# A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up

@article{Bellomo2016ADC, title={A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up}, author={Nicola Bellomo and Michael Winkler}, journal={Communications in Partial Differential Equations}, year={2016}, volume={42}, pages={436 - 473} }

ABSTRACT This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller–Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic–elliptic system under the initial condition and no-flux boundary conditions in balls Ω⊂ℝn, where χ>0 and . The main results assert… Expand

#### 53 Citations

On a parabolic–elliptic system with gradient dependent chemotactic coefficient

- Mathematics
- 2018

Abstract We consider a second order PDEs system of Parabolic–Elliptic type with chemotactic terms. The system describes the evolution of a biological species “ u ” moving towards a higher… Expand

A critical blow-up exponent in a chemotaxis system with nonlinear signal production

- Mathematics
- 2018

This paper is concerned with radially symmetric solutions of the Keller–Segel system with nonlinear signal production, as given by in the ball for and R > 0, where f is a suitably regular function… Expand

Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation

- Mathematics
- 2020

This manuscript deals with the three-dimensional version of a flux-limited Keller-Segel system coupled to the incompressible Stokes equations through transport and buoyancy.
The main goal consists… Expand

Local and global solutions for a hyperbolic–elliptic model of chemotaxis on a network

- Computer Science
- Mathematical Models and Methods in Applied Sciences
- 2019

A hyperbolic–elliptic system on a network which arises in biological models involving chemotaxis and suitable transmission conditions at internal points of the network are considered. Expand

Global existence and boundedness in a chemotaxis–haptotaxis system with signal-dependent sensitivity

- Mathematics
- 2018

Abstract This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v… Expand

Facing Low Regularity in Chemotaxis Systems

- Computer Science
- Jahresbericht der Deutschen Mathematiker-Vereinigung
- 2019

This note discusses some approaches addressing the design of solution theories which are able to adequately cope with the possible destabilizing effects of chemotactic cross-diffusion, from the context of rigorous blow-up detection. Expand

Finite-time blow-up in a degenerate chemotaxis system with flux limitation

- Physics
- 2017

This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by ( ) ⎧⎨ ⎩ ut = ∇ · ( u∇u √ u2 + |∇u|2 ) − χ∇ ·… Expand

Global existence and asymptotic behavior of the fractional chemotaxis system with signal-dependent sensitivity

- Mathematics, Computer Science
- Comput. Math. Appl.
- 2019

A suitable mathematical framework is developed for a unified treatment of the existence and decay estimates of the global classical solutions to the problem simultaneously under the smallness initial assumptions. Expand

Keller-Segel Chemotaxis Models: A Review

- Mathematics
- 2021

We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global… Expand

Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with gradient-dependent flux limitation

- Physics
- 2021

Abstract In a bounded domain Ω ⊂ R 3 , we are concerned with the evolution system ( ⋆ ) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n f ( | ∇ c | 2 ) ∇ c ) , c t + u ⋅ ∇ c = Δ c − c + n , u t + ( u ⋅ ∇ ) u = Δ u + ∇… Expand

#### References

SHOWING 1-10 OF 48 REFERENCES

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

- Mathematics
- 2015

This paper deals with the parabolic–elliptic Keller–Segel system with signal-dependent chemotactic sensitivity function,
under homogeneous Neumann boundary conditions in a smooth… Expand

A blow-up mechanism for a chemotaxis model

- Mathematics
- 1997

We consider the following nonlinear system of parabolic equations: (1) ut =Δu−χ∇(u∇v), Γvt =Δv+u−av for x∈B R, t>0. Here Γ,χ and a are positive constants and BR is a ball of radius R>0 in R2. At the… Expand

The Keller-Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2006

The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller–Segel model of chemotaxis with volume filling effect. In the… Expand

Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect

- Mathematics
- 2010

Abstract We consider the elliptic–parabolic PDE system { u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − ∇ ⋅ ( ψ ( u ) ∇ v ) , x ∈ Ω , t > 0 , 0 = Δ v − M + u , x ∈ Ω , t > 0 , with nonnegative initial data u 0 having… Expand

Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

- Mathematics
- 2010

We consider the classical parabolic–parabolic Keller–Segel system
{ut=Δu−∇⋅(u∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,
under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn.
It is… Expand

Finite-time blow-up in a degenerate chemotaxis system with flux limitation

- Physics
- 2017

This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by ( ) ⎧⎨ ⎩ ut = ∇ · ( u∇u √ u2 + |∇u|2 ) − χ∇ ·… Expand

Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation

- Mathematics
- 2008

This paper deals with the phenomenon of gradient blow-up of nonnegative classical solutions of the Dirichlet problem for (*) u t = u p u xx + Ku r u x 2 + u q in Ω x (0,T) in a bounded interval Ω c… Expand

Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces

- Computer Science, Mathematics
- Math. Comput. Model.
- 2008

It is proved that when the equation is set in the whole space R^d and dimension d>=3 the critical spaces for the initial bacteria density and the chemical gradient are respectively L^a(R^d), a>d/2, and L^d(R*d). Expand

Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

- Mathematics
- 2010

Consider the diffusive Hamilton-Jacobi equation ut = Δu + |∇u|p, p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known… Expand

The Porous Medium Equation

- Mathematics
- 2006

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions… Expand