Keller Segel System Research Articles

Improvements and generalizations of results concerning attraction‐repulsion chemotaxis models

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers “Boundedness in a nonlinear attraction‐repulsion Keller–Segel system with production and consumption,” by S. Frassu, C. van der Mee and G. Viglialoro [J. Math. Anal. Appl. 504(2):125428, 2021] and “Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent,” by S. Frassu and G. Viglialoro [Nonlinear Anal. 213:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements, and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.

Existence, uniqueness and L∞-bound for weak solutions of a time fractional Keller-Segel system

We study the global existence, uniqueness and L∞-bound for the weak solutions to a time fractional Keller-Segel systems with logistic source∂αu∂tα=Δu−∇⋅u∇v+ua−bu,x∈ℝn,t>00=Δv+u,x∈ℝn,t>0where α ∈ (0,1), a ≥ 0, b > 0 with u(x,0) = u0, v(x,t) is represented by the Newton potentialvxt=1nn−2ωn∫ℝn1x−yn−2uydyWe divide the damping coefficient into different cases and use different methods to prove the existence of weak solutions: (i) when b>1−2n, for any initial value u0 and birth rate a ≥ 0, weak solutions exist globally. (ii) when 0<b≤1−2n, weak solutions have global existence under the condition of small initial data u0 and small birth rate a. Furthermore, by establishing fractional differential inequalities, the L∞-bound of weak solutions is obtained. Finally, we also prove that the weak solution must be unique when the damping effect is strong.

Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions

This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller–Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order to prove our theorem, we deal with some auxiliary identity as an evolution equation with a time dependent operator. In view of this new perspective, the direct consequence of the abstract theory is rich enough to establish global existence of the system.

Optimal time-decay rates of the Keller–Segel system coupled to compressible Navier–Stokes equation in three dimensions

Recently, Hattori–Lagha established the global existence and asymptotic behavior of the solutions for a three-dimensional compressible chemotaxis system with chemoattractant and repellent (Hattori and Lagha in Discrete Contin. Dyn. Syst. 41(11):5141–5164, 2021). Motivated by Hattori–Lagha’s work, we further investigated the optimal time-decay rates of strong solutions with small perturbation to the three-dimensional Keller–Segel system coupled to the compressible Navier–Stokes equations, which models for the motion of swimming bacteria in a compressible viscous fluid. First, we reformulate the system into a perturbation form. Then we establish a prior estimates of solutions and prove the existence of the global-in-time solutions based on the local existence of unique solutions. Finally, we will establish the optimal time-decay rates of the nonhomogeneous system by the decomposition technique of both low and high frequencies of solutions as in (Wang and Wen in Sci. China Math., 2020, https://doi.org/10.1007/s11425-020-1779-7). Moreover, the decay rate is optimal since it agrees with the solutions of the linearized system.

Slow Grow-up in a Quasilinear Keller–Segel System

In a ball Ω=BR(0)⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$\\end{document}, n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 2$$\\end{document}, the chemotaxis system ut=∇·(D(u)∇u)-∇·(uS(u)∇v),0=Δv-μ+u,μ=1|Ω|∫Ωu,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \ abla \\cdot \\big ( D(u) \ abla u \\big ) - \ abla \\cdot \\big ( uS(u)\ abla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered under no-flux boundary conditions, with a focus on nonlinearities S∈C2([0,∞))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\in C^2([0,\\infty ))$$\\end{document} which exhibit super-algebraically fast decay in the sense that with some KS>0,β∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_S>0, \\beta \\in [0,1)$$\\end{document} and ξ0>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi _0>0$$\\end{document}, S(ξ)>0andS′(ξ)≤-KSξ-βS(ξ)forallξ≥ξ0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} S(\\xi )>0 \\quad \ ext{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \ ext{ for } \ ext{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\\end{document}It is, inter alia, shown that if furthermore D∈C2((0,∞))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\in C^2((0,\\infty ))$$\\end{document} is positive and suitably small in relation to S by satisfying ξS(ξ)D(ξ)≥KSDξλforallξ≥ξ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \ ext{ for } \ ext{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\\end{document}with some KSD>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{SD}>0$$\\end{document} and λ>2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >\\frac{2}{n}$$\\end{document}, then throughout a considerably large set of initial data, (⋆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\star $$\\end{document}) admits global classical solutions (u, v) fulfilling z(t)C≤‖u(·,t)‖L∞(Ω)≤Cz(t)forallt>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \ ext{ for } \ ext{ all } t>0, \\end{aligned}$$\\end{document}with some C=C(u,v)≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=C^{(u,v)}\\ge 1$$\\end{document}, where z denotes the solution of z′(t)=z2(t)·S(z(t)),t>0,z(0)=ξ0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\\end{document}which is seen to exist globally, and to satisfy z(t)→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z(t)\\rightarrow +\\infty $$\\end{document} as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow \\infty $$\\end{document}. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in (⋆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\star $$\\end{document}) at logarithmic and doubly logarithmic rates, respectively.

Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller–Segel systems

Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller–Segel systems

Spectral Analysis for Singularity Formation of the Two Dimensional Keller–Segel System

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

Well‐posedness and asymptotic behavior for the fractional Keller–Segel system in critical Besov–Herz‐type spaces

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.

Existence of generalized solutions to an attraction-repulsion Keller-Segel system with degradation

We are concerned with an attraction-repulsion Keller-Segel system with a degradation source of a subquadratic power in a bounded domain with smooth boundary in dimensions two and higher. There are three different cases depending on correlations of parameters of the system, which are attraction-dominating, repulsion-dominating, and attraction-repulsion balancing, respectively. We establish the existence of generalized solutions to the system with admissible powers of degradation corresponding to each case. It turns out that, as expected, the attraction-dominating case requires strictly larger powers of degradation than repulsion-dominating one. It is, however, noteworthy that our result presents the power of degradation for the attraction-repulsion balancing case to be the smallest compared to other cases.

A family of mass‐critical Keller–Segel systems

The no-flux initial-boundary value problem for the quasilinear Keller–Segel system u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( S ( u ) ∇ v ) , v t = Δ v − v + u , \begin{equation} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{l}u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \hspace*{-6pc}\\[3pt] v_t=\Delta v-v+u, \end{array} \right.} \end{equation} (*)is considered in smoothly bounded domains Ω ⊂ R n $\Omega \subset \mathbb {R}^n$ , n ⩾ 3 $n\geqslant 3$ , where D ∈ C 2 ( [ 0 , ∞ ) ) $D\in C^2([0,\infty ))$ and S ∈ C 2 ( [ 0 , ∞ ) ) $S\in C^2([0,\infty ))$ are such that D > 0 $D>0$ on [ 0 , ∞ ) $[0,\infty )$ and that S ( 0 ) = 0 < S ( s ) $S(0)=0<S(s)$ for all s > 0 $s>0$ . A particular focus is on cases in which there exist κ > 0 , C S D > 0 $\kappa >0, C_{SD}>0$ and f ∈ L 1 ( ( 1 , ∞ ) ) $f\in L^1((1,\infty ))$ such that − f ( s ) ⩽ D ( s ) S ( s ) − κ s 2 / n ⩽ C S D s for all s ⩾ 1 . \begin{equation} \hspace*{6pc}- f(s) \leqslant \frac{D(s)}{S(s)} - \frac{\kappa }{s^{2/n}} \leqslant \frac{C_{SD}}{s} \quad \mbox{for all } s\geqslant 1.\hspace*{-6pc} \end{equation} (**)It is first shown that then there exists m 0 > 0 $m_0>0$ such that whenever u 0 $u_0$ and v 0 $v_0$ are reasonably regular and nonnegative with ∫ Ω u 0 < m 0 $\int _\Omega u_0 < m_0$ , within a suitably generalized concept of solvability one can always find a global solution for which u $u$ remains bounded with respect to the norm in L 2 ( n − 1 ) / n ( Ω ) $L^{2(n-1)/{n}}(\Omega )$ . Second, in radially symmetric settings this is complemented by a converse result on nonexistence of such solutions for some appropriately large initial data, hence leading to the conclusion that any pair ( D , S ) $(D,S)$ from the family of nonlinearities fulfilling (**) gives rise to a critical mass phenomenon in (*). Remarkably, this does not only include cases of arbitrarily strong diffusion degeneracies due to fast decay of D ( s ) $D(s)$ as s → ∞ $s\rightarrow \infty$ , but according to the considerably wide funnel described by (*) this moreover indicates that mass criticality in Keller–Segel systems is far more than a nongeneric phenomenon limited to precise functional forms of model ingredients hardly to be found in nature. Applications to concrete scenarios include the detection of mass-critical probability distribution functions of the form 0 ⩽ s ↦ exp ( − s ( n − 2 ) / n ) $0\leqslant s \mapsto \exp (-s^{(n-2)/{n}})$ in the version of (**) accounting for so-called volume-filling effects.

Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its $$C^{2}$$ -norm with certain initial data.

Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping

This paper is concerned with a class of signal-dependent motility Keller-Segel systems{ut=∇⋅(D(v)∇u−S(v)u∇v)+f(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under no-flux boundary conditions in a smooth bounded domain Ω⊂Rn. 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 satisfyD,S∈C3((0,+∞)), and there exist constants k,K,x0>0 such thatk≤D(s)≤K,|S(s)|≤Kfor alls∈(x0,+∞). The generalized logistic source f∈C1([0,+∞)) satisfies f(0)=0 and there exists l>max⁡{2(n+2)n+4,2(n−1)n} such thatlims→+∞⁡f(s)sl=−∞. 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.

Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.

A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller–Segel systems

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling.

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.

On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $ L^{p} $-space

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion

&lt;p style='text-indent:20px;'&gt;We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ N $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional unit ball "&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ B $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;" describing the behavior of a biological species "&lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;" and a chemical stimuli "&lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;". The system presents a sub-linear dependence of "&lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \nabla v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;" in the chemotactic coefficient and a nonlinear diffusive term. The evolution of &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is described by the equation&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ u_t - \Delta u^m = - div (\chi u |\nabla v|^{p-2} \nabla v), \quad \mbox{ for } \ m &amp;gt;2, \quad p \in ( 1,2), \quad N \geq 1 $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;for a positive constant &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \chi $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The concentration of the chemical substance &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfies the linear elliptic equation&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ - \Delta v = u - \frac{1}{|B|} \int_{B} u_0dx. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We consider the radially symmetric case and we prove the local existence of weak solutions for the mass accumulation function under assumption&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE3"&gt; \begin{document}$ - \frac{1}{m}+ \frac{1}{N} + 1-\frac{pm}{4(m-1)} \geq 0, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;for radial and regular initial data. Additionally, if the constrain&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE4"&gt; \begin{document}$ \frac{m }{m- 2} \left[ \frac{pm}{2(m-1)}-1\right] \leq 1 $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;is satisfied, the solution globally exists in time.&lt;/p&gt;

Traveling wave solutions of a singular Keller-Segel system with logistic source.

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.

A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion

&lt;p style='text-indent:20px;'&gt;Many problems arising in biology display a complex system dynamics at different scales of space and time. For this reason, multiscale mathematical models have attracted a great attention as they enable to take into account phenomena evolving at several characteristic lengths. However, they require advanced model reduction techniques to reduce the computational cost of solving all the scales.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;In this work, we present a novel version of the Keller-Segel model of chemotaxis on embedded multiscale geometries, &lt;i&gt;i.e.&lt;/i&gt;, one-dimensional networks embedded in three-dimensional bulk domains. Applying a model reduction technique based on spatial averaging for geometrical order reduction, we reduce a fully three-dimensional Keller-Segel system to a coupled 3D-1D multiscale model. In the reduced model, the dynamics of the cellular population evolves on a one-dimensional network and its migration is influenced by a three-dimensional chemical signal evolving in the bulk domain. We propose the multiscale version of the Keller-Segel model as a realistic approach to describe the invasion of malignant cancer cells along the collagen fibers that constitute the extracellular matrix. Performing several numerical simulations, we investigate how the invasive abilities of the cells are affected by the topology of the network (&lt;i&gt;i.e.&lt;/i&gt;, matrix fibers orientation and alignment) as well as by three-dimensional spatial effects. We discuss these results in light of biological evidences.&lt;/p&gt;

Global existence and decay estimates for the classical solution of fractional attraction–repulsion chemotaxis system

The purpose of this paper is devoted to investigate the generalized Keller–Segel system with a fractional parabolic equation and two classical elliptic equations in Rn with n⩾2. When the chemotaxis sensitivities ξ1 and ξ2 are constants, the global well-posedness of solution to (1.1) is derived if repulsion prevails over attraction in the sense that ξ1μ1−ξ2μ2>0. In the process of proving the above results, we introduce Riesz transform and Kato–Ponce’s commutator inequalities to deal with the challenge arising from fractional diffusion. However, there are few results in the case that ξ1 and ξ2 are functions. We develop a unified framework to deal with the existence, uniqueness and decay estimates in such case. At the same time, the global existence, uniqueness and decay estimates of classical solution to the generalized system (1.2) for small initial data are obtained by selecting an appropriate functional space.