Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion

2018 ◽  
Vol 28 (07) ◽  
pp. 1413-1451 ◽  
Author(s):  
Dan Li ◽  
Chunlai Mu ◽  
Pan Zheng
Keyword(s):  
Tumor Invasion ◽  
Large Time ◽  
System Modeling ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Matrix ◽  
Initial Boundary Value

This paper deals with the quasilinear chemotaxis system modeling tumor invasion [Formula: see text] under homogenous Neumann boundary conditions in a smoothly convex bounded domain [Formula: see text] [Formula: see text], where [Formula: see text] is a given function satisfying [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text]. Here the matrix-valued function [Formula: see text] fulfills [Formula: see text] for all [Formula: see text] with some [Formula: see text] and [Formula: see text]. It is shown that for all reasonably regular initial data, a corresponding initial-boundary value problem for this system possesses a globally defined weak solution under some assumptions. Based on this boundedness property, it can finally be proved that in the large time limit, any such solution approaches the spatially homogenous equilibrium [Formula: see text] in an appropriate sense, where [Formula: see text], [Formula: see text] and [Formula: see text] provided that merely [Formula: see text] on [Formula: see text]. To the best of our knowledge, there are the first results on boundedness and asymptotic behavior of the system.

Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food

2019 ◽  
Vol 29 (11) ◽  
pp. 2151-2182 ◽  
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Large Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Food Concentration ◽  
Large Time Behavior ◽  
Space Distribution ◽  
Time Behavior ◽  
Population Number ◽  
Initial Boundary Value ◽  
Two Populations

This work deals with a taxis cascade model for food consumption in two populations, namely foragers directly orienting their movement upward the gradients of food concentration and exploiters taking a parasitic strategy in search of food via tracking higher forager densities. As a consequence, the dynamics of both populations are adapted to the space distribution of food which is dynamically modified in time and space by the two populations. This model extends the classical one-species chemotaxis-consumption systems by additionally accounting for a second taxis mechanism coupled to the first in a consecutive manner. It is rigorously proved that for all suitably regular initial data, an associated Neumann-type initial-boundary value problem for the spatially one-dimensional version of this model possesses a globally defined bounded classical solution. Moreover, it is asserted that the considered two populations will approach spatially homogeneous distributions in the large time limit, provided that either the total population number of foragers or that of exploiters is appropriately small.

scholarly journals LARGE-TIME BEHAVIOR OF DISCRETE KINETIC EQUATIONS WITH NON-SYMMETRIC INTERACTIONS

2002 ◽  
Vol 12 (11) ◽  
pp. 1555-1564 ◽  
Author(s):  
ANTON ARNOLD ◽  
JOSE A. CARRILLO ◽  
MOULAY D. TIDRIRI
Keyword(s):  
Steady State ◽  
Large Time ◽  
Exponential Convergence ◽  
Kinetic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Velocity Models ◽  
Initial Boundary Value

We consider the initial-boundary value problem for general linear discrete velocity models appearing in kinetic theory. With time-independent inflow boundary data we prove the existence of a unique steady state and the exponential convergence in time towards the steady state. The proof is based on the construction of suitable multiplyers used in a weighted L2-norm.

ASYMPTOTIC BEHAVIOR OF THE MOTION OF A VISCOUS HEAT-CONDUCTING ONE-DIMENSIONAL GAS WITH RADIATION: THE PURE SCATTERING CASE

2013 ◽  
Vol 11 (01) ◽  
pp. 1350003 ◽  
Author(s):  
BERNARD DUCOMET ◽  
ŠÁRKA NEČASOVÁ
Keyword(s):  
Large Time ◽  
Transport Coefficients ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Heat Conducting ◽  
Time Behavior ◽  
Static State ◽  
Viscous Heat ◽  
Initial Boundary Value

We study the large-time behavior of the solution of an initial-boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation. Assuming only scattering processes between matter and photons (neglecting absorption and emission) and suitable hypotheses on the transport coefficients, we prove that the unique weak solution of the problem converges toward the static state.

scholarly journals On global classical solutions to one-dimensional compressible Navier–Stokes/Allen–Cahn system with density-dependent viscosity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Menglong Su
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Time Behavior ◽  
One Dimensional ◽  
Global Classical Solutions ◽  
Initial Boundary Value

AbstractIn this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

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