 3 is a global weak solution of (1).□

Acknowledgments

The authors are grateful to all of the anonymous reviewers for their carefully reading and valuable comments on how to improve the paper.

  1. Funding information: This work was supported by the Applied Fundamental Research Plan of Sichuan Province (No. 2018JY0503), the Scientific Research Fund of the Education Department of Sichuan Province (Grant No. 15233448), and the Key Scientific Research Fund of Xihua University (Grant No. z1412621).

  2. Conflict of interest: Authors state no conflict of interest.

References

[1] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), no. 3, 399–415, https://doi.org/10.1016/0022-5193(70)90092-5. 10.1016/0022-5193(70)90092-5Search in Google Scholar

[2] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ. 252 (2012), no. 3, 2520–2543, https://doi.org/10.1016/j.jde.2011.07.010. 10.1016/j.jde.2011.07.010Search in Google Scholar

[3] L. Wang, C. Mu, and S. Zhou, Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys. 65 (2014), no. 6, 1137–1152, https://doi.org/10.1007/s00033-013-0375-4. 10.1007/s00033-013-0375-4Search in Google Scholar

[4] J. Zheng and Y. Wang, A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 2, 669–686, https://doi.org/10.3934/dcdsb.2017032. 10.3934/dcdsb.2017032Search in Google Scholar

[5] P. Zheng and C. Mu, Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source, Math. Nachr. 288 (2015), no. 5, 710–720, https://doi.org/10.1002/mana.201300105. 10.1002/mana.201300105Search in Google Scholar

[6] J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. 37 (2017), no. 12, 6099–6121, https://doi.org/10.3934/dcds.2017262. 10.3934/dcds.2017262Search in Google Scholar

[7] J. Zheng, Global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant, arxiv:1801.01774v1, (2018). Search in Google Scholar

[8] L. Wang, C. Mu, X. Hu, and P. Zheng, Boundedness and asymptotic stability of solutions to a two species chemotaxis system with consumption of chemoattractant, J. Differ. Equ. 264 (2018), no. 5, 3369–3401, https://doi.org/10.1016/j.jde.2017.11.019. 10.1016/j.jde.2017.11.019Search in Google Scholar

[9] X. Bai and M. Winkler, Equilibration in a fully parabolic two species chemotaxis system with competitive kinetics, Indian Univ. Math. J. 65 (2016), no. 2, 553–583. 10.1512/iumj.2016.65.5776Search in Google Scholar

[10] K. Lin and C. Mu, Convergence of global and bounded solutions of a two species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 6, 2233–2260, https://doi.org/10.3934/dcdsb.2017094. Search in Google Scholar

[11] M. Negreanu and I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differ. Equ. 258 (2015), no. 5, 1592–1617, https://doi.org/10.1016/j.jde.2014.11.009. Search in Google Scholar

[12] M. Negreanu and I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal. 46 (2014), no. 6, 3761–3781, https://doi.org/10.1137/140971853. Search in Google Scholar

[13] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two species chemotaxis system with any chemical diffusion, J. Differ. Equ. 261 (2016), no. 5, 2650–2669, https://doi.org/10.1016/j.jde.2016.05.008. Search in Google Scholar

[14] X. Li, Y. Wang, and Z. Xiang, Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci. 14 (2016), no. 7, 1889–1910, https://doi.org/10.4310/CMS.2016.v14.n7.a5. 10.4310/CMS.2016.v14.n7.a5Search in Google Scholar

[15] Y. Wang, Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 2, 329–349, https://doi.org/10.3934/dcdss.2020019. Search in Google Scholar

[16] Y. Wang and Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Zeitschrift für angewandte Mathematik und Physik 66 (2015), 3159–3179, https://doi.org/10.1007/s00033-015-0557-3. Search in Google Scholar

[17] Y. Wang and L. Zhao, A 3D self-consistent chemotaxis-fluid system with nonlinear diffusion, J. Differ. Equ. 269 (2020), no. 1, 148–179, https://doi.org/10.1016/j.jde.2019.12.002. Search in Google Scholar

[18] C. Stinner, C. Surulescu, and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal. 46 (2014), no. 3, 1969–2007, https://doi.org/10.1137/13094058X. Search in Google Scholar

[19] Y. Wang, M. Winkler, and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 18 (2018), no. 2, 421–466. 10.2422/2036-2145.201603_004Search in Google Scholar

[20] C. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differ. Equ. 263 (2017), no. 9, 5759–5772, https://doi.org/10.1016/j.jde.2017.06.034. Search in Google Scholar

[21] C. Jin, Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. London Math. Soc. 50 (2018), no. 4, 598–618, https://doi.org/10.1112/blms.12160. Search in Google Scholar

[22] Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. 32 (2012), no. 5, 1901–1914, https://doi.org/10.3934/dcds.2012.32.1901. Search in Google Scholar

[23] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ. 35 (2010), no. 8, 1516–1537, https://doi.org/10.1080/03605300903473426. Search in Google Scholar

[24] M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl. 348 (2008), no. 2, 708–729, https://doi.org/10.1016/j.jmaa.2008.07.071. Search in Google Scholar

[25] M. Winkler, Global large-data solutions in a Chemotaxis-(Navier-)Stokes system modeling celluar swimming in fluid drops, Comm. Partial Differ. Equ. 37 (2012), no. 2, 319–351, https://doi.org/10.1080/03605302.2011.591865. Search in Google Scholar

[26] Z. Wu, J. Yin, and C. Wang, Elliptic and Parabolic Equations, World Scientific Publishing Co. Pvt. Ltd, Singapore, 2006. 10.1142/6238Search in Google Scholar
Received: 2021-03-05
Revised: 2021-06-24
Accepted: 2021-07-05
Published Online: 2021-08-31

© 2021 Ting Huang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0074
Keywords for this article
chemotaxis system; nonlinear diffusion; boundedness; logistic source
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2021-0074/html
Scroll to top button