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This work is concerned with the fractional-in-time parabolic–parabolic Keller–Segel system in a bounded domain Ω ⊂ R d ( d ≥ 2 ), for distinct fractional diffusions of the cells and chemoattractant. We prove results on existence, uniqueness, continuous dependence on the initial data and its robustness, continuation, and a blow-up alternative of solutions in Lebesgue spaces. Then, we use those results to show the existence of global solutions to the problem, when the chemoattractant diffusion is not slower than the cell diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and asymptotic behavior to a diffusion system which is non-local in time. As consequence of our theorems we deduce new results for the fractional-in-time Keller-Segel model. Our approach is intimately related with the Sonine kernels. [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 work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of RN$\mathbb {R}^N$. The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution. [ABSTRACT FROM AUTHOR]

In this article, a class of systems of cross-diffusion chemotaxis is studied. Using the entropy dissipation method and assuming that chemotactic sensitivity mainly separates cell density and chemical signal, it is established the existence of global solutions with cross-diffusion effects. The global existence and uniqueness of the classical solutions of this system is proved by the contraction mapping principle using a priori estimates and Schauder type estimates for parabolic equations. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using q -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function χ ρ = 1 , ρ and ρ 2 to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme. [ABSTRACT FROM AUTHOR]

The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed.

In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initial‐boundary value problem) of one‐dimensional Keller–Segel model. For the Cauchy problem, we prove that the solutions time‐asymptotically converge to the nonlinear diffusion wave whose profile is self‐similar solution to the corresponding parabolic equation, which is derived by Darcy's law. For the initial‐boundary value problem, we consider two cases: Dirichlet boundary condition and null‐Neumann boundary condition on (u,ρ)$$ \left(u,\rho \right) $$. In the case of Dirichlet boundary condition, similar to the Cauchy problem, the asymptotic profile is still the self‐similar solution of the corresponding parabolic equation, which is derived by Darcy's law; thus, we only need to deal with boundary effect. In the case of null‐Neumann boundary condition, the global existence and asymptotic behavior of solutions near constant steady states are established. The proof is based on the elementary energy method and some delicate analysis of the corresponding asymptotic profiles. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of traveling wave solutions for the Keller-Segel model with logarithmic sensitivity. By the Hopf-Cole transformation and traveling wave transformation, the degenerate Keller-Segel system is transformed into a singularly perturbed system. By constructing an invariant region to prove the existence of the traveling wave solutions for the degenerate system, we obtain the traveling wave solutions for Keller-Segel system with small parameter by using geometric singular perturbation theory and Fredholm theory. Finally, we discuss the asymptotic behaviors of the traveling wave solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation... [ABSTRACT FROM AUTHOR]

We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L 2 and L ∞ convergence rates are obtained. We first convert the model into a 2 × 2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions. [ABSTRACT FROM AUTHOR]

This work is concerned with the time-fractional doubly parabolic Keller-Segel system in ℝN (N ≽ 1), and we derive some refined results on the large time behavior of solutions which are presupposed to enjoy some uniform boundedness properties. Moreover, the well-posedness and the asymptotic stability of solutions in Marcinkiewicz spaces are studied. The results are achieved by means of an appropriate estimation of the system nonlinearities in the course of an analysis based on Duhamel-type representation formulae and the Kato-Fujita framework which consists in constructing a fixed-point argument by using a suitable time-dependent space. [ABSTRACT FROM AUTHOR]

The quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization (⁎) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (x , n , c) ∇ c) − n m , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + m , x ∈ Ω , t > 0 , m t + u ⋅ ∇ m = Δ m − n m , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u + ∇ P = Δ u + (n + m) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 is considered in a bounded domain Ω ⊂ R 3 with smooth boundary, where ϕ ∈ W 2 , ∞ (Ω) , S denotes a given tensor-valued function fulfilling | S (x , n , c) | ≤ S 0 (c) (1 + n) − α with α ≥ 0 , and S 0 is a nonnegative nondecreasing function. It is shown that if α > 0 , then for any κ ∈ R , the corresponding initial-boundary value problem (⁎) possesses at least one global weak solution, which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. Additionally, we also show that, after some time, these solutions become classical solutions, that is, there exists a positive constant T such that (n , c , m , u) is smooth in Ω × (T , ∞). Finally, we can also prove that when t → ∞ , any such solution (n , c , m , u) would stabilize to the spatially uniform equilibrium (n ˆ , m ˆ , m ˆ , 0) in L ∞ (Ω) , where n ˆ = 1 | Ω | { ∫ Ω n 0 − ∫ Ω m 0 } + and m ˆ = 1 | Ω | { ∫ Ω m 0 − ∫ Ω n 0 } +. The results can provide an important reference for solving the existence of classical solutions of three-dimensional Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the volume-filling effect on global solvability and stabilization in a parabolic-elliptic Keller-Segel-Stokes systems ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (n) ∇ c) , x ∈ Ω , t > 0 , u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 (K S F) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (n) ∇ c) , x ∈ Ω , t > 0 , u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 (K S F) with no-flux boundary conditions for n n and c c as well as no-slip boundary condition for u u in a bounded domain Ω ⊆ R 3 Ω ⊆ R 3 with smooth boundary. Here the nonnegative function S ∈ C 2 (Ω ¯) S ∈ C 2 (Ω ¯) denotes the chemotactic sensitivity which fulfills | S (n) | ≤ C S (1 + n) − α for all n ≥ 0 | S (n) | ≤ C S (1 + n) − α for all n ≥ 0 with some C S > 0 C S > 0 and α > 0 α > 0. Imposing no restriction on the size of the initial data, by seeking some new functionals and using the bootstrap arguments on the system, we establish the existence and boundedness of global classical solutions to parabolic-elliptic Keller-Segel-Stokes system under the assumption α > 1 2 α > 1 2. On the basis of this, we further prove that if the chemotactic coefficient C S C S is appropriately small, the obtained solutions are shown to approach the spatially homogeneous steady state (n ¯ 0 , n ¯ 0 , 0) (n ¯ 0 , n ¯ 0 , 0) in the large time limit, where n ¯ 0 = 1 | Ω | ∫ Ω n 0 n ¯ 0 = 1 | Ω | ∫ Ω n 0 , provided that merely n 0 ≢ 0 n 0 ≢ 0 on Ω Ω. [ABSTRACT FROM AUTHOR]

In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic–elliptic system modeling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit–implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points. [ABSTRACT FROM AUTHOR]

One of the most important models for understanding chemotactic aggregation is the Keller–Segel system. We analyze the time‐fractional Keller–Segel system in a new framework; we derive the global well‐posedness and the asymptotic behavior of solutions in critical Besov‐weak‐Herz spaces that consist in Besov spaces based on weak‐Herz spaces. [ABSTRACT FROM AUTHOR]

In [1] , a finite element method for the (Patlak-) Keller-Segel equations, with additional cross-diffusion and logistic source terms in the elliptic equation, has been introduced. Some a priori estimates of the regularized solutions have been derived, independently of the regularization parameters. Moreover, the existence of the finite element solutions has been shown by using the fixed point theorem and some stability bounds on the fully discrete approximations are obtained. The convergence of the approximate solutions in space has been shown. In this article, we have completed what has been achieved in [1] , where we have completed the convergence of time, where this article starts from where the study in [1] ended. The current study aims to prove the existence of weak solutions to this model, which includes passing the limit to the remaining regulation parameter and achieving the convergence in time. We firstly show that the solutions, independent of regulation parameter, can be bounded. Then these bounds are used to derive bounds on the time-derivatives which are also independent of regulation parameter. In addition, compactness arguments have been used to discuss the convergence of the finite element the approximate problem. We demonstrated the existence of a weak solution for the model under study. Finally, we use the implicit scheme to conduct simulations in one and two space dimensions. [ABSTRACT FROM AUTHOR]

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes. We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O = [ 0 , 1 ]. In particular, let W 1 , W 2 be two time-homogeneous spatial Wiener processes over a filtered probability space A. Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system { d u − (r u Δ u − χ div (u ∇ v)) d t = u ∘ d W 1 , d v − (r v Δ v − α v) d t = β u d t + v ∘ d W 2 , with initial conditions (u (0) , v (0)) = (u 0 , v 0). The positive terms r u and r v are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α ≥ 0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T > 0 , we will prove the existence of a martingale solution on [ 0 , T ]. [ABSTRACT FROM AUTHOR]

We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemo- taxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes. [ABSTRACT FROM AUTHOR]

This paper is concerned with a parabolic‐elliptic Keller–Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel to describe the aggregation phase of Dictyostelium discoideum cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal‐dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non‐constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller–Segel system with signal‐dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy. [ABSTRACT FROM AUTHOR]

A class of parabolic-elliptic Keller–Segel–Stokes systems generalizing the prototype is considered under boundary conditions of homogeneous Neumann type for n (the density of the cell population) and c (the chemical concentration), and Dirichlet type for u (the velocity field), in a bounded domain Ω ⊆ R 2 with smooth boundary, where C S > 0 and ϕ is a given sufficiently smooth function. The model is proposed to describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. Moreover, the chemical diffuses much faster than the cells move. It is shown that under the condition that α > 0 , for any sufficiently smooth initial data (n 0 , u 0) satisfying some compatibility conditions, the associated initial-boundary-value problem (KSF) possesses a global bounded classical solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on α here is optimal. Building on this boundedness property, it can finally even be proved that the corresponding solution of the system decays to ( n ¯ 0 , n ¯ 0 , 0) exponentially if C S is smaller, where n ¯ 0 = 1 | Ω | ∫ Ω n 0 . Our main tool is consideration of the energy functional ∫ Ω n 1 + α , which is a new energy-like functional. [ABSTRACT FROM AUTHOR]

This article concerns the high-dimensional chemotaxis system with consumption of chemoattractant $$\displaylines{ u_t=\Delta u-\nabla\cdot(u\nabla v),\cr v_t=\Delta v-uv, }$$ under homogeneous boundary conditions of Neumann type, in a bounded domain $\Omega\subset\mathbb{R}^n~(n\geq 4)$ with smooth boundary. We prove that that if the initial data satisfy $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,q}(\Omega)$ for some $q>n$, this model possesses at least one global renormalized solution.

In this paper, the similarity method has been used to solve fractional Keller‐Segel model where the diffusion is represented by a nonlocal fractional Laplace operator. In order to verify the results, fractional centred difference method and weighted shifted Grüwald‐Letnikov difference method have been employed for reduction equations. The comparison of results establishes the accuracy and efficiency of the proposed two numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of traveling wave solutions for a Keller-Segel model with bacterial population growth. We show the existence of exactly two heteroclinic orbits. The result is verified by using appropriate numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization (*) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (x , n , c) ∇ c) − n m , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + m , x ∈ Ω , t > 0 , m t + u ⋅ ∇ m = Δ m − n m , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u + ∇ P = Δ u + (n + m) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions in a bounded domain Ω ⊂ R 3 with smooth boundary, where ϕ ∈ W 2 , ∞ (Ω). Here S (x , n , c) denotes the rotational effect which satisfies S ∈ C 2 (Ω ¯ × [ 0 , ∞) 2 ; R 3 × 3) and | S (x , n , c) | ≤ S 0 (c) (1 + n) − α with α ≥ 0 and some nonnegative nondecreasing function S 0. Based on a new weighted estimate and some careful analysis, if α > 0 , then for any κ ∈ R , system (⁎) possesses a global weak solution. Furthermore, for any p > 1 , this solution is uniformly bounded with respect to the norm in L p (Ω) × L ∞ (Ω) × L ∞ (Ω) × L 2 (Ω ; R 3). Moreover, if κ = 0 , then system (⁎) admits a classical solution which is global in time and bounded. [ABSTRACT FROM AUTHOR]

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition: {εΔw − λw1+x= 0, in Ω ∇w ⋅ ñ + λw = 0, on ∂Ω (*) where Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary, ñ denotes the unit outward normal vector of ∂Ω and λ ∈ R / {0}. ε and γ are positive constants. The problem (*) is derived from the well-known singular Keller-Segel system. When γ > 0, we show there is only trivial solution w = 0. When γ < 0 and Ω = BR0 is a ball, we show that problem (*) has a non-constant solution which converges to zero uniformly as ε tends to zero. The main idea of this paper is to transform the Robin problem (*) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (*), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system. [ABSTRACT FROM AUTHOR]

This paper deals with a two-chemical reaction type Keller-Segel system coupled with incompressible viscous fluid equations which models the dynamics of cells in fluid in a two dimensional bounded domain Ω { ∂ t n + u ⋅ ∇ n = Δ n − ∇ ⋅ (n χ (n , v , w , x) ∇ v) , ∂ t v + u ⋅ ∇ v = Δ v − v + w , ∂ t w + u ⋅ ∇ w = Δ w − w + n , ∂ t u + ∇ P = Δ u + n ∇ ψ , ∇ ⋅ u = 0. Here, χ (n , v , w , x) represents the saturated sensitivity. Our result suggests that suitable saturation can prevent the blow-up arising from the classical Keller-Segel type signal production mechanism without any smallness condition on initial mass. [ABSTRACT FROM AUTHOR]

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational fluxin a bounded domain with smooth boundary, where is given constant, is a matrix-valued sensitivity satisfying with some and. As the case (with or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with admits at least one global weak solution if. To the best of our knowledge, this is the first analytical work for the full three-dimensional four-component chemotaxis-Navier-Stokes system. [ABSTRACT FROM AUTHOR]

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation (K S N S) { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n S (x , n , c) ∇ c) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + κ (u ⋅ ∇) u + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , which describes chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells, where κ ∈ R , ϕ ∈ W 2 , ∞ (Ω) and S is a given function with values in R 2 × 2 which fulfills | S (x , n , c) | ≤ C S with some C S > 0. If m > 1 and Ω ⊂ R 2 is a bounded domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (K S N S) possesses a global (weak) solution which is bounded. Our main tool is consideration of the energy functional ∫ Ω n m + ∫ Ω | ∇ c | 2. [ABSTRACT FROM AUTHOR]

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.

In this paper, we consider the vanishing viscosity limit problem for a system arising from the Keller‐Segel equations in three space dimensions. First, we construct an accurate approximate solution that incorporates the effects of boundary layers. Then, we prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero. Our approach is based on the method of matched asymptotic expansions of singular perturbation theory and the classical energy estimates. [ABSTRACT FROM AUTHOR]

We consider the time‐fractional Keller‐Segel system of order α∈(0,1). Interesting properties of solutions are highlighted, like regularity and large time behavior in Lebesgue spaces, which depend on the fractional exponent α. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-boundary value problem for the classical Keller–Segel system {ρt − Δρ = −∇ ⋅ (ρ∇c), x ∈ Ω, t > 0 γct − Δc + c = ρ, x ∈ Ω, t > 0 (1) in a bounded domain Ω⊂Rd with d ≥ 2 under homogeneous Neumann boundary conditions, where γ ≥ 0. We study the existence of non-trivial global classical solutions near the spatially homogeneous equilibria ρ = c ≡ M > 0 with M being any given large constant which is an open problem proposed in [2,p. 1687]. More precisely, we prove that if 0 < M < 1 + λ1 with λ1 being the first positive eigenvalue of the Neumann Laplacian operator, one can find ε0 > 0 such that for all suitable regular initial data (ρ0,γc0) satisfying 1|Ω| ∫Ωρ0dx − M = γ (1|Ω|∫Ωc0dx− M) = 0 (2) and ρ0 − MLd/2(Ω) + γ∇c0Ld(Ω) < ε0, (3) problem (1) possesses a unique global classical solution which is bounded and converges to the trivial state (M,M) exponentially as time goes to infinity. The key step of our proof lies in deriving certain delicate Lp−Lq decay estimates for the semigroup associated with the corresponding linearized system of (1) around the constant steady states. It is well-known that classical solution to system (1) may blow up in finite or infinite time when the conserved total mass m ≜ ∫Ωρ0dx exceeds some threshold number if d = 2 or for arbitrarily small mass if d ≥ 3. In contrast, our results indicates that non-trivial classical solutions starting from initial data satisfying (2)-(3) with arbitrarily large total mass m exists globally provided that |Ω| is large enough such that m < (1 + λ1)|Ω|. [ABSTRACT FROM AUTHOR]

We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal L p decay rates for p ≥ 2. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following chemotaxis-growth system u t = Δ u − ∇ ⋅ u ∇ v + μ (u − u α) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , w t = Δ w − w + u , x ∈ Ω , t > 0 , in a smooth bounded domain Ω ⊂ R n (n ⩾ 2) with nonnegative initial data and null Neumann boundary condition, where μ > 0 , α > 1. It is stated that if α > n 4 + 1 2 , the solution is globally bounded. Moreover, if μ > 0 is sufficiently large, the solution (u , v , w) emanating from nonnegative initial data u 0 , v 0 , w 0 with u 0 ⁄ ≡ 0 is globally bounded and satisfies ‖ u ⋅ , t − 1 ‖ L ∞ Ω + ‖ v ⋅ , t − 1 ‖ L ∞ Ω + ‖ w ⋅ , t − 1 ‖ L ∞ Ω → 0 as t → ∞. [ABSTRACT FROM AUTHOR]

This paper investigates the following Keller-Segel-Stokes system with nonlinear diffusion (K S F) { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n ∇ c) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under homogeneous boundary conditions of Neumann type for n and c , and of Dirichlet type for u in a three-dimensional bounded domains Ω ⊆ R 3 with smooth boundary, where ϕ ∈ W 2 , ∞ (Ω) , m > 0. It is proved that if m > 4 3 , then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system (K S F) , which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on m. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the global stability of homogeneous equilibria in Keller–Segel–Navier–Stokes equations in scaling-invariant spaces. We prove that for any given 0 < M < 1 + μ 1 with μ 1 being the first eigenvalue of Neumann Laplacian, the initial–boundary value problem of the Keller–Segel–Navier–Stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to (M , M , 0) in the norm of L d / 2 (Ω) × W ˙ 1 , d (Ω) × L d (Ω) and satisfies a natural average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller–Segel–Navier–Stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly. [ABSTRACT FROM AUTHOR]

We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise. • Travelling wave solutions are transiently unstable for a range of parameter values. • The point eigenvalue of order two at the origin persists for small diffusion. • The constant consumption model is spectrally stable in the weighted function space. • The spectral gap vanishes for linear consumption. • Eigenfunctions for the origin are contained in the weighted function space. [ABSTRACT FROM AUTHOR]

In the recent decades, the biological models have been noticed to find a suitable numerical procedure. Among the biological models, equations in tumor growth have many applications. In the current paper, we consider some equations in chemotaxis and haptotaxis models. The studied models have blow-up phenomena in their solutions. In addition, the proposed numerical technique is based on a meshless method that is well-known generalized smoothed particle hydrodynamics (SPH) method. [ABSTRACT FROM AUTHOR]

In this paper, we study well-posedness, existence of a lower finite time blow-up bound and variants of controllability of the classical chemotaxis model in Ω × (0 , T) {\Omega\times(0,T)} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} , N = 1 , 2 , 3 {N=1,2,3}. The spatial domain restrictions allow the system with initial data in L 2 ⁢ (Ω) {L^{2}(\Omega)} to admit a solution in L ∞ ⁢ [ 0 , T ; L 2 ⁢ (Ω)) ∩ L 2 ⁢ (0 , T ; H 1 ⁢ (Ω)) L^{\infty}[0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega)) and to have the property that the gradient chemical solutions are uniformly bounded in Ω × (0 , T) {\Omega\times(0,T)}. A lower finite time blow-up bound of solutions in the norm of L 2 ⁢ (Ω) {L^{2}(\Omega)} is proved using the differential inequality technique. Furthermore, using Carleman estimates and appropriate energy functionals, we show that the model is null and approximate controllable at any finite time T > 0 {T>0} with a single control in L 2 ⁢ (ω × (0 , T)) {L^{2}(\omega\times(0,T))} acting on the cell-density equation, linearized through a priori uniform boundedness of the chemical drift solutions, where ω ⊂ Ω {\omega\subset\Omega} is a non-empty open subset of Ω. Lastly, bang-bang-type controls for the problem are constructed. [ABSTRACT FROM AUTHOR]

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order α∈(0,1). We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result. [ABSTRACT FROM AUTHOR]

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.

This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.