847 research outputs found
Large mass boundary condensation patterns in the stationary Keller-Segel system
We consider the boundary value problem in
with Neumann boundary condition, where is a bounded smooth
domain in , This problem is equivalent to the
stationary Keller-Segel system from chemotaxis. We establish the existence of a
solution which exhibits a sharp boundary layer along the entire
boundary as . These solutions have large mass in
the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.
A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis
We consider two models which were both designed to describe the movement of
eukaryotic cells responding to chemical signals. Besides a common standard
parabolic equation for the diffusion of a chemoattractant, like chemokines or
growth factors, the two models differ for the equations describing the movement
of cells. The first model is based on a quasilinear hyperbolic system with
damping, the other one on a degenerate parabolic equation. The two models have
the same stationary solutions, which may contain some regions with vacuum. We
first explain in details how to discretize the quasilinear hyperbolic system
through an upwinding technique, which uses an adapted reconstruction, which is
able to deal with the transitions to vacuum. Then we concentrate on the
analysis of asymptotic preserving properties of the scheme towards a
discretization of the parabolic equation, obtained in the large time and large
damping limit, in order to present a numerical comparison between the
asymptotic behavior of these two models. Finally we perform an accurate
numerical comparison of the two models in the time asymptotic regime, which
shows that the respective solutions have a quite different behavior for large
times.Comment: One sentence modified at the end of Section 4, p. 1
A one-dimensional Keller-Segel equation with a drift issued from the boundary
We investigate in this note the dynamics of a one-dimensional Keller-Segel
type model on the half-line. On the contrary to the classical configuration,
the chemical production term is located on the boundary. We prove, under
suitable assumptions, the following dichotomy which is reminiscent of the
two-dimensional Keller-Segel system. Solutions are global if the mass is below
the critical mass, they blow-up in finite time above the critical mass, and
they converge to some equilibrium at the critical mass. Entropy techniques are
presented which aim at providing quantitative convergence results for the
subcritical case. This note is completed with a brief introduction to a more
realistic model (still one-dimensional).Comment: short version, 8 page
A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for
the kinetic chemotaxis system with random inputs, which will converge to the
modified Keller-Segel model with random inputs in the diffusive regime. Based
on the generalized Polynomial Chaos (gPC) approach, we design a high order
stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time
discretization with a macroscopic penalty term. The new schemes improve the
parabolic CFL condition to a hyperbolic type when the mean free path is small,
which shows significant efficiency especially in uncertainty quantification
(UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property
will be shown asymptotically and verified numerically in several tests. Many
other numerical tests are conducted to explore the effect of the randomness in
the kinetic system, in the aim of providing more intuitions for the theoretic
study of the chemotaxis models
A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension
We study radial solutions in a ball of of a semilinear,
parabolic-elliptic Patlak-Keller-Segel system with a nonlinear sensitivity
involving a critical power. For , the latter reduces to the classical
linear model, well-known for its critical mass . We show that a critical
mass phenomenon also occurs for , but with a strongly different
qualitative behaviour. More precisely, if the total mass of cells is smaller or
equal to the critical mass M, then the cell density converges to a regular
steady state with support strictly inside the ball as time goes to infinity. In
the case of the critical mass, this result is nontrivial since there exists a
continuum of stationary solutions and is moreover in sharp contrast with the
case where infinite time blow-up occurs. If the total mass of cells is
larger than M, then all solutions blow up in finite time. This actually follows
from the existence (unlike for ) of a family of self-similar, blowing up
solutions with support strictly inside the ball.Comment: 35 page
On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis
We consider the phenomenon of collapse in the critical Keller-Segel equation
(KS) which models chemotactic aggregation of micro-organisms underlying many
social activities, e.g. fruiting body development and biofilm formation. Also
KS describes the collapse of a gas of self-gravitating Brownian particles. We
find the fluctuation spectrum around the collapsing family of steady states for
these equations, which is instrumental in derivation of the critical collapse
law. To this end we develop a rigorous version of the method of matched
asymptotics for the spectral analysis of a class of second order differential
operators containing the linearized Keller-Segel operators (and as we argue
linearized operators appearing in nonlinear evolution problems). We explain how
the results we obtain are used to derive the critical collapse law, as well as
for proving its stability.Comment: 22 pages, 1 figur
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