37 research outputs found
Constructing solutions for a kinetic model of angiogenesis in annular domains
We prove existence and stability of solutions for a model of angiogenesis set
in an annular region. Branching, anastomosis and extension of blood vessel tips
are described by an integrodifferential kinetic equation of Fokker-Planck type
supplemented with nonlocal boundary conditions and coupled to a diffusion
problem with Neumann boundary conditions through the force field created by the
tumor induced angiogenic factor and the flux of vessel tips. Our technique
exploits balance equations, estimates of velocity decay and compactness results
for kinetic operators, combined with gradient estimates of heat kernels for
Neumann problems in non convex domains.Comment: to appear in Applied Mathematical Modellin
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant
We study the behavior of two biological populations âuâ and âvâ 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 âwâ is a non-diffusive substance and satisfies an ODE, more precisely,
{ut=Îuâââ
(uÏ1(w)âw)+ÎŒ1u(1âu),xâΩ,t>0,vt=Îvâââ
(vÏ2(w)âw)+ÎŒ2v(1âv),xâΩ,t>0,wt=h(u,v,w),xâΩ,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 âuâ and âvâ 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, Ïi and the size of ÎŒi. Finally, some examples of the theoretical results are presented for particular functions h and Ïi
Dynamics in a Chemotaxis Model with Periodic Source
We consider a system of two differential equations modeling chemotaxis. The system consists of a parabolic equation describing the behavior of a biological species âuâ coupled to an ODE patterning the concentration of a chemical substance âvâ. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function h, which increases as âuâ increases. Under suitable assumptions we prove that the solution is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a certain periodic asymptotic behavior. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system showing that the model considered here can provide very interesting and complex dynamics
An inverse problem for the compressible Reynolds equation
We study the existence of solutions to a system of equations for equilibrium positions in lubricated journal bearings under load effects. The mechanism under consideration consists of two parallel cylinders, one inside the other, in close distance and relative motion. The unknowns of the problem are the equilibrium position of the inner cylinder and the pressure of the lubricant described by the compressible Reynolds equation. To complete the system, Newton's second law gives the equilibrium of forces. We present results on existence of solutions for a range of applied forces F
Uniform boundedness of solutions for a predator-prey system with diffusion and chemotaxis
In this Note we study a nonlinear system of reaction-diffusion differential equations consisting of an ordinary differential equation coupled to a fully parabolic chemotaxis system. This system constitutes a mathematical model for the evolution of a prey-predator biological population with chemotaxis and dormant predators. Under suitable assumptions we prove the global in time existence and boundedness of classical solutions of this system in any space dimension
Uniform boundedness of solutions for a predator-prey system with diffusion and chemotaxis
In this Note we study a nonlinear system of reaction-diffusion differential equations consisting of an ordinary differential equation coupled to a fully parabolic chemotaxis system. This system constitutes a mathematical model for the evolution of a prey-predator biological population with chemotaxis and dormant predators. Under suitable assumptions we prove the global in time existence and boundedness of classical solutions of this system in any space dimension
On a parabolic-elliptic chemotactic system with non-constant chemotactic sensitivity
We study a parabolicâelliptic chemotactic system describing the evolution of a populationâs density âuâ and a chemoattractantâs concentration âvâ. The system considers a non-constant chemotactic sensitivity given by âÏ(Nâu)â, for Nâ„0, and a source term of logistic type âλu(1âu)â. The existence of global bounded classical solutions is proved for any Ï>0, Nâ„0 and λâ„0. By using a comparison argument we analyze the stability of the constant steady state u=1, v=1, for a range of parameters.
â For N>1 and Nλ>2Ï, any positive and bounded solution converges to the steady state.
â For Nâ€1 the steady state is locally asymptotically stable and for ÏN<λ, the steady state is globally asymptotically stable
Asymptotic stability of a mathematical model of cell population
We consider a simplified system of a growing colony of cells described as a free boundary problem. The system consists of two hyperbolic equations of first order coupled to an ODE to describe the behavior of the boundary. The system for cell populations includes non-local terms of integral type in the coefficients. By introducing a comparison with solutions of an ODE's system, we show that there exists a unique homogeneous steady state which is globally asymptotically stable for a range of parameters under the assumption of radially symmetric initial data