Xuegang Hu

Boundedness in the higher dimensional attractionrepulsion chemotaxis-growth system

This paper deals with an attractionrepulsion chemotaxis system with logistic source {ut=u(uv)+(uw)+f(u),(x,t)(0,),vt=v1v+1u,(x,t)(0,),wt=w2w+2u,(x,t)(0,), under homogeneous Neumann boundary conditions in a smooth bounded domain Rn(n3) with nonnegative initial data (u0,v0,w0)W1,()W2,()W2,(), where >0, >0, i>0, i>0(i=1,2) and f(u)auu2 with a0 and >0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n3, 1=2 and there exists 0>0 such that 1+2<0. The main aim of this paper is to solve the higher-dimensional boundedness question addressed by Xie and Xiang in [IMA J. Appl. Math. 81 (2016) 165198].

Boundedness in a quasilinear chemotaxis model with consumption of chemoattractant and logistic source

This paper is devoted to the chemotaxis system with logistic source endowed with homogeneous Neumann boundary conditions and suitable initial conditions, where ( ) is a bounded and smooth domain. Here we assume that where , , and . Moreover, we will require for all with some , and . Then if one of the following cases hold:(i) , where (ii) ;(iii) and b is sufficiently large, it is proved that the system possesses a unique global bounded classical solution.

Persistence property in a two-species chemotaxis system with two signals

This paper deals with a two-species chemotaxis system with two different signals under homogeneous Neumann boundary conditions in a bounded convex domain with the non-negative initial data. This system is a generalization of the classical Keller-Segel chemotaxis models to the case of two species which are attracted by two different chemical signals. Under suitable conditions, it is proved that for any non-negative global classical solutions, the masses of two species do not extinct at any time.

A note on the boundedness in a chemotaxis-growth system with nonlinear sensitivity

This paper deals with a parabolic-elliptic chemotaxis-growth system with nonlinear sensitivity { ut = ∆u− χ∇ · (ψ(u)∇v) + f (u), (x, t) ∈ Ω× (0, ∞), 0 = ∆v− v + g(u), (x, t) ∈ Ω× (0, ∞), under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ Rn (n ≥ 1), where χ > 0, the chemotactic sensitivity ψ(u) ≤ (u + 1)q with q > 0, g(u) ≤ (u + 1)l with l ∈ R and f (u) is a logistic source. The main goal of this paper is to extend a previous result on global boundedness by Zheng et al. [J. Math. Anal. Appl. 424(2015), 509–522] under the condition that 1 ≤ q + l < 2 n + 1 to the case q + l < 1.

Dead-core rate for the fast diffusion equation with strong absorption in higher dimensional cases

This paper deals with the dead-core rate for the fast diffusion equation with a strong absorption in several space dimensions ut=Δum−up,(x,t)∈Ω×(0,∞),where 0 < p < m < 1 and Ω=B(0,1)={x∈RN:|x|<1} with N ≥ 1. By using the self-similar transformation technique and the Zelenyak method, in higher dimensional radially symmetric cases, we prove that the dead-core rate is not self-similar. Moreover, we also give the precise estimates on the single-point final dead-core profile. Finally, when the absorption term −up is replaced by −a(x,t)up, then we derive that the dead-core rate can turn into the corresponding ODE rate if the coefficient function a(x, t) is a suitable uniformly bounded positive function, which implies that a(x, t) plays an important role in the study of the dead-core rate. The main aim of this paper is to extend the results obtained by Guo et al. (2010) to the higher dimensional radially symmetric case.

Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

Abstract This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity u t = Δ u − χ ∇ ⋅ u ∇ ln v + r u − μ u 2 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t = Δ v − v + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 , where the parameters χ , μ > 0 and r ∈ R . Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.