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}
}
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… Expand

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
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 theExpand
Boundary Effect on Asymptotic Behaviour of Solutions to the p-System with Linear Damping
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 NeumannExpand
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 isExpand
Asymptotic Convergence of Solutions for One-Dimensional Keller-Segel Equations.
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, whenExpand
Critical space for the parabolic-parabolic Keller–Segel model in Rd
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 ellipticExpand
Optimal Convergence Rates to Diffusion Waves for Solutions of the Hyperbolic Conservation Laws with Damping
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}_ +Expand
Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant
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 valueExpand
Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System
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 areExpand
A blow-up mechanism for a chemotaxis model
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 theExpand
ASYMPTOTIC BEHAVIOR OF SOLUTION TO NONLINEAR DAMPED p-SYSTEM WITH BOUNDARY EFFECT
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 ,Expand
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