# Convergence to diffusion waves for solutions of 1D Keller-Segel model

@inproceedings{Liu2021ConvergenceTD,
title={Convergence to diffusion waves for solutions of 1D Keller-Segel model},
author={F. L. Liu and N. G. Zhang and C. J. Zhu},
year={2021}
}
• Published 23 September 2021
• Mathematics, Physics
In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initial-boundary value problem) of one-dimensional Keller-Segel model. For the Cauchy problem, we prove that the solutions time-asymptotically converge to the nonlinear diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which is derived by Darcy’s law, as in [11, 28]. For the initial-boundary value problem, we consider two cases: Dirichlet boundary…

## References

SHOWING 1-10 OF 42 REFERENCES
Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping
• Physics
• 2000
In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane (x, t) e R+ x R+. By means of a suitable shift function, which will play a key role to overcome the
Boundary Effect on Asymptotic Behaviour of Solutions to the p-System with Linear Damping
• Mathematics
• 1999
We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞),vt−ux=0,ut+p(v)x=−αu, with the Dirichlet boundary condition u|x=0=0 or the Neumann
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
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
Asymptotic Convergence of Solutions for One-Dimensional Keller-Segel Equations.
• Mathematics
• 2020
The second and third authors of this paper have constructed in [14] finite-dimensional attractors for the one-dimensional Keller-Segel equations. They have also remarked in [14, Section 7] that, when
Critical space for the parabolic-parabolic Keller–Segel model in Rd
• Mathematics
• 2006
Abstract We study the Keller–Segel system in R d when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic
Optimal Convergence Rates to Diffusion Waves for Solutions of the Hyperbolic Conservation Laws with Damping
• Mathematics
• 2005
Abstract.This paper is devoted to study the asymptotic behaviors of the solutions to a model of hyperbolic balance laws with damping on the quarter plane \$(x,t) \in \mathbb{R}_ + \times \mathbb{R}_ +
Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant
• Mathematics
• 2009
In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v0(x),u0(x)), the corresponding initial–boundary value
Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System
• Mathematics
• 2010
We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes system for ideal polytropic fluids, where the far field states are
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
ASYMPTOTIC BEHAVIOR OF SOLUTION TO NONLINEAR DAMPED p-SYSTEM WITH BOUNDARY EFFECT
• Mathematics
• 2010
For the initial-boundary value problem to the 2 × 2 damped p-system with nonlinear source, 8 > : vt − ux = 0, ut + p(v)x = −�u − �|u| q 1 u, q ≥ 2, (v, u)|t=0 = (v0,u0)(x) → (v+,u+) as x → +∞, 4 2 ,