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In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality. [ABSTRACT FROM AUTHOR]

In this study, we consider a diffusion system of the Keller-Segel type, which is pivotal in describing the underlying mechanisms of pattern formation. First, the heteroclinic orbit for the degenerate case is investigated by constructing an invariant region. Furthermore, the persistence of travelling waves in Keller-Segel systems in the presence of local or nonlocal delay is respectively established. The geometric singular perturbation theory along with Fredholm theory are the foundational analytical tools utilized in this study. Eventually, the asymptotic behaviors of the travelling waves are elucidated, offering new insights into their dynamic properties. [ABSTRACT FROM AUTHOR]

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]

In this paper, we investigate the existence and uniqueness of almost periodic solutions for the parabolic-elliptic Keller–Segel system on the whole space R n (n ⩾ 4) . We work in the framework of critical spaces such as on the weak-Lorentz space L n 2 , ∞ (R n) . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. [ABSTRACT FROM AUTHOR]

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space Rn(wheren⩾4)$\mathbb {R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space Hn(wheren⩾2)$\mathbb {H}^n\,\, (\hbox{where }n \geqslant 2)$. We work in framework of critical spaces such as on weak‐Lorentz space Ln2,∞(Rn)$L^{\frac{n}{2},\infty }(\mathbb {R}^n)$ to obtain the results for the Keller–Segel system on Rn$\mathbb {R}^n$ and on Lp2(Hn)$L^{\frac{p}{2}}(\mathbb {H}^n)$ for n

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝd for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝd. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model. [ABSTRACT FROM AUTHOR]

In this article we provide new insights into the well-posedness and maximal regularity of systems of abstract evolution equations, in the framework of periodic Lebesgue spaces of vector-valued functions. Our abstract model is flexible enough as to admit time-fractional derivatives in the sense of Liouville-Grünwald. We characterize the maximal regularity property solely in terms of R -boundedness of a block operator-valued symbol, and provide corresponding estimates. In addition, we show practical criteria that imply the R -boundedness part of the characterization. We apply these criteria to show that the Keller-Segel system, as well as a reactor model system, have L q − L p maximal regularity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}^N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta +{\left(-\Delta \right)}^s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}^s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N+4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the sub-critical case of an anisotropic fully parabolic Keller–Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without restriction on the size of the initial data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}^N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta +{\left(-\Delta \right)}^s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}^s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N+4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

The global existence of solutions to a degenerate Keller-Segel system of parabolic-parabolic type is considered, under certain smallness conditions of initial functions. The exponent is chosen in between critical cases and sufficient condition for the global existence of solutions are already known by Kimijima, Nakagawa and Ogawa in 2015 for the parabolic-elliptic case. Their result is extended to parabolic-parabolic case by applying the minimizing movement scheme. [ABSTRACT FROM AUTHOR]

We complete previous results about the incompressible limit of both the n -dimensional (n ≥ 3) compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish two novel uniform estimates in L 3 of the pressure gradient and in L 1 for the time derivative of the pressure. We also prove regularity à la Aronson-Bénilan. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density. [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]

In this work, symmetry groups are used to determine symmetry reductions of a (2 + 1)‐dimensional Keller–Segel system depending on two arbitrary functions. We show that the point symmetries of the considered Keller–Segel system comprise an infinite‐dimensional Lie algebra which involves three arbitrary functions. By way of example, we have used these point symmetries to reduce straightaway the given system of second‐order partial differential equations to a system of second‐order ordinary differential equations. Moreover, we are allowed to substitute one of the dependent variables from one of the equations into the other, leading to an equivalent fourth‐order nonlinear ordinary differential equation. This equation is reduced through the use of solvable symmetry subalgebras, and some exact solutions are obtained for a particular case. [ABSTRACT FROM AUTHOR]

This paper concerns the chemotaxis-Navier-Stokes system$\begin{equation} \begin{cases} n_t+u\cdot {\nabla } n = \Delta n - {{\nabla }\cdot}(n\chi(c) {\nabla } c),\\ c_t+u\cdot {\nabla } c = \Delta c -nf(c),\\ u_t+\kappa (u\cdot {\nabla }) u = \Delta u + {\nabla } P +n {\nabla }\Phi, \quad {{\nabla }\cdot} u = 0 \end{cases} \end{equation} \;\;\;\;\;\;\;\;\;\;\;\;(\star)$in a smoothly convex bounded domain under Neumann/Neumann/Dirichlet boundary conditions. The existing literature reveals that $ (\star) $ possesses the global solution under some mild assumptions on $ \chi $ and $ f $ [25,28]. It is shown in this paper that even though $ \chi $ admits some singularity (e.g. $ \chi(c) = c^{-\alpha} $ for some $ \alpha>0 $), appropriate use of some quantity involving $ \chi $ and $ f $ guarantees the existence of the global solution to $ (\star) $ for any choice of suitably regular (nonnegative) initial data. [ABSTRACT FROM AUTHOR]

The Cauchy problems of Keller-Segel system with logistic source seem much less throughly understood than associated initial-boundary problems in bounded domains. This paper is concerned with the Cauchy problem of Keller-Segel system with generalized logistic source given by$ \begin{eqnarray*} \left\{ \begin{array}{llc} \label{188} u_t = \Delta u - \nabla \cdot (u\nabla v)+\lambda u-\mu u^{k}, \\ 0 = \Delta v+u, \end{array} \right. \end{eqnarray*} $in $ \mathbb{R}^{n} $ for $ n\ge 3 $, where $ \lambda\in \mathbb{R} $, $ \mu >0 $, $ k>1 $.Under the assumption $ k<\frac{3}{2}-\frac{1}{n} $, it is shown that there exists $ m_{*}>0 $ such that if the radially symmetric initial data satisfies $ \int_ {B_{\frac{1}{2}}(0)}u_{0}\geq m_{*} $, then the problem admits a finite-time blowup solution. [ABSTRACT FROM AUTHOR]

This paper deals with classical solutions $ (u, v) $ to chemotaxis stystems with local sensing. In Fujie-Jiang (Calc. Var. Partial Differential Equations, 2021), some auxiliary function $ w $ is introduced and the comparison estimate: $ d_1v\leq w $ ($ d_1>0 $) plays an important role in the analysis. Recently, in Fujie-Senba (Nonlinearity, 2022) this comparison estimate is refined: $ d_1 v\leq w \leq d_2v $ ($ d_1, d_2>0 $) for some restricted case. In this paper, it is pointed out that in light of the lower bound for $ v $, some part of the proof in Fujie-Senba (Nonlinearity, 2022) can be applied in more general setting. [ABSTRACT FROM AUTHOR]

We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows: $ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $ in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $. In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.

It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller–Segel system can be induced by initial data having large negative energy if the total mass $ \Lambda \in (4\pi, \infty)\setminus 4\pi \cdot \mathbb{N} $ and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann–Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller–Segel model.

We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $ L^{\frac{n}{2}}(\mathbb{R}^{n}) $ for the parabolic-elliptic Keller-Segel system, $ n\geq4 $. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-$ L^{\frac{n}{2}} $ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $ L^{q}((0, T);L^{p}) $-norms with $ q\neq\infty $), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $ C([0, T);L^{\frac{n}{2}}) $, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $ \gamma $ of the chemoattractant is also analyzed.

This article is devoted to the analysis of a fractional chemotaxis model in RN with a time fractional variation in the Caputo sense and a fractional spatial diffusion. This model encompasses the fractional Keller-Segel system [9] which describes the movement of living organisms towards higher concentration regions of chemical attractants, and a fractional Lotka-Volterra competition model [16] describing the competition interspecies in which one of the competing species avoids encounters with rivals by means of chemorepulsion. We prove product estimates in Besov-Morrey spaces and derive global estimates for mild solutions of the fractional heat equation. We use these results to prove the existence and uniqueness of global mild solutions for the differential system in a framework of Besov-Morrey spaces. [ABSTRACT FROM AUTHOR]

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well. [ABSTRACT FROM AUTHOR]

In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space Rd{{\mathbb{R}}}^{d}, a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.

In this paper, we give a new construction of u 0 ∈ B p , ∞ σ such that the corresponding solution to the hyperbolic Keller-Segel model starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ σ (R d) with d ≥ 1 and 1 ≤ p ≤ ∞ , which implies the ill-posedness for this equation in B p , ∞ σ . Our result generalizes the recent work in Zhang et al. (J Differ Equ 334:451-489, 2022) where the case d = 1 and p = 2 was considered. [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 the Keller-Segel system with gradient dependent chemotactic sensitivity, ... where Ω ⊂ R n (n ∈ N) is a bounded domain with smooth boundary, and χ > 0, p ∈ (1,∞) are constants. The purpose of this paper is to establish stability of constant equilibria under some smallness conditions for the initial data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the maximal regularity of the heat equation in Morrey spaces based on the 2010 paper by Ogawa and Shimizu. What differs from their work is that the Fourier multipliers are used instead of interpolation. Some recent studies on the real interpolation show that Morrey spaces do not interpolate well. The estimate needed in this paper depends on the local means. The norm estimate of local means is transformed into the form which is suitable in the context of the maximal regularity. As an application, the Cauchy problems for the Keller-Segel system are studied. The function spaces used in this paper correspond to the scaling critical case for the Keller-Segel system. An observation shows why Besov-Morrey spaces are suitable for the study of maximal regularity and that some other related spaces are not. [ABSTRACT FROM AUTHOR]

This paper considers the following two-species chemotaxis system with two chemicals u t = Δ u − ∇ ⋅ (u ∇ v) , x ∈ Ω , t > 0 , 0 = Δ v − α v + w , x ∈ Ω , t > 0 , w t = Δ w − ∇ ⋅ (w ∇ z) , x ∈ Ω , t > 0 , 0 = Δ z − γ z + u , x ∈ Ω , t > 0 subject to the homogeneous Neumann boundary condition with α , γ > 0 , where Ω ⊂ R 2 is a smooth bounded domain. In the previous paper [Yu et al, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity 31 (2018) 502–514], we proved that the system possesses finite-time blow-up solutions provided (0.1) m u m w − 2 π (m u + m w) > 0 , where m u and m w denote the initial masses of u and w respectively. In this paper, we first establish the critical mass curve m u m w − 2 π (m u + m w) = 0 by proving that solutions are globally bounded whenever (0.2) m u m w − 2 π (m u + m w) < 0 , which means the blow-up condition (0.1) is optimal. Based on this, we further show for the blow-up region (0.1) that (u (x , t) d x , w (x , t) d x) forms a delta function singularity at the blow-up point x 0 as t → T max with the collapse mass (M x 0 u , M x 0 w) satisfying M x 0 u M x 0 w − M ∗ (M x 0 u + M x 0 w) ≥ 0 , where M ∗ = 4 π for x 0 ∈ Ω and M ∗ = 2 π for x 0 ∈ ∂ Ω. [ABSTRACT FROM AUTHOR]

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In this paper, a one-dimensional Keller-Segel system of parabolic-parabolic type which is defined on the bounded interval with the Dirichlet boundary condition is considered. Under the assumption that initial data is sufficiently small, a unique mild solution to the system is constructed and the continuity of solution for the initial data is shown, by using an argument of successive approximations. [ABSTRACT FROM AUTHOR]

We consider the global existence and decay of integral solutions to the parabolic-parabolic Keller-Segel system in d-dimension. On the one hand, by Banach fixed point theorem and some properties of heat kernel, we prove the local existence and the global existence of integral solutions for the different initial data under some conditions that involve the size of the initial data. On the other hand, in the case of global solutions, we obtain their optimal time decay by Gronwall's lemma. [ABSTRACT FROM AUTHOR]

We consider the Keller-Segel system with gradient dependent chemotactic sensitivity $$\displaylines{ u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,\quad x\in\partial\Omega,\; t>0,\cr u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in\Omega }$$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$. We shown that for all reasonably regular initial data $u_0\geq 0$ and $v_0\geq0$, the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that $1

In this paper, we derive a variant of the classical Keller–Segel model of chemotaxis incorporating a growth term of logistic type for the cell population n(t,x), say νn(1−n) with ν>0, and a nonstandard chemical production–degradation mechanism involving first‐ and second‐order derivatives of the logarithm of the cell density, say fλab(n,nx,nxx)=λn+anxxn+bnx2n2 with λ,a,b∈R, via the (n,S)‐hydrodynamical system associated with a modified Ginzburg–Landau equation governing the evolution of the complex wavefunction ψ=neiS. In a chemotactic context, S(t,x) will play the role of the concentration of chemical substance. Then, after carrying out a detailed analysis of the modulational stability of uniform‐in‐space plane waves, dark soliton‐shaped traveling wave densities of the former system are constructed from solitary wave solutions of the latter. [ABSTRACT FROM AUTHOR]

We construct local strong solutions of Keller–Segel system of parabolic–elliptic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also show that the solution exists globally in time for small initial data. The solutions belong to the Lorentz space in time direction since our method relies on the maximal Lorentz regularity of heat equations. [ABSTRACT FROM AUTHOR]

We consider the well-posedness of the Keller–Segel system in uniformly local Lebesgue spaces. It is well known that the parabolic-elliptic Keller–Segel system is one of diffusion equations involving a nonlocal term. In this paper, we study the parabolic-elliptic Keller–Segel system by using only local properties of the initial data. Moreover, the unconditional uniqueness of mild solutions to the Keller–Segel system is studied using uniformly local Lebesgue spaces. We also consider the uniformly local almost periodicity of mild solutions. [ABSTRACT FROM AUTHOR]

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat equations. [ABSTRACT FROM AUTHOR]

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N (N ≥ 4). Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C (R , L s , ∞ (R N)) when S (n) = n − 1 2 . Next for the case S (n) = 0 , we study the mild solution and point out it can become strong solution in B C (R , L s , ∞ (R N)) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations. [ABSTRACT FROM AUTHOR]

The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square n(t,x) and the argument S(t,x) of a wavefunction ψ = √neiS that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct a family of quasi-stationary solutions to the Keller-Segel system under the influence of no-flux and anti-Fick laws. [ABSTRACT FROM AUTHOR]

In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix A (u) are given, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since A (u) is typically neither symmetric nor positive definite. In particular, the normal ellipticity of A (u) for all u and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If A is constant or constant up to nonlinear perturbations, the existence of an entropy structure is equivalent to the normal ellipticity of A. The results are applied to various examples from physics and biology. Finally, the normal ellipticity of the n -species population model of Shigesada, Kawasaki, and Teramoto is proved. [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]

This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant C~{\tilde{C}} such that the initial data (u0,n0,c0):=(u0h,u03,n0,c0){(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})} satisfy

We establish the existence of nonnegative weak solutions to time fractional Keller–Segel system with Dirichlet boundary condition in a bounded domain with smooth boundary. Since the considered system has a cross‐diffusion term and the corresponding diffusion matrix is not positive definite, we first regularize the system. Then under suitable assumptions on the initial conditions, we establish the existence of solutions to the system by using the Galerkin approximation method. The convergence of solutions is proved by means of compactness criteria for fractional partial differential equations. The nonnegativity of solutions is proved by the standard arguments. Furthermore, the existence of the weak solution to the system with Neumann boundary condition is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of traveling pulse solutions of one-dimensional generalized Keller-Segel system with nonlinear chemical gradients and small cell diffusion by using the dynamical systems approach, especially based on geometric singular perturbation theory and Poincaré-Bendixson theorem. To show the existence of traveling pulse solutions, we first analyze the dynamics of the system by geometric singular perturbation theory. And then we seek an invariant region for the associated traveling wave equation. Finally, we apply Poincaré-Bendixson theorem to analyze the flow on this invariant region to obtain the existence of traveling pulse solutions in this bounded invariant region. As applications, we present two examples to illustrate our main results. [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]