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In this paper, we investigate the Keller–Segel system coupled with the Navier–Stokes equations in a bounded domain Ω ⊂ R 2 as follows n t + u · ∇ n = Δ n - ∇ · (n ∇ c) - κ n 2 , c t + u · ∇ c = Δ c - n c , v t + u · ∇ v = Δ v - γ v + n , u t + (u · ∇) u + ∇ π = Δ u - n f , ∇ · u = 0. We establish the global existence of classical solutions for the initial-boundary value problem in a bounded domain under mild assumption on the initial data. [ABSTRACT FROM AUTHOR]

This paper investigates the fast chemical diffusion limit from a parabolic-parabolic Keller-Segel system to the corresponding parabolic-elliptic Keller-Segel system by constructing approximate solutions with an appropriate order via an asymptotic expansion. Nonlinear stability of the precise initial layer is characterized with an exact convergence rate by using basic energy method. [ABSTRACT FROM AUTHOR]

This paper is concerned with a quasilinear parabolic–parabolic–elliptic chemotaxis system u t = ∇ · (φ (u) ∇ u - ψ (u) ∇ v) + a u - b u γ , x ∈ Ω , t > 0 , v t = Δ v - v + w γ 1 , x ∈ Ω , t > 0 , 0 = Δ w - w + u γ 2 , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R n (n ≥ 1) , where a , b , γ 1 , γ 2 > 0 , γ > 1 , φ and ψ are nonlinear functions satisfying φ (s) ≥ a 0 (s + 1) α and | ψ (s) | ≤ b 0 s (1 + s) β - 1 for all s ≥ 0 with a 0 , b 0 > 0 and α , β ∈ R. When β + γ 1 γ 2 < max { n + 2 n + α , γ } , then the system has a classical solution which is globally bounded in time. Moreover, when β + γ 1 γ 2 = max { n + 2 n + α , γ } , it has been shown that the existence of global bounded classical solution depends on the size of coefficient b and initial data u 0. Furthermore, we consider a specific system with γ 1 = 1 , γ 2 = κ and γ = κ + 1 for κ > 0. If b > 0 is sufficiently large, the global classical solution(u, v, w) exponentially converges to the steady state ((a b) 1 κ , a b , a b) in L ∞ norm as t → ∞ , where convergence rate is explicitly expressed in terms of the system parameters. [ABSTRACT FROM AUTHOR]

This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Segel system with signal-dependent diffusion and chemotaxis { u t = ∇ ⋅ (γ (v) ∇ u − u ϕ (v) ∇ v) , x ∈ Ω , t > 0 , v t = d Δ v − v + u , x ∈ Ω , t > 0 in a bounded domain Ω ⊂ R N (N ≤ 4) with smooth boundary, where the density-dependent motility functions γ (v) and ϕ (v) denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in (J. Theor. Biol. 30:225–234, 1970) to describe the aggregation phase of Dictyostelium discoideum cells, where the two motility functions satisfy a proportional relation ϕ (v) = (α − 1) γ ′ (v) with α > 0 denoting the ratio of effective body length (i.e. distance between receptors) to the step size. The major technical difficulty in the analysis is the possible degeneracy of diffusion. In this work, we show that if γ (v) > 0 and ϕ (v) > 0 are smooth on [ 0 , ∞) and satisfy inf v ≥ 0 d γ (v) v ϕ (v) (v ϕ (v) + d − γ (v)) + > N 2 , then the above Keller-Segel system subject to Neumann boundary conditions admits classical solutions uniformly bounded in time. The main idea of proving our results is the estimates of a weighted functional ∫ Ω u p v − q d x for p > N 2 by choosing a suitable exponent p depending on the unknown v , by which we are able to derive a uniform L ∞ -norm of v and hence rule out the diffusion degeneracy. [ABSTRACT FROM AUTHOR]

We are concerned with the Keller–Segel–Navier–Stokes system ρ t + u · ∇ ρ = Δ ρ - ∇ · (ρ S (x , ρ , c) ∇ c) - ρ m , (x , t) ∈ Ω × (0 , T) , m t + u · ∇ m = Δ m - ρ m , (x , t) ∈ Ω × (0 , T) , c t + u · ∇ c = Δ c - c + m , (x , t) ∈ Ω × (0 , T) , u t + (u · ∇) u = Δ u - ∇ P + (ρ + m) ∇ ϕ , ∇ · u = 0 , (x , t) ∈ Ω × (0 , T) subject to the boundary condition (∇ ρ - ρ S (x , ρ , c) ∇ c) · ν = ∇ m · ν = ∇ c · ν = 0 , u = 0 in a bounded smooth domain Ω ⊂ R 3 . It is shown that this problem admits a global classical solution with exponential decay properties when S ∈ C 2 (Ω ¯ × [ 0 , ∞) 2) 3 × 3 satisfies | S (x , ρ , c) | ≤ C S for some C S > 0 , and the initial data satisfy certain smallness conditions. [ABSTRACT FROM AUTHOR]

For n≥2 let Ω⊂Rn be a bounded domain with smooth boundary as well as some nonnegative functions 0≢u0∈W1,∞(Ω) and v0∈W1,∞(Ω). With ε∈(0,1) we want to know in which sense (if any!) solutions to the parabolic-parabolic system⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩ut=∇⋅((u+1)m−1∇u)−∇⋅(u∇v)in Ω×(0,∞),εvt=Δv−v+uin Ω×(0,∞),∂u∂ν=∂v∂ν=0on ∂Ω×(0,∞),u(⋅,0)=u0, v(⋅,0)=v0in Ωconverge to those of the system where ε=0 and where the initial condition for v has been removed. We will see in our theorem that indeed the solutions of these systems converge in a meaningful way if m>1+n−2n without the need for further conditions, e. g. on the size of ∥u0∥Lp(Ω) for some p∈[1,∞]. [ABSTRACT FROM AUTHOR]

In this paper we consider a d-dimensional (d=1,2) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order α∈(0,2). We prove uniform in time boundedness of its solution in the supercritical range α>d(1−c), where c is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for ∥u(t)−u∞∥L∞→0, where u∞≡1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result. [ABSTRACT FROM AUTHOR]

We consider a two dimensional parabolic–elliptic Keller–Segel equation with a fractional diffusion of order α ∈ ( 0 , 2 ) and a logistic term. In the case of an analogous problem with standard diffusion, introduction of the logistic term, well motivated by biological applications, results in global smoothness of solutions (i.e. suppression of blowup ), compare Tello & Winkler [48] . We show that this phenomenon extends into potentially less regular case of fractional diffusions. Namely, we obtain existence of global in time regular solutions emanating from initial data with no size restrictions for c < α < 2 , where c ∈ ( 0 , 2 ) depends on the equation's parameters. For an even wider range of α ′ s , we prove existence of global in time weak solution for general initial data. [ABSTRACT FROM AUTHOR]

We consider the parabolic-parabolic Keller-Segel system with singular sensitivity and logistic source: $$ u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v) +ru-\mu u^2$$ , $$v_t=\Delta v-v+u$$ under the homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset \mathbb {R}^2$$ , $$\chi ,\mu >0$$ and $$r\in \mathbb {R}$$ . It is proved that the system exists globally bounded classical solutions if $$r>\frac{\chi ^2}{4}$$ for $$0<\chi \le 2$$ , or $$r>\chi -1$$ for $$\chi >2$$ . [ABSTRACT FROM AUTHOR]

The present paper deals with the parabolic–parabolic Keller–Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial data with finite massM<8π, finite second log-moment, and finite entropy. The aim of the paper is twofold: (1) We prove the uniqueness of the “free energy” solution. The proof uses a DiPerna–Lions renormalizing argument, which makes possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the criticalL4∕3Lebesgue norm similarly as for the 2dvorticity Navier–Stokes equation. (2) We prove a radially symmetric and polynomial weightedexponential stability of the self-similar profile in the quasiparabolic–elliptic regime. The proof is based on a perturbation argument, which takes advantage of the exponential stability of the self-similar profile for the parabolic–elliptic Keller–Segel equation established by Campos–Dolbeault and Egana–Mischler. [ABSTRACT FROM AUTHOR]

In the present article we consider several issues concerning the doubly parabolic Keller–Segel system (1.1)–(1.2) in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε > 0 is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term ( α = 0 ) the solutions behave like self-similar solutions, while in the presence of the degradation term ( α > 0 ) the global solutions behave like the heat kernel. [ABSTRACT FROM AUTHOR]