Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion (original) (raw)
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This paper investigates the well-posedness of a reaction-diffusion system of chemotaxis type, with a nonlinear diffusion coefficient and a dynamics (growth-death) of the cell population b, and a stationary equation for the chemoattractant c. With respect to other works in which a nonlinear diffusion for cells has been considered, we treat here two distinct cases for this diffusion coefficient, the first in which it is a positive bounded function on R, and the other in which it may display a singularity at a finite value of the cell density. Essentially, the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient. The chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities. For homogeneous Neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the L 2-norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth. Under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data. In a case when the diffusion coefficient is singular, we focus on a model expressed by a variational inequality, describing the saturation of the cell population at the blowing-up diffusion value. Here, the proof requires the study of an intermediate problem with Robin boundary conditions, which may be interesting by itself. In all situations, uniqueness follows on a time interval included (not necessarily strictly) in that of the solution existence, under sufficient conditions.
We consider a general model of chemotaxis with finite speed of propagation in one space dimension. For this model we establish a general result of stability of some constant states both for the Cauchy problem on the whole real line and for the Neumann problem on a bounded interval. These results are obtained using the linearized operators and the accurate analysis of their onlinear perturbations. Numerical schemes are proposed to approximate these equations, and the expected qualitative behavior for large times is compared to several numerical tests.
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SIAM Journal on Applied Mathematics, 2003
A simple model of chemotactic cell migration gives rise to travelling wave solutions. By varying the cellular growth rate and chemoattractant production rate, travelling waves with both smooth and discontinuous fronts are found using phase plane analysis. The phase plane exhibits a curve of singularities whose position relative to the equilibrium points in the phase plane determines the nature of the heteroclinic orbits, where they exist. Smooth solutions have trajectories connecting the steady states lying to one side of the singular curve. Travelling shock waves arise by connecting trajectories passing through a special point in the singular curve and recrossing the singular curve, by way of a discontinuity. Hyperbolic partial differential equation theory gives the necessary shock condition. Conditions on the parameter values determine when the solutions are smooth travelling waves versus discontinuous travelling wave solutions. These conditions provide bounds on the travelling wave speeds, corresponding to bounds on the chemotactic velocity or bounds on cellular growth rate. This analysis gives rise to the possibility of representing sharp fronts to waves of invading cells through a simple chemotactic term, without introducing a nonlinear diffusion term. This is more appropriate when cell populations are sufficiently dense.
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give a proof of global existence for the classical chemotaxis model in one space dimension with use of energy estimates. Here we present an alternative proof which uses the regularity properties of the heat-equation semigroup. With this method we can identify a large selection of admissible spaces, such that the chemotaxis model deÿnes a global semigroup on these spaces.
Metastability in Chemotaxis Models
Journal of Dynamics and Differential Equations, 2005
We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280-301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultra-long transients and merging or coarsening. We study the underlying bifurcation structure and show that the existence time for the pseudostructures exponentially grows with the size of the system. We give approximations for one-step steady state solutions. We show that patterns with two or more steps are metastable and we approximate the two-step interaction using asymptotic expansions. This covers the basic effects of coarsening/merging and dissolving of local maxima. These effects are similar to pattern dynamics in other chemotaxis models, in spinodal decomposition of Cahn-Hilliard models, or to metastable patterns in microwave heating models. , (6) with (u(x, t), S(x, t)) ∈ , for M = M 1 , 0 M 1 1, then (1 − u, η/β − S) is a solution of the same system for M 2 = 1 − M 1 .
On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding
Mathematical Methods in the Applied Sciences, 2009
This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a p-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.
Stationary Solutions and Stability of Constant Equilibrium Solutions for a Chemotaxis System
2023
Stationary Solutions and Stability of Constant Equilibrium Solutions for a Chemotaxis System title Awatif Zawali We consider a one-dimensional Chemotaxis model describing the dynamics of the cell density n, cell velocity u, chemoattractant c 1 , and chemorepellent c 2 , respectively. The model is related to the angiogenesis in cancer cells. n represents the cell density of the vessel, the chemoattractant c 1 is vascular endothelial growth factor (VEGF) and the chemorepellent c 2 is antiangiogenic drug. We study the existence of a nonconstant steady-state solution using the singular perturbation method. We also provide numerical examples of steady-state solutions for the fast system to illustrate the idea. For the dynamical problem in part II, we study the stability of constant stationary solutions for the initial boundary value problem. We discuss the existence of global solutions based on the existence of local solution and the a-priori estimates. I owe a great amount of gratitude to my family for their continued support and encouragement throughout my research. I am extremely thankful for the patience and understanding of my mother, father, husband, and children who experienced all of the ups and downs of my research. I could not imagine the completion of this dissertation without their love.