On a nonlocal nonlinear ODE arising in magnetic recording.

Abstract In this paper we study the uniqueness of the solution for a nonlinear ODE with nonlocal terms. We consider a limit case of a one-dimensional equation arising in magnetic recording. The equation models the tape deflection where the magnetic head profile, with trenches to control the tape position, is a known function

Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation

Abstract We consider the Cauchy problem ut = u + eu, x ∈ RN , t ∈ (0, T ), u(x, 0) = u0, x ∈ RN , where u0 ∈ C(RN ) and T > 0. We first study the radial steady states of the equation and the number of intersections distinguishing four different cases: N = 1, N = 2, 3 N 9 and N 10, writing explicitly every steady state for N = 1 and N = 2. Then we study the large time behavior of solutions of the parabolic problem

Regularity of solutions to a lubrication problem with discontinuous separation data

We study the regularity ofthe solution to the Reynolds equation for incompressible and compressible fluids when the gap between the lubricated surfaces, “h(x; y)”, presents a discontinuity in a two-dimensional bounded domain. As in the one-dimensional problem studied by Rayleigh, the solution P does not belong to C1.

Mathematical analysis and stability of a chemotaxis model with logistic term.

In this paper we study a non-linear system of dierential equations arising in chemotaxis. The system consists of a PDE that describes the evolution of a population and an ODE which models the concentration of a chemical substance. We study the number of steady states under suitable assumptions, the existence of one global solution to the evolution problem in terms of weak solutions and the stability of the steady states

Mathematical analysis of a model of Morphogenesis: steady states

We consider a simple mathematical model of distribution of morphogens (signaling molecules responsible for the differentiation of cells and the creation of tissue patterns) proposed by Lander, Nie and Wang in 2002. The model consists of a system of two equations: a PDE of parabolic type modeling the distribution of free morphogens and an ODE describing the evolution of bound receptors. Three biological processes are taken into account: diffusion, degradation and reversible binding. We present results concerning the steady states

Stabilization in a two-species chemotaxis system with logistic source

We study a system of three partial differential equations modelling the spatiotemporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. For a range of the parameters the system possesses a uniquely determined spatially homogeneous positive equilibrium (u?, v?) globally asymptotically stable within a certain nonempty range of the logistic growth coefficients

On a mathematical model of tumor growth based on cancer stem cells

We consider a simple mathematical model of tumor growth based on cancer stem cells. The model consists of four hyperbolic equations of first order to describe the evolution of different subpopulations of cells: cancer stem cells, progenitor cells, differentiated cells and dead cells. A fifth equation is introduced to model the evolution of the moving boundary. The system includes non-local terms of integral type in the coefficients. Under some restrictions in the parameters we show that there exists a unique homogeneous steady state which is stable

On a two species chemotaxis model with slow chemical diffucion.

In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results

Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

We study the behavior of two biological populations “u” and “v” attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations. The model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production. The system is completed with a third equation modeling the evolution of chemical. We assume that the chemical “w” is a non-diffusive substance and satisfies an ODE, more precisely, {ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u),x∈Ω,t>0,vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v),x∈Ω,t>0,wt=h(u,v,w),x∈Ω,t>0, under appropriate boundary and initial conditions in an n-dimensional open and bounded domain Ω. We consider the cases of positive chemo-sensitivities, not necessarily constant elements. The chemical production function h increases as the concentration of the species “u” and “v” increases. We first study the global existence and uniform boundedness of the solutions by using an iterative approach. The asymptotic stability of the homogeneous steady state is a consequence of the growth of h, χi and the size of μi. Finally, some examples of the theoretical results are presented for particular functions h and χi

Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non-negative solution of a chemotaxis system with non constant chemotaxis sensitivity function X. This system appears as a limit case of a model formorphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75, 2007).Under suitable boundary conditions, modeling the presence of a morphogen source at x=0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover,we prove the convergence of the solution to the unique steady state provided that ? is small and ? is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements