Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity

1 Introduction and sketch of the main results

In 1970, Keller and Segel proposed a mathematical model concerning about cell’s life cycle, especially an aggregation process. In this model, “chemotaxis” plays an essential role to induce the aggregation process of the cell. When cells are starving, cells move toward increasing concentrations of the signal substance that is produced by cells [12]:

where n denotes the cell density and w is the chemical concentration. The simplified model of Model (1.1) was proposed by Nanjundiah [24]:

The mathematical analysis of (1.2) and the variants thereof mainly concentrates on the boundedness and blow-up of the solutions (refer to, e.g., [3,11,22,23,25,32,36,38] and references therein). When τ w t = Δ w − w + n is replaced by w t = Δ w , Li et al. [22] proved that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 N , r − 3 2 × ( B ˙ 2 N , r − 1 2 ) N as 1 ≤ r < N due to the lack of continuity of the solution for the Cauchy problem. When n t = Δ n − χ ∇ ⋅ ( n ∇ w ) is replaced by n t = Δ ( γ ( w ) n ) + a n − b n θ and τ = 0 , Lyu and Wang [23] explored how strong the logistic damping can warrant the global boundedness of solutions and further established the asymptotic behavior of solutions on top of the conditions. Keller and Segel [13] introduced a phenomenological model of wave-like solution behavior without any type of cell kinetics, a prototypical version of which is given by:

where u represents the density of bacteria and v denotes the concentration of the nutrient. The second equation models the consumption of the signal. In the first equation, the chemotactic sensitivity is determined according to the Weber-Fechner law, which says that the chemotactic sensitivity is proportional to the reciprocal of signal density. Winkler [39] proved that if initial data satisfies appropriate regularity assumptions, System (1.3) possesses at least one global generalized solution in two-dimensional bounded domains. Moreover, he took asymptotic behavior of solutions to System (1.3) into account and proved that v ( ⋅ , t ) ⇀ * 0 in L ∞ ( Ω ) and v ( ⋅ , t ) → 0 in L p ( Ω ) as t → ∞ provided ∫ Ω u 0 ≤ m , − ∫ Ω ln v 0 ‖ v 0 ‖ L ∞ ( Ω ) ≤ M , where m and M are positive constants. When u v is replaced by g ( u ) v , g ∈ C 1 ( R ) and 0 ≤ g ( u ) ≤ u β , β ∈ ( 0 , 1 ) , χ ∈ ( 0 , 1 ) and any sufficiently regular initial data, Lankeit and Viglialoro [20] showed that System (1.3) has a global classical solution. Moreover, if additionally m = ‖ u 0 ‖ L 1 ( Ω ) is sufficiently small, then also their boundedness is achieved. When System (1.3) has a logistic source f ( u ) , Lankeit and Lankeit [17] showed that System (1.3) possesses a global generalized solution for any χ ≥ 0 , r ≥ 0 , and μ > 0 if f ( u ) = κ u − μ u 2 . As f ( u ) = r u − μ u k and 1 v is replaced by ϕ ( v ) , ϕ ( v ) ∈ C 1 ( 0 , ∞ ) satisfying ϕ ( v ) → 0 as v → ∞ . Zhao and Zheng [50] proved that System (1.3) possesses a unique positive global classical solution provided k > 1 with N = 1 or k > 1 + N 2 with N ≥ 2 . For more recent outcomes, one can see [16,19,33,34,40,45].

When v does not stand for a nutrient consumed but for a signaling substance produced by the bacteria themselves, which is given by:

and when 1 v is replaced by χ ( v ) , χ ( v ) ≤ χ 0 ( 1 + α v ) k with some χ 0 > 0 , α > 0 , and k > 1 , Winkler [35] proved that for any choice of appropriate initial data, System (1.4) possesses a unique global classical solution that is bounded in Ω × ( 0 , ∞ ) for N ≥ 1 . Stinner and Winkler [31] showed that for any λ ∈ ( 0 , min { 1 , 1 χ 2 } ) , System (1.4) admits at least one couple ( u , v ) of nonnegative functions defined in Ω × ( 0 , ∞ ) such that ( u , v ) is a global weak power- λ solution of (1.4) for N ≥ 2 . Winkler [37] proved that 0 < χ < 2 N and then for any such data, there exists a global-in-time classical solution. Moreover, global existence of weak solutions is established whenever 0 < χ < N + 2 3 N − 4 . The boundedness of solution is left as an open problem. Fujie [7] solved the open problem of uniform-in-time boundedness of solutions for 0 < χ < 2 N , which was conjectured by Winkler [37]. Recently, Winkler and Yokota [44] proved that System (1.4) possesses a uniquely determined global classical solution if χ ∈ ( 0 , χ 0 ] and χ 2 ≤ δ , where χ 0 ∈ ( 0 , 2 N ) , δ > 0 are the constants. Furthermore, the solution of (1.4) converges to the homogeneous steady state ( u ¯ 0 , u ¯ 0 ) at an exponential rate with respect to the norm in ( L ∞ ( Ω ) ) 2 as t → ∞ , where u ¯ 0 = 1 ∣ Ω ∣ ∫ Ω u 0 . Lankeit and Winkler [21] introduced an apparently novel type of generalized solution and proved that under the hypothesis: (i) χ < ∞ if N = 2 ; (ii) χ < 8 if N = 3 ; (iii) χ < N N − 2 if N ≥ 4 , for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that u ∈ L loc 1 ( Ω ¯ × [ 0 , ∞ ) ) . Recently, when u t = Δ u − χ ∇ ⋅ u v ∇ v is replaced by ε u t = Δ u − χ ∇ ⋅ u v ∇ v , Winkler [43] proved that if N ≥ 3 and χ > N N − 2 , a statement on spontaneous emergence of arbitrarily large values of u for appropriately small ε is derived. When the second equation degenerates into an elliptic equation, ∇ v v is replaced by ∇ χ ( v ) , χ ∈ C loc 2 + ω ( ( 0 , ∞ ) ) with some ω ∈ ( 0 , 1 ) , χ ′ > 0 , χ ′ ( v ) → 0 as v → ∞ , Fujie and Senba [8] showed that System (1.4) admits a global classical positive solution that is uniformly bounded. Black [4] showed that System (1.4) admits at least one global generalized solution if 0 < χ < N N − 2 . Zhigun [53] showed that System (1.4) admits a generalized supersolution with appropriate condition and assumed that ( u , v ) satisfies additional conditions and that the generalized supersolution is a classical solution. When System (1.4) has a logistic source f ( u ) = r u − μ u 2 , Fujie et al. [9] proved that System (1.4) possesses global bounded classical solutions if r > χ 2 4 for 0 < χ ≤ 2 , or r > χ − 1 for χ > 2 in two-dimensional bounded domains. When v t = Δ v − v + u is replaced by 0 = Δ v − α 1 v + β 1 u , r , and μ are replaced by r ( x , t ) and μ ( x , t ) , respectively, and r ( x , t ) and μ ( x , t ) are Hölder continuous in t ∈ R with exponent γ > 0 uniformly with respect to x ∈ Ω ¯ , continuous in x ∈ Ω ¯ uniformly with respect to t ∈ R , and there are positive constants r i , μ i ( i = 1 , 2 ) such that 0 < r 1 ≤ r ( x , t ) ≤ r 2 and 0 < μ 1 ≤ μ ( x , t ) ≤ μ 2 , Kurt and Shen [14] showed that in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large χ . In addition, the solutions are shown to be uniformly bounded under the conditions r inf > α 1 χ 2 4 for 0 < χ ≤ 2 , or r inf > α 1 ( χ − 1 ) for χ > 2 , where r inf = inf x ∈ Ω ¯ , t ∈ R r ( x , t ) . For the fully parabolic system, Zhao and Zheng [49] proved that System (1.4) possesses a global bounded classical solution if r > χ 2 4 for 0 < χ ≤ 2 , or r > χ − 1 for χ > 2 in two-dimensional bounded domains. Based on existent results in [49], Zheng et al. [52] showed that the global bounded solution ( u , v ) exponentially converges to the steady state ( r μ , r μ ) . Zhao and Zheng [51] generalized their own work [49] to the higher-dimensional case. When r u − μ u 2 is replaced by r u − μ u k , Zhao [48] proved that System (1.4) admits globally bounded classical solutions whenever χ ∈ ( 0 , min { 1 2 , 1 2 ( N − 1 ) } ) . This means that the singular sensitive coefficient χ suitably small does benefit the global boundedness of classical solutions, which is independent of the dampening exponent k > 1 in the logistic kinetics. For more recent outcomes, one can see [1,5,6,10,15,18,28,42,46,47].

This article is concerned with the following chemotaxis system with singular sensitivity and signal production:

where Ω ⊂ R 2 is a bounded convex domain with smooth boundary ∂ Ω and ∂ ∂ ν denotes the derivative with respect to the outer normal of ∂ Ω , u represents the cell density, and v denotes the concentration of the chemical signal. χ > 0 , θ > 1 is a given parameter and initial data u 0 and v 0 are the known functions satisfying

The additional external production of the signal chemical

Recently, Ren and Ma [29] have proved that if θ > 2 , System (1.5) possesses at least one global weak solution, which is locally bounded in the sense that

When the first equation in (1.6) is replaced by u t = ∇ ⋅ ( u θ − 1 ∇ u ) − χ ∇ ⋅ u v ∇ v − u v + B 1 ( x , t ) and the second equation in (1.6) is replaced by v t = Δ v − v + u v + B 2 ( x , t ) , Rodríguez and Winkler [30] obtained the only result on the global boundedness of weak solution to urban crime nonlinear diffusion system in the two-dimensional case when θ > 3 2 .

Now, we state the main result in this article.

In this article, we use symbols C i and c i ( i = 1 , 2 , … ) as some generic positive constants that may vary in the context. For simplicity, u ( x , t ) is written as u , the integral ∫ Ω u ( x ) d x is written as ∫ Ω u ( x ) , and ∫ 0 t ∫ Ω u ( x , t ) d x d t is written as ∫ 0 t ∫ Ω u ( x , t ) .

The contents of this article are as follows. In Section 2, we first introduce the concept of weak solutions and then give the global existence result for System (1.5). In Section 3, we give some fundamental estimates for the solution to System (1.5) and prove Theorem 1.1.

2 Preliminaries

Under the assumptions of u , the first equation of System (1.5) may be degenerate at u = 0 . Therefore, System (1.5) does not allow for classical solvability in general as the well-known porous medium equations. We introduce the following definition of weak solutions.

In order to construct weak solutions by an approximation procedure, we introduce the following regularized problems:

for ε ∈ ( 0 , 1 ) . All of these problems (2.3) are, indeed, globally solvable in the classical sense.

3 Proof of Theorem 1.1

In this section, we will obtain the integral inequalities of u ε over Ω , whose core is to improve the regularity of v ε with regard to W 1 , q ( Ω ) for any q > 2 and to obtain the L ∞ -boundedness of u ε by applying a standard Moser-type recursive argument.