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This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + μ 1 u (1 − u − a 1 v) , t > 0 , x ∈ Ω , v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + μ 2 v (1 − v − a 2 u) , t > 0 , x ∈ Ω , w t = Δ w − w + u + v , t > 0 , x ∈ Ω , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , t > 0 , x ∈ ∂ Ω , u (0 , x) = u 0 (x) , v (0 , x) = v 0 (x) , w (0 , x) = w 0 (x) , x ∈ Ω , where Ω ⊂ R N (N ≥ 2) is a bounded smooth convex domain, and the parameters χ 1 , χ 2 , μ 1 , μ 2 , a 1 and a 2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ 1 , χ 2 ∈ (0 , 1 2) for N = 2 , 3 or χ 1 , χ 2 ∈ (0 , N − 2 N − 1) for N ≥ 4. Moreover, if min ⁡ { μ 1 , μ 2 } > max ⁡ { χ 1 , χ 2 } 2 4 , we obtain the uniformly lower bound for w. Finally, when χ 1 , χ 2 are suitably small, it is proved that if 0 < a 1 , a 2 < 1 , then the solution (u , v , w) converges to (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 2 − a 1 − a 2 1 − a 1 a 2 ) in L ∞ norm as t → ∞ ; if 0 < a 2 < 1 ≤ a 1 , then the solution (u , v , w) converges to (0 , 1 , 1) in L ∞ norm as t → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for a quasi-linear hyperbolic-parabolic chemotaxis system modelling vasculogenesis. As Liu, Peng and Wang pointed out in [20] , the smooth solutions of Cauchy problem for this system globally exist and converge to the shifted nonlinear diffusion waves. It is worth noting that due to the difficulty in constructing a group of correction functions to eliminate the gaps between the original solutions and the diffusion waves at infinity, they got their results under the stiff conditions m ± = 0 and ϕ ± = a b ρ ±. However, by a deep observation, we realize that these two conditions can be removed. In this paper, by making full use of the results obtained in [20] , and with the help of a group of new correction functions, we get some more general results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide the analysis result for the partial differential equations arising from the rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. The rigorous derivation is divided into two articles. In this paper, we establish the solution theory of the degenerate parabolic-elliptic Keller-Segel problem and its non-local version, which will be used in the second paper for the discussion of the mean-field limit. A parabolic regularized system is introduced to bridge the stochastic particle model and the degenerate Keller-Segel system. We derive the existence of the solution to this regularized system by constructing approximate solutions, giving uniform estimates and taking the limits, where a crucial step is to obtain the L ∞ Bernstein type estimate for the gradient of the approximate solution. Based on this, we obtain the well-posedness of the corresponding non-local equation through perturbation method. Finally, the weak solution of the degenerate Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions lemma. [ABSTRACT FROM AUTHOR]

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics { (n 1) t + u ⋅ ∇ n 1 = Δ n 1 − χ ∇ ⋅ (n 1 ∇ w) + n 1 (λ 1 − μ 1 n 1 + a n 2) , x ∈ Ω , t > 0 , (n 2) t + u ⋅ ∇ n 2 = Δ n 2 + ξ ∇ ⋅ (n 2 ∇ z) + n 2 (λ 2 − μ 2 n 2 − b n 1) , x ∈ Ω , t > 0 , w t + u ⋅ ∇ w = Δ w − w + n 2 , x ∈ Ω , t > 0 , u ⋅ ∇ z = Δ z − z + n 1 , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + (n 1 + n 2) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions for n 1 , n 2 , w and z in three-dimensional bounded domains and no-slip boundary conditions for u , this is ∂ n 1 ∂ ν = ∂ n 2 ∂ ν = ∂ w ∂ ν = ∂ z ∂ ν = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , where χ > 0 , ξ > 0 , μ 1 ≥ 0 , μ 2 ≥ 0 , λ 1 ≥ 0 , λ 2 ≥ 0 , a ≥ 0 , b ≥ 0 and ϕ ∈ W 2 , ∞ (Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution. [ABSTRACT FROM AUTHOR]

In this paper, I consider a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the center-stable and center unstable manifolds, one gets the existence of a positive traveling wave connecting the two constant steady states (0 , 0) and (b , α b β) with a small wave speed ϵc. In addition, the traveling wave is monotone for b ≥ 1 and is not monotone for 0 < b < 1. Moreover, by the spectral analysis it shows that the above traveling wave is spectrally unstable in some exponentially weighted spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis system with power-type singular sensitivities: u t = Δ u − χ ∇ ⋅ (u v α ∇ v) and v t = Δ v − v + u in a bounded domain Ω ⊂ R 2 , with χ > 0 and α > 0. It is notable that, whenever α = 0 , this system reduces to the minimal Keller-Segel model; the persistent Dirac-type singularities are known to occur at least in radial parabolic-elliptic setting. In this work, our first result shows that for any χ > 0 , if either α ≥ 1 4 , or α ∈ (0 , 1 4) and ‖ u 0 ‖ L 1 is small appropriately, then the corresponding initial-boundary value problem possesses a global generalized solution (u , v) with the property that u belongs to L 1 (Ω × (0 , T)) for any T > 0 , which further implies that the general sensitivities 1 v α , with α ≥ 1 4 , can rule out the emergence of Dirac-type singularities. Our second result indicates that, if α ∈ (0 , 1 2) and ‖ u 0 ‖ L 1 is small properly, then such global generalized solution becomes bounded and smooth at least eventually, and approaches the spatial equilibria in the large time limit. [ABSTRACT FROM AUTHOR]

In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n c ∇ c) + n c , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + (u ⋅ ∇) u = Δ u + ∇ P + n ∇ Φ , x ∈ Ω , t > 0 , subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain Ω ⊂ R 2 , where + n c in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach (1 | Ω | ∫ Ω (n 0 + c 0) , 0 , 0) as t → ∞. As compared to a precedent result obtained without cell proliferation, the complexity caused by + n c in the first equation requires some conditions on the initial data c 0 rather only on n 0 in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by + n c. [ABSTRACT FROM AUTHOR]

We consider classical solutions to the chemotaxis system with logistic source, a u − μ u 2 , under nonlinear Neumann boundary conditions ∂ u ∂ ν = | u | p with p > 1 in a smooth convex bounded domain Ω ⊂ R n , where n ≥ 2. This paper aims to show that if p < 3 2 , and μ > 0 , n = 2 , or μ is sufficiently large when n ≥ 3 , then the parabolic-elliptic chemotaxis system admits a unique nonnegative global-in-time classical solution that is bounded in Ω × (0 , ∞). The similar result is also true if p < 3 2 , n = 2 , and μ > 0 or p < 7 5 , n = 3 , and μ is sufficiently large for the parabolic-parabolic chemotaxis system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the degenerate system { u t = ∇ ⋅ (f (u , w) ∇ u − g (u) ∇ v) , x ∈ Ω , t > 0 , v t = Δ v + w z , x ∈ Ω , t > 0 , w t = − w z , x ∈ Ω , t > 0 , z t = Δ z − z + u , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R N (N ≥ 2) under the no-flux boundary condition for u and the homogeneous Neumann boundary condition for v , z with non-negative initial data u 0 , v 0 , w 0 , z 0. Here, the diffusivity f and the sensitivity g are assumed to fulfill f (u , w) ≥ u m − 1 (m > 1), 0 ≤ g (u) ≤ u α (α ∈ R). It is shown that if α + 1 < m + 4 N (N ≥ 2) or α + 1 = m + 4 N (N ≥ 3) with small mass of u 0 , then the system possesses a global bounded weak solution which converges to the constant equilibrium in the weak⁎ topology in L ∞ (Ω) as t → ∞. [ABSTRACT FROM AUTHOR]

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model. [ABSTRACT FROM AUTHOR]

This paper deals with the singular chemotaxis-fluid system with indirect signal production or consumption n t + u ⋅ ∇ n = Δ n − χ ∇ ⋅ (n v ∇ v) ; v t + u ⋅ ∇ v = Δ v + g (v , w) ; w t + u ⋅ ∇ w = Δ w − w + n ; u t + (u ⋅ ∇) u = Δ u − ∇ P + n ∇ Φ ; ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in a bounded and smooth domain Ω ⊂ R 2 with no-flux/no-flux/no-flux/Dirichlet boundary conditions, where Φ ∈ W 2 , ∞ (Ω). In the cases signal production (i.e., g (v , w) = − v + w) and consumption (i.e., g (v , w) = − v w), it is proved that for any appropriately regular initial data, the associated initial-boundary value problem possesses a global classical solution for any χ > 0. It is worth mentioning that the global existence of classical solution for χ ∈ (0 , 1) or generalized solutions for arbitrary χ > 0 was obtained in the previously known results on the corresponding system with direct signal production, and that only the global generalized solutions for arbitrary χ > 0 were established in a precedent concerning the corresponding system with direct signal consumption. In comparison to these results for the case of direct signal production or consumption, our results rigorously confirm that both the indirect signal production and consumption mechanisms authentically contribute to the global solvability of the two-dimensional singular chemotaxis-fluid system. [ABSTRACT FROM AUTHOR]

This paper aims to provide the rigorous convergence analysis of the chemotaxis system { ∂ n ϵ ∂ t = Δ n ϵ − ∇ ⋅ (n ϵ ∇ c ϵ) , τ 1 ∂ c ϵ ∂ t = Δ c ϵ − c ϵ + w ϵ , τ 2 ∂ w ϵ ∂ t = − w ϵ + n ϵ in a smoothly bounded planar domain Ω as τ 1 → 0 and (or) τ 2 → 0. Precisely, we reveal that actually the indirect signal properties are strong enough to allow for a uniformly global-in-time convergence in terms of τ 1 = τ 2 = ϵ ↘ 0 and τ 1 = 1 , τ 2 = ϵ ↘ 0 for ∫ Ω n ϵ (⋅ , 0) < 4 π , which correspond to the classical Keller-Segel systems with direct signal production, and of τ 1 = ϵ ↘ 0 , τ 2 = 1 for suitable small initial data. As a byproduct, we show that for any p ≥ 2 , the L p -norm of n ϵ blows up if ∫ Ω n ϵ (⋅ , 0) ∈ (4 π , ∞) ∖ 4 π N , which improves the existing results that the solution is unbounded in the sense of L ∞ -norm of n ϵ. [ABSTRACT FROM AUTHOR]

This paper is concerned with the two-species chemotaxis-fluid system (1.0) { v 1 t + u ⋅ ∇ v 1 = Δ v 1 − χ ∇ ⋅ (v 1 ∇ w) + v 1 (λ 1 − μ 1 v 1 + a v 2) , x ∈ Ω , t > 0 , v 2 t + u ⋅ ∇ v 2 = Δ v 2 + ξ ∇ ⋅ (v 2 ∇ z) + v 2 (λ 2 − μ 2 v 2 − b v 1) , x ∈ Ω , t > 0 , u ⋅ ∇ w = Δ w − w + v 2 , x ∈ Ω , t > 0 , u ⋅ ∇ z = Δ z − z + v 1 , x ∈ Ω , t > 0 , u t + ∇ P + κ (u ⋅ ∇) u = Δ u + (v 1 + v 2) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω is a bounded domain in R N (N ∈ { 2 , 3 }) with smooth boundary ∂Ω, κ ∈ R and ϕ ∈ W 2 , ∞ (Ω). The parameters χ , ξ , a , b , λ 1 , λ 2 , μ 1 and μ 2 are positive. Under appropriate regularity assumptions on the initial data, we obtain the global existence, uniqueness, and boundedness of classical solution to this system for any small μ 1 > 0 and μ 2 > 0. Our results generalized and improved previous results, and partially results are new. [ABSTRACT FROM AUTHOR]

This note is to fix some small gaps in the proofs of some of the results in our paper [R.B. Salako and W. Shen, Journal of Differential Equations 262 (2017) 5635–5690]. [ABSTRACT FROM AUTHOR]

This paper deals with a doubly degenerate nutrient taxis system given by (⋆) { u t = ∇ ⋅ (u v ∇ u) − χ ∇ ⋅ (u α v ∇ v) + ℓ u v , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , in a smoothly bounded convex domain Ω ⊂ R n , n = 2 , 3 , with zero-flux boundary conditions, where α > 0 , χ > 0 and ℓ ≥ 0. It is known that when α = 2 , for which, in fact, the system was proposed by Leyva et al. (2013) to model the aggregation patterns of colonies of Bacillus subtilis identified on the surface of thin agar plates, a globally defined weak solution to (⋆) can be obtained, either in the case n = 1 , or for its corresponding spatially two-dimensional version, the latter case additionally requiring certain restrictions on the size of initial data. The present paper complements the results, asserting that under the assumption α ∈ { (1 , 3 2) if n = 2 , and (7 6 , 13 9) if n = 3 , for all sufficiently smooth initial data, (⋆) possesses such a global weak solution (u , v). [ABSTRACT FROM AUTHOR]

This work considers the chemotaxis-growth system ∂ t u = Δ u − ∇ ⋅ (u ∇ v) + u − u α , ∂ t v = Δ v − v + w , ∂ t w = u − δ w , } in R d × R + , where d ≥ 2 , α ≥ 2 and δ > 0. In a smooth bounded domain Ω, the global boundedness of smooth solutions to this system depends on | Ω | , which implies that the methods on bounded domain are not feasible for the Cauchy problem. Recently, the second author of this paper and Zheng [13] proved boundedness of classical Keller-Segel system with logistic term in R d by a nice coupled structure. However, such a nice coupled structure disappears to this system. Using the coupled structure and damping effects, involving with special time weighted local-in-space energy-type functional and Littlewood-Paley decomposition, we prove the global-in-time existence and boundedness of classical solutions to this system in R d for α > d 2 , which gives new approaches on dealing with some complex equations by the idea of localization. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + u (a 1 − b 1 u − c 1 v) , x ∈ Ω v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + v (a 2 − b 2 v − c 2 u) , x ∈ Ω 0 = Δ w − μ w + ν u + λ v , x ∈ Ω ∂ u ∂ n = ∂ v ∂ n = ∂ w ∂ n = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, and χ i , a i , b i , c i (i = 1 , 2) and μ , ν , λ are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data u 0 , v 0 ∈ C 0 (Ω ¯) with u 0 + v 0 ≢ 0 , (0.1) has a unique globally defined classical solution (u (t , x ; u 0 , v 0) , v (t , x ; u 0 , v 0) , w (t , x ; u 0 , v 0)) with u (0 , x ; u 0 , v 0) = u 0 (x) and v (0 , x ; u 0 , v 0) = v 0 (x) in any space dimensional setting with any positive constants χ i , a i , b i , c i (i = 1 , 2) and μ , ν , λ. Moreover, we prove that there is χ ⁎ (μ , χ 1 , χ 2) > 0 satisfying χ ⁎ (μ , χ 1 , χ 2) = { μ χ 2 4 if 0 < χ < 2 μ (χ − 1) if χ ≥ 2 , when χ 1 = χ 2 : = χ and χ ⁎ (μ , χ 1 , χ 2) ≤ min ⁡ { μ χ 2 + μ (χ 1 − χ 2) 2 4 , μ χ 1 + μ (χ 2 − χ 1) 2 4 } , when χ 1 ≠ χ 2 such that the condition min ⁡ { a 1 , a 2 } > χ ⁎ (μ , χ 1 , χ 2) implies lim sup t → ∞ ‖ u (t , ⋅ ; u 0 , v 0) + v (t , ⋅ ; u 0 , v 0) ‖ ∞ ≤ M ⁎ and lim inf t → ∞ inf x ∈ Ω ⁡ (u (t , x , u 0 , v 0) + v (t , x ; u 0 , v 0)) ≥ m ⁎ for some positive constants M ⁎ , m ⁎ independent of u 0 , v 0 , the latter is referred to as combined pointwise persistence. [ABSTRACT FROM AUTHOR]

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain Ω. This system is a simplified version of the chemotaxis model introduced by Keller and Segel in [14]. With the aid of priori L ∞ -estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset ω ⊂ Ω at some time T > 0 , then the solution is identically zero. [ABSTRACT FROM AUTHOR]

This paper deals with the initial value problem for a parabolic attraction-repulsion chemotaxis system in R 2 : { ∂ t u = Δ u − ∇ ⋅ (u ∇ (β 1 v 1 − β 2 v 2)) , t > 0 , x ∈ R 2 , ∂ t v j = Δ v j − λ j v j + u , t > 0 , x ∈ R 2 (j = 1 , 2) , u (0 , x) = u 0 (x) , v j (0 , x) = v j 0 (x) , x ∈ R 2 (j = 1 , 2) with positive parameters β 1 , β 2 , λ 1 , λ 2 and nonnegative initial data u 0 , v 10 , v 20. It is well known that the sign of β 1 − β 2 and the mass on u 0 play a crucial role in the global boundedness of the system. Specifically, in the attractive dominant case β 1 > β 2 , the uniform boundedness of nonnegative solutions has been guaranteed under the condition ∫ R 2 u 0 d x < 4 π. Also, in the balance case β 1 = β 2 , it has been proven that every nonnegative solution is bounded uniformly in time without any restriction on the mass u 0. In this paper, we discuss the global existence and boundedness of nonnegative solutions to the system in the repulsive dominant case β 1 < β 2. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of signal-dependent motility Keller-Segel systems { u t = ∇ ⋅ (D (v) ∇ u − S (v) u ∇ v) + f (u) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under no-flux boundary conditions in a smooth bounded domain Ω ⊂ R n. This model describes the aggregation phase of Dictyostelium discoideum cells in response to the secreted chemical signal cyclic adenosine monophosphate. The parameter functions D and S satisfy D , S ∈ C 3 ((0 , + ∞)) , and there exist constants k , K , x 0 > 0 such that k ≤ D (s) ≤ K , | S (s) | ≤ K for all s ∈ (x 0 , + ∞). The generalized logistic source f ∈ C 1 ([ 0 , + ∞)) satisfies f (0) = 0 and there exists l > max ⁡ { 2 (n + 2) n + 4 , 2 (n − 1) n } such that lim s → + ∞ ⁡ f (s) s l = − ∞. The main result of the paper is to show that the existence of the global-in-time solution in an appropriately generalized sense to the model for all sufficiently smooth initial data which indicates that superlinear damping can rule out blow-up for such kind of models. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Keller-Segel-Stokes system (⁎) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n f (| ∇ v | 2) ∇ v) , v t + u ⋅ ∇ v = Δ v − v + w , w t + u ⋅ ∇ w = Δ w − w + n , u t = Δ u + ∇ P + n ∇ ϕ , ∇ ⋅ u = 0 , under no-flux/no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded domains Ω ⊂ R 3 , with given suitably regular functions f and ϕ , as well as f satisfies f (| ∇ v | 2) ≤ K f (1 + | ∇ v | 2) − α 2 . It is shown that for all suitably regular initial data an associated initial-boundary value problem (⁎) possesses a globally defined bounded classical solution provided α > 1 4. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if α > 1 2 in [50]. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the three-dimensional Keller-Segel-Stokes system. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis consumption model { u t = Δ (u ϕ (v)) + a u − b u γ , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1). Here the parameters a > 0 , b > 0 and γ > 1 , and the motility function ϕ satisfies ϕ ∈ C 3 ([ 0 , ∞)) with ϕ (s) > 0 for all s ≥ 0. For all suitably regular initial data, if one of the following cases holds: (i) n ≤ 2 , γ > 1 ; (ii) n ≥ 3 , γ > 2 ; (iii) n ≥ 3 , γ = 2 and b is large, then this system possesses global bounded classical solutions, while if n ≥ 3 and γ ∈ (1 , 2 ] , then this system admits at least one global weak solution, which becomes smooth after some waiting time and which will converge to the constant equilibrium ((a b) 1 γ − 1 , 0). [ABSTRACT FROM AUTHOR]

We study the global existence of the parabolic-parabolic Keller–Segel system in R d , d ≥ 2. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter τ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler et al. (2015) [7] and Corrias et al. (2014) [12]. Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in τ , when τ ≫ 1 : we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis-(Navier-)Stokes system { n t + u ⋅ ∇ n = ∇ ⋅ (φ (c) ∇ n) − ∇ ⋅ (n χ (c) ∇ c) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u = Δ u − ∇ P + n ∇ ψ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , subject to no-flux boundary conditions in a bounded domain Ω ⊂ R 2 with smooth boundary, with a given smooth potential ψ : Ω → R and κ ∈ R. The density-dependent motility φ (c) and the chemotactic function χ (v) are assumed to satisfy some further technical conditions. It is shown that the problem possesses a global classical solution (n κ , c κ , u κ). Furthermore, we prove that these solutions stabilize uniformly to ( n 0 ‾ , 0 , 0) exponentially with respect to t , where n 0 ‾ : = 1 | Ω | ∫ Ω n 0. Finally, it is also shown that there exists a constant μ > 0 such that for all κ ∈ (0 , 1) ‖ n (κ) (⋅ , t) − n (0) (⋅ , t) ‖ L ∞ (Ω) + ‖ c (κ) (⋅ , t) − c (0) (⋅ , t) ‖ L ∞ (Ω) + ‖ u (κ) (⋅ , t) − u (0) (⋅ , t) ‖ L ∞ (Ω) ≤ C κ 1 3 e − μ t for all t > 0 , where C > 0 is a constant independent of κ and t. [ABSTRACT FROM AUTHOR]

In this paper, we study the following reaction-diffusion-advection system { u t = d 1 ∇ ⋅ (D (u) ∇ u) − χ ∇ ⋅ (S (u) ∇ v) + F (u , w) , (x , t) ∈ Ω × (0 , ∞) , v t = d 2 Δ v − μ 1 v + s 1 g (u) w , (x , t) ∈ Ω × (0 , ∞) , w t = d 3 Δ w + s 2 − μ 2 w − λ w h (u) w , (x , t) ∈ Ω × (0 , ∞) , under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R n , which describes a cross-talk between antigens and immune cells via chemokines in the immune system, where d 1 , d 2 , d 3 , s 1 , s 2 , λ w , μ 1 , μ 2 , χ are positive parameters, the functions D , S , g , h ∈ C 2 ([ 0 , + ∞)) and F belongs to C 2 ([ 0 , + ∞) × [ 0 , + ∞)). When n ≥ 1 , we study the global existence and boundedness of classical solutions for the above system under some suitable conditions of the functions D , S , F , g and h. When n ≤ 3 , we consider the stability and instability of the above system based on the spectral analysis, energy estimate and bootstrap technique. Moreover, some numerical simulations are carried out to support our theoretical results and also find some new interesting phenomena, such as complex spatial-temporal patterns. Our results not only extend the previous ones of [25] , but also involve some new conclusions. [ABSTRACT FROM AUTHOR]

Keller–Segel system models prototype of the population-based chemotaxis, which describes the oriented or partially oriented movement of cells in response to a chemical signal produced by the cells themselves. This paper focuses on dynamics of a class of Keller–Segel systems with a large diffusion. We obtain several different kinds of traveling waves and characterize their spectral stability. The main tools are geometric singular perturbation theory, qualitative theory and spectral analysis via the method of nonlocal eigenvalue problem. [ABSTRACT FROM AUTHOR]

This paper deals with a chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u − χ ∇ ⋅ (u v ∇ v) + r u − μ u k , v t = Δ v − v + u in a smooth bounded domain Ω ⊂ R n with n ≥ 2 , where χ , r , μ > 0 and k > 1. It is proved that the system 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, independent of the dampening exponent k > 1 in the logistic kinetics. Moreover, the properties of the mass persistence for u and uniformly lower bound for v have been established. [ABSTRACT FROM AUTHOR]

In this paper, we study a chemotaxis system with singular sensitivity, indirect signal production and logistic source: u t = Δ u − χ ∇ ⋅ (u v − 1 ∇ v) + r u − μ u 2 ; τ v t = Δ v − α 1 v + β 1 w ; τ w t = Δ w − α 2 w + β 2 u , x ∈ Ω , t > 0 in a bounded and smooth domain Ω ⊂ R N (N ≥ 1) with no-flux boundary conditions, where α 1 , α 2 , β 1 , β 2 , χ , λ , μ are positive constants, τ = 0. Compared with previous work K. Fujie, M. Winkler, T. Yokota (2014) [7] , the novelty here is that logistic source is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in any space dimensional setting are not only in the space two dimensional setting. For any appropriately regular initial data, we prove that the system has a unique globally defined classical solution in any space dimensional setting even for arbitrarily large χ , and our result significantly improves and extends the result obtained in previously known ones. [ABSTRACT FROM AUTHOR]

In this paper, we study an attraction-repulsion Navier-Stokes system with consumption of chemoattractant: n t + u ⋅ ∇ n = Δ n − χ ∇ ⋅ (n ∇ c) + ξ ∇ ⋅ (n ∇ v) ; c t + u ⋅ ∇ c = Δ c − n c ; τ v t + u ⋅ ∇ v = Δ v − v + n ; u t + κ (u ⋅ ∇) u = Δ u + ∇ P + n ∇ ϕ ; ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in a bounded and smooth domain Ω ⊂ R 2 with no-flux/Dirichlet boundary conditions, where χ , ξ are positive constants, τ ∈ { 0 , 1 }. We present the global existence of attraction-repulsion Navier-Stokes system with τ = 1 and global boundedness of attraction-repulsion Navier-Stokes system with τ = 0 , and our result generalize the result obtained in previously known ones and partly result is new. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of bounded classical solutions to the attraction-repulsion chemotaxis system with logistic source (⋆) { u t = Δ u − χ ∇ ⋅ (u ∇ v) + ξ ∇ ⋅ (u ∇ w) + f (u) , x ∈ Ω , t > 0 , 0 = Δ v − β v + α u , x ∈ Ω , t > 0 0 = Δ w − δ w + γ u , x ∈ Ω , t > 0 in a smooth bounded domain Ω ⊆ R N (N ≥ 1) , subject to nonnegative initial data and homogeneous Neumann boundary conditions, where f (u) ≤ a − b u r for all u ≥ 0 with some a ≥ 0 , b > 0 and r ≥ 1. Here χ , α , ξ , β as well as γ and δ are positive constants. It is proved that the corresponding system (⋆) possesses a unique global bounded classical solution in the balance case χ α = ξ γ with r > 2 N − 2 N or, the attraction domination case ξ α > ξ γ with b ≥ (N − 2) + N (χ α − ξ γ) and r = 2 , respectively. The study of this paper improves the results in Li-Xiang (2016) [12] , Xu-Zheng (2018) [39] , Wang (2016) [26] as well as Zhao et al. (2017) [42] and Tello-Winkler (2007) [23] , in which, the assumption b > (N − 2) + N (χ α − ξ γ) (see Li-Xiang (2016) as well as Wang (2016) and Zhao et al. (2017)) or b > (N − 2) + N χ (see Tello-Winkler (2007)) or r > 2 N + 2 N + 2 (see Xu-Zheng (2018)) or r > N 2 + 4 N − N 2 (see Li-Xiang (2016)) are intrinsically required. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quasilinear two-species chemotaxis model with nonlinear signal production and nonlinear sensitivity. Under appropriate regularity assumption on the initial data, the global boundedness of classical solution is obtained. Our results partially generalize and improve some results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a chemotaxis-Stokes system with porous-media type cell diffusion ∇ ⋅ (n m − 1 ∇ n) in a bounded domain Ω ⊂ R N (N = 2 , 3) with smooth boundary. We prove that for the no-flux–saturation–no-slip boundary value and suitable regular initial data, the mild assumption m > 3 N − 2 2 N is sufficient for the global existence and boundedness of weak solutions, and in particular confirm the experimental observation in Tuval et al. (2005, PNAS). In comparison to the considerable literature, the novelty here is that we require signal to attain a prescribed saturation value throughout the boundary. This result will be the first step towards a qualitative theory for the boundary layer. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis. Under some suitable structural assumption on the pressure function, we first predict and derive the system admits a nonlinear diffusion wave in R driven by the damping effect. Then we show that the solution of the concerned system will locally and asymptotically converge to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergence rate of the nonlinear diffusion wave is algebraic. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain Ω ⊂ R 2 provided that the initial data is partially small. We first get a new blow-up criterion and then we use bootstrap argument to obtain the global existence of strong solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of traveling waves for a Keller-Segel model with singular chemotactic term and zero chemo-attractant diffusion, where the model and the waves are established to explain the propagation of bacteria pulses along a capillary tube observed in Adler's experiment [1]. By applying the detailed spectral analysis, Evans function method and special transformations and combining with some numerical simulations, in some range of the parameters all the waves are shown to be spectrally stable in some exponentially weighted spaces, and in other range of parameters all the waves are shown to be unstable in any exponentially weighted spaces. The local well-posedness of the classical positive solution to the Cauchy problem of the model is also obtained by applying semigroup argument and some special transformations. [ABSTRACT FROM AUTHOR]

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity. [ABSTRACT FROM AUTHOR]