Related papers: The fractional Keller-Segel model
The Keller-Segel model is a system of partial differential equations that describes the movement of cells or organisms in response to chemical signals, a phenomenon known as chemotaxis. In this study, we analyze a doubly parabolic…
We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<\alpha\leq 2$. We prove some features related to the classical two-dimensional…
A Keller-Segel model describes macroscopic dynamics of bacterial colonies and biological cells. Bacteria secret chemical which attracts other bacteria so that they move towards chemical gradient creating nonlocal attraction between…
This paper is concerned with the fractional Keller-Segel system in the temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion,…
We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due a substance concentration. Considering the initial-boundary value problem for the…
We introduce stochastic models of chemotaxis generalizing the deterministic Keller-Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean's…
We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we…
Chemotaxis systems of Keller--Segel type constitute one of the central mathematical frameworks for understanding aggregation phenomena in biological and ecological systems. Over the past decades, the theory has evolved from the classical…
A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy…
We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then…
In this paper, we are mainly concerned with the well-posed problem of the fractional Keller--Segel system in the framework of variable Lebesgue spaces. Based on carefully examining the algebraical structure of the system, we reduced the…
We investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic…
We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to…
As a class of nonlinear partial differential equations, the Keller-Segel system is widely used to model chemotaxis in biology. In this paper, we present the construction and analysis of a decoupled linear, mass-conservative, block-centered…
The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an…
This paper investigates the Keller-Segel model with quadratic cellular diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its nontrivial patterns. We obtain explicit formulas of radially symmetric stationary solutions…
The Keller-Segel partial differential equation is a two-dimensional model for chemotaxis. When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to the…
The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyper-bolic form and some special parabolic forms have been derived from…
The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven.…
This article studies the aggregation diffusion equation \[ \partial_t\rho = \Delta^\frac{\alpha}{2} \rho + \lambda\,\mathrm{div}((K*\rho)\rho), \] where $\Delta^\frac{\alpha}{2}$ denotes the fractional Laplacian and $K =…