On explosions of solutions to a system of partial differential equations modelling chemotaxis

1992/01/01 by W. Jäger, Willi Jäger, S. Luckhaus +1 · 63 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · Medicine · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.1090/s0002-9947-1992-1046835-6

Abstract

A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of parameters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values.

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• Complete infinite-time mass aggregation in a quasilinear Keller–Segel system
• Global existence for a kinetic model of pattern formation with density-suppressed motilities
• Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
• Instantaneous regularization of measure-valued population densities in a Keller–Segel system with flux limitation
• Spectral instability and non-uniqueness for the Keller-Segel system
• Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate di ffusion
• A degenerate chemotaxis system with flux limitation: Finite-time blow-up
• Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source
• Uniform in time L^∞-estimates for nonlinear aggregation-diffusion equations
• Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation
• Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system
• Parabolic-elliptic reduction suppresses Hopf bifurcation in a forager-exploiter system with cascade taxis
• A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
• Stationary states of quadratic diffusion equations with long-range attraction
• Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces
• Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes
• Dissipation enhancement of planar helical flows and applications to three-dimensional Kuramoto-Sivashinsky and Keller-Segel equations
• Global Existence and Finite Time Blow-Up for Critical\n Patlak-Keller-Segel Models with Inhomogeneous Diffusion
• Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions
• Blow up of solutions to generalized Keller--Segel model
• Large global solutions of the parabolic-parabolic Keller-Segel system in\n higher dimensions
• Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system
• Boundedness and asymptotically stability to chemotaxis system with competitive kinetics and nonlocal terms
• Equilibration of aggregation-diffusion equations with weak interaction forces
• Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime
• Existence and stability of infinite time blow-up in the Keller-Segel system
• Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation
• Fully Adaptive Newton-Galerkin Time Stepping Methods for Singularly Perturbed Parabolic Evolution Equations
• Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity
• Local Null Controllability of a Chemotaxis System of Parabolic-Elliptic Type
• Suppression of blow-up in Patlak-Keller-Segel via shear flows
• Residual vanishing for blowup solutions to 2D Smoluchowski-Poisson equation
• Auto-chemotactic micro-swimmer suspensions: modeling, analysis and\n simulations
• Well-posedness and Gevrey Analyticity of the Generalized Keller-Segel System in Critical Besov Spaces
• Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
• Exponential attractor for a chemotaxis-growth system of equations
• Multi-species Patlak-Keller-Segel system
• A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant
• Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity
• Blow-up dynamics of self-attracting diffusive particles driven by competing convexities
• How far do chemotaxis-driven forces influence regularity in the\n Navier-Stokes system?
• Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity
• Equilibria of aggregation-diffusion models with nonlinear potentials
• Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models
• Upper bound on heat kernels of finite particle systems of Keller-Segel type
• Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system
• Global Existence for a Kinetic Model of Pattern Formation with Density-suppressed Motilities
• The microscopic derivation and well-posedness of the stochastic Keller-Segel equation
• A new result for boundedness of solutions to a quasilinear higher-dimensional chemotaxis -- haptotaxis model with nonlinear diffusion
• Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation
• Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian\n Operators
• On a two-species cross attraction system in higher dimensions
• Solvability of a Keller-Segel system with signal-dependent sensitivity\n and essentially sublinear production
• Inhomogeneous Patlak-Keller-Segel models and Aggregation Equations with Nonlinear Diffusion in \Real^d
• Sharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system
• A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators
• Analysis of a non local model for spontaneous cell polarisation
• Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel\n and Navier-Stokes in R2 with Measure-valued Initial Data
• Ground states in the diffusion-dominated regime. [europepmc]
• The Microscopic Derivation and Well-Posedness of the Stochastic Keller-Segel Equation. [europepmc]
• Stable Singularity Formation for the Keller-Segel System in Three Dimensions. [europepmc]
• An additive-noise approximation to Keller-Segel-Dean-Kawasaki dynamics: local well-posedness of paracontrolled solutions. [europepmc]

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