Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis. (original) (raw)
Related papers
Global smooth solutions for a hyperbolic chemotaxis model on a network
2014
In this paper we study a semilinear hyperbolic-parabolic system modeling biological phenomena evolving on a network composed by oriented arcs. We prove the existence of global (in time) smooth solutions to this problem. The result is obtained by using energy estimates with suitable transmission conditions at nodes.
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.
Discrete and Continuous Dynamical Systems-series B, 2009
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 global 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 nonlinear perturbations. Numerical schemes are proposed to approximate these equations, and the expected qualitative behavior for large times is compared to several numerical tests.
The one-dimensional chemotaxis model: global existence and asymptotic profile
Mathematical Methods in the Applied Sciences, 2004
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.
A coupled chemotaxis-fluid model: Global existence
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2011
We consider a model arising from biology, consisting of chemotaxis equations coupled to viscous incompressible fluid equations through transport and external forcing. Global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the chemotaxis-Navier-Stokes system in two space dimensions, we obtain global existence for large data. In three space dimensions, we prove global existence of weak solutions for the chemotaxis-Stokes system with nonlinear diffusion for the cell density.
Existence of Weak Solutions for a Hyperbolic Model of Chemosensitive Movement
Journal of Mathematical Analysis and Applications, 2001
A hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered. In contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account. This qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment. The model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation. We assume Ž . that the speed depends on the external stimulus only and not on its gradients . In that case solutions with steep gradients are expected which have the interpretation of moving swarms. A notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown. The proof relies on the vanishing viscosity method; i.e., we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter. Numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients. ᮊ
Existence of global solutions to chemotaxis fluid system with logistic source
Electronic Journal of Qualitative Theory of Differential Equations, 2021
We establish the existence of global solutions and L q time-decay of a three dimensional chemotaxis system with chemoattractant and repellent. We show the existence of global solutions by the energy method. We also study L q time-decay for the linear homogeneous system by using Fourier transform and finding Green's matrix. Then, we find L q time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimate.
Local Well-posedness of Kinetic Chemotaxis Models
Journal of Evolution Equations, 2008
This paper presents a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is wellposed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper: the reaction-diffusion system is (1) a single linear equation, (2) a FitzHugh-Nagumo system with cubic nonlinearity and (3) a FitzHugh-Nagumo system with a piecewise linear approximation of a cubic nonlinearity. We use a limit argument to obtain solutions in L 1 ∩L ∞. The presented results are a first step in further analysis of these coupled systems: global existence of solutions, positivity, attractors and limit approximations (e.g. diffusion and hydrodynamic limits).