ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS

2011 ◽  
Vol 21 (08) ◽  
pp. 1631-1650 ◽  
Author(s):  
DONG LI ◽  
TONG LI ◽  
KUN ZHAO
Keyword(s):  
Steady State ◽  
Parabolic System ◽  
Smooth Solution ◽  
System Modeling ◽  
Fourier Method ◽  
Classical Solutions ◽  
Time Behavior ◽  
Time Dynamics ◽  
Blowup Criterion ◽  
Long Time

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration. Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.

Initial Value Problems in Viscoelasticity

1988 ◽  
Vol 41 (10) ◽  
pp. 371-378 ◽  
Author(s):  
W. J. Hrusa ◽  
J. A. Nohel ◽  
M. Renardy
Keyword(s):  
Classical Solutions ◽  
Linear Wave ◽  
Time Behavior ◽  
Initial Value ◽  
One Dimensional ◽  
Nonlinear Viscoelastic ◽  
Linear Wave Propagation ◽  
Long Time ◽  
Nonlinear Viscoelastic Materials ◽  
Time Existence

We review some recent mathematical results concerning integrodiff erential equations that model the motion of one-dimensional nonlinear viscoelastic materials. In particular, we discuss global (in time) existence and long-time behavior of classical solutions, as well as the formation of singularities in finite time from smooth initial data. Although the mathematical theory is comparatively incomplete, we make some remarks concerning the existence of weak solutions (i e, solutions with shocks). Some relevant results from linear wave propagation will also be discussed.

scholarly journals A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes

2016 ◽  
Vol 26 (06) ◽  
pp. 1149-1189 ◽  
Author(s):  
Harald Garcke ◽  
Johannes Kampmann ◽  
Andreas Rätz ◽  
Matthias Röger
Keyword(s):  
Numerical Simulations ◽  
Lipid Raft ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Lipid Phase ◽  
Qualitative Behavior ◽  
Time Behavior ◽  
Relaxation Dynamics ◽  
Unsaturated Lipids ◽  
Long Time

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes the relaxation dynamics for an energy which takes lipid–phase separation and lipid–cholesterol interaction energy into account. It takes the form of an (extended) Cahn–Hilliard equation. Different laws for the exchange term represent equilibrium and nonequilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the nonequilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.

scholarly journals Long-time behavior of global weak solutions for a Beris-Edwards type model of nematic liquid crystals

2020 ◽  
Author(s):  
Blanca CLIMENT-EZQUERRA ◽  
Francisco Guillen-Gonzalez
Keyword(s):  
Liquid Crystal ◽  
Stokes System ◽  
System Modeling ◽  
Fluid Velocity ◽  
Global Solutions ◽  
Navier Stokes ◽  
Type Model ◽  
Time Behavior ◽  
Infinite Time ◽  
Long Time

We consider a generalization of the standard Beris-Edwards system modeling incompressible liquid crystal flows of nematic type. This couples a Navier-Stokes system for the fluid velocity with an evolution equation for the Q-tensors variable describing the direction of liquid crystal molecules. The convergence at infinite time for global solutions is studied and we prove that whole trajectory goes to a single equilibrium by using a Lojasiewicz-Simon’s result.

scholarly journals Spontaneous fine-tuning to environment in many-species chemical reaction networks

2017 ◽  
Vol 114 (29) ◽  
pp. 7565-7570 ◽  
Author(s):  
Jordan M. Horowitz ◽  
Jeremy L. England
Keyword(s):  
Steady State ◽  
Fine Tuning ◽  
Reaction Networks ◽  
Environmental Forcing ◽  
Time Dynamics ◽  
Long Time ◽  
State Chemical ◽  
Far From Equilibrium ◽  
Environmental Work ◽  
Random Connectivity

A chemical mixture that continually absorbs work from its environment may exhibit steady-state chemical concentrations that deviate from their equilibrium values. Such behavior is particularly interesting in a scenario where the environmental work sources are relatively difficult to access, so that only the proper orchestration of many distinct catalytic actors can power the dissipative flux required to maintain a stable, far-from-equilibrium steady state. In this article, we study the dynamics of an in silico chemical network with random connectivity in an environment that makes strong thermodynamic forcing available only to rare combinations of chemical concentrations. We find that the long-time dynamics of such systems are biased toward states that exhibit a fine-tuned extremization of environmental forcing.

scholarly journals Asymptotic Analysis for a nonlinear Reaction-Diffusion System Modeling an Infectious Disease

2021 ◽  
Author(s):  
Hong-Ming Yin ◽  
Jun Zou
Keyword(s):  
Infectious Disease ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Time Behavior ◽  
Strongly Coupled ◽  
Nonlinear Reaction ◽  
Open Questions ◽  
Long Time

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as cholera. One of the features in this model is that a certain portion of the recovered human hosts lost a lifetime immunity and could be infected again. Another feature in the model is that the mobility for each species is assumed to be dependent upon location and time. We also assume that the whole group is susceptible with the bacteria. This leads to a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on known parameters and functions. Moreover, the long-time behavior and stability analysis for the solution are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability for all steady-state solutions. These results obtained in this paper answered several open questions raised in the previous literature

GLOBAL BIFURCATION AND LONG TIME BEHAVIOR OF THE VOLTERRA–LOTKA ECOLOGICAL MODEL

2001 ◽  
Vol 11 (01) ◽  
pp. 133-142 ◽  
Author(s):  
KAITAI LI ◽  
LIZHOU WANG
Keyword(s):  
Steady State ◽  
Global Attractor ◽  
Ecological Model ◽  
Global Bifurcation ◽  
Time Behavior ◽  
Order Interval ◽  
Long Time Behavior ◽  
Positive Steady State ◽  
Long Time ◽  
Steady State Solutions

The structure of the positive steady-state solutions for the Volterra–Lotka ecological model of two cooperating species is investigated. Some new results of existence, nonexistence and uniqueness are presented. The existence of a global attractor which is contained in an order interval is shown and the bounds of various dimensions of the attractor are given.

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