Parabolic Elliptic Keller Segel System Research Articles

On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds

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.

Well‐posedness and time decay of fractional Keller–Segel–Navier‐Stokes equations in homogeneous Besov spaces

AbstractIn this paper, we consider the parabolic–elliptic Keller–Segel system, which is coupled to the incompressible Navier–Stokes equations through transportation and friction. It is shown that when the system is diffused by Lévy motion, the well‐posedness of the mild solution to the corresponding Cauchy problem in homogeneous Besov spaces is established by means of the Banach fixed point theorem. Furthermore, we prove the Lorentz regularity in time direction and the maximal regularity of solutions. In addition, we obtain the additional regularity and explore the time decay property of global mild solutions.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn 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 and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

We consider the parabolic–elliptic Keller–Segel system in dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document}. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term

On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term

A Keller–Segel system involving mixed local and nonlocal operators with logistic term on ℝN

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on , involving mixed local and nonlocal operators with . At first, we prove that the semigroup generated by 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 . 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.

Global existence of large solutions for the parabolic–elliptic Keller–Segel system in Besov type spaces

In this paper, we investigate global existence of large solutions for the parabolic–elliptic Keller–Segel system in the homogeneous Besov type spaces. A class of initial data was presented, generating a global smooth solution although the Ḃ∞,∞−2-norm of the initial data may be chosen arbitrarily large.

Critical mass capacity for two-dimensional Keller–Segel model with nonlocal reaction terms

This paper deals with the parabolic–elliptic Keller–Segel system on , involving a source term of logistic type defined in terms of the mass capacity M 0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M 0 and the initial mass m 0. For general solutions, the existence of a global weak solution is proved under the assumption that both M 0 and m 0 are less than 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either with any integrable initial data or accompanied with large initial data. Moreover, gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by with λ > 0. Subsequently we prove that if the initial data is strictly below for some λ > 0, then the solution vanishes in as . If the initial data is strictly above for some λ > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.

Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. Part I Partial differential equation

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.

Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system ut=Δu−χ∇⋅(uf(|∇v|)∇v),0=Δv−M+u,with homogeneous Neumann boundary condition is considered in a radially symmetric domain Ω=BR(0)⊂RN(N≥3), where f(ξ)=(ξp−2(1+ξp)q−pp),ξ≥0,p≥2,1<q≤p<∞,and BR(0) is a N-dimensional ball of radius R>0. We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when NN−1<q<2.

Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two

This paper deals with the parabolic-elliptic Keller–Segel system with weakly singular sensitivity and logistic source under the homogeneous Neumann boundary: , in a smooth bounded domain , with , . It is proved that the system possesses a globally bounded classical solution for with µ > 0 suitably large, without establishing the uniformly positive bound for v from below. In addition, we give the explicit expression of the upper bound for solution u with respect to the parameters via a recursive argument on α. This concludes that weakly singular sensitivity benefits to obtain the global boundedness of classical solution in dimension two.

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a sphere that shrinks to a point. The singularity is in particular of type II, non self-similar and resembles a traveling wave imploding at the origin in renormalized variables. We show the stability of this dynamics among spherically symmetric solutions, and to our knowledge, this is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

Critical mass in a quasilinear parabolic-elliptic Keller-Segel model

The parabolic-elliptic Keller-Segel system{ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇v),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,(⋆) is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥3) with smooth boundary. Here ϕ and ψ are positive functions which satisfy ψ(s)/ϕ(s) grows like s2n as s→∞. We show that there exist m⁎,m⁎>0 such that if the total mass ∫Ωu0<m⁎, (⋆) admits a global bounded solution; if Ω is radially symmetric and ∫Ωu0>m⁎, we can find initial data u0(x)=u0(|x|) such that the corresponding solution must be unbounded.

Existence of blow-up solutions for a degenerate parabolic-elliptic Keller–Segel system with logistic source

This paper deals with existence of finite-time blow-up solutions to a degenerate parabolic–elliptic Keller–Segel system with logistic source. Recently, finite-time blow-up was established for a degenerate Jäger–Luckhaus system with logistic source. However, blow-up solutions of the aforementioned system have not been obtained. The purpose of this paper is to construct blow-up solutions of a degenerate Keller–Segel system with logistic source.

Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems

This paper presents a stabilization result for weak solutions of degenerate parabolic equations in divergence form. More precisely, the result asserts that the global-in-time weak solution converges to the average of the initial data in some topology as time goes to infinity. It is also shown that the result can be applied to a degenerate parabolic-elliptic Keller-Segel system.

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;

Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source

This paper deals with the quasilinear parabolic–elliptic Keller–Segel system with logistic source, ut=Δ(u+1)m−χ∇⋅(u(u+1)α−1∇v)+λ(|x|)u−μ(|x|)uκ,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,where Ω≔BR(0)⊂Rn(n≥3) is a ball with some R>0; m>0, χ>0, α>0 and κ≥1; λ and μ are continuous nonnegative functions. About this problem, Winkler (2018) found the condition for κ such that solutions blow up in finite time when m=α=1. In the case that m=1 and α∈(0,1) as well as λ and μ are constants, some conditions for α and κ such that blow-up occurs were obtained in a previous paper (Tanaka and Yokota, 2020). Moreover, in the case that m≥1 and α=1 Black et al. (2021) showed that there exist initial data such that the corresponding solution blows up in finite time under some conditions for m and κ. The purpose of the present paper is to give conditions for m≥1, α>0 and κ≥1 such that solutions blow up in finite time.

Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic–elliptic Keller–Segel system in the whole n-dimensional space is studied. For this model, every constant A in {mathbb {R}} is a stationary solution. The main goal of this work is to show that A < 1 is a stable steady state while A > 1 is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic\u2013elliptic Keller\u2013Segel systems

We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider JLut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-1|Ω|∫Ωu+u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - \\frac{1}{|\\Omega |} \\int \\limits _\\Omega u + u \\end{array}\\right. } \\end{aligned}$$\\end{document}and PEut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-v+u,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - v + u, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda and mu are given spatially radial nonnegative functions and m, kappa > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for m,kappa leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant lambda , mu > 0, we find that there are initial data which lead to blow-up in (JL) if 0≤κ<min12,n-2n-(m-1)+ifm∈2n,2n-2nor0≤κ<min12,n-1n-m2ifm∈0,2n,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 0 \\le \\kappa&< \\min \\left\\{ \\frac{1}{2}, \\frac{n - 2}{n} - (m-1)_+ \\right\\}&\\quad \\text {if } m\\in \\left[ \\frac{2}{n},\\frac{2n-2}{n}\\right) \\\\ \\text { or }\\quad 0 \\le \\kappa&<\\min \\left\\{ \\frac{1}{2},\\frac{n-1}{n}-\\frac{m}{2}\\right\\}&\\quad \\text {if } m\\in \\left( 0,\\frac{2}{n}\\right) , \\end{aligned}$$\\end{document}and in (PE) if m in [1, frac{2n-2}{n}) and 0≤κ<min(m-1)n+12(n-1),n-2-(m-1)nn(n-1).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 0 \\le \\kappa < \\min \\left\\{ \\frac{(m-1) n + 1}{2(n-1)}, \\frac{n - 2 - (m-1) n}{n(n-1)} \\right\\} . \\end{aligned}$$\\end{document}

On the Existence of Global Smooth Solutions to the Parabolic–Elliptic Keller–Segel System with Irregular Initial Data

We consider the parabolic–elliptic Keller–Segel system *ut=Δu-χ∇·(u∇v),0=Δv-v+u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned} u_t&= \\Delta u - \\chi \ abla \\cdot (u \ abla v), \\\\ 0&= \\Delta v - v + u \\end{aligned} \\right. \\end{aligned}$$\\end{document}in a smooth bounded domain Omega subseteq {mathbb {R}}^n, nin {mathbb {N}}, with Neumann boundary conditions. We look at both chemotactic attraction (chi > 0) and repulsion (chi < 0) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data mu in {mathcal {M}}_+({overline{Omega }}) that, if eithern = 2, chi < 0 orn = 2, chi > 0 and the initial mass is small orn = 3, chi < 0 and mu = f in L^p(Omega ), p > 1 holds, it is still possible to construct global classical solutions to (star ), which are continuous in t = 0 in the vague topology on {mathcal {M}}_+({overline{Omega }}).