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On a Model of Leukemia Development with a Spatial Cell Distribution

A. Ducrot, V. Volpert (2010)

Mathematical Modelling of Natural Phenomena

In this paper we propose a mathematical model to describe the evolution of leukemia in the bone marrow. The model is based on a reaction-diffusion system of equations in a porous medium. We show the existence of two stationary solutions, one of them corresponds to the normal case and another one to the pathological case. The leukemic state appears as a result of a bifurcation when the normal state loses its stability. The critical conditions of leukemia development are determined by the proliferation...

On Chemotaxis Models with Cell Population Interactions

Z. A. Wang (2010)

Mathematical Modelling of Natural Phenomena

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. The extended chemotaxis models have nonlinear diffusion and chemotactic sensitivity depending on cell population density, which is a modification of the classical Keller-Segel model in which the diffusion and chemotactic sensitivity are constants (linear). The existence and boundedness of global solutions of these models are discussed and...

On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

M. Prizzi, K. P. Rybakowski (2003)

Studia Mathematica

We study a family of semilinear reaction-diffusion equations on spatial domains Ω ε , ε > 0, in l lying close to a k-dimensional submanifold ℳ of l . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by H ¹ s ( Ω ) . The definition of H ¹ s ( Ω ) , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...

On the asymptotic behavior for convection-diffusion equations associated to higher order elliptic operators in divergence form.

Mokhtar Kirane, Mahmoud Qafsaoui (2002)

Revista Matemática Complutense

We consider the linear convection-diffusion equation associated to higher order elliptic operators⎧  ut + Ltu = a∇u   on Rnx(0,∞)⎩  u(0) = u0 ∈ L1(Rn),where a is a constant vector in Rn, m ∈ N*, n ≥ 1 and L0 belongs to a class of higher order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on Rn. The aim of this paper is to study the asymptotic behavior, in Lp (1 ≤ p ≤ ∞), of the derivatives Dγu(t) of the solution of the convection-diffusion equation...

On the cost of null-control of an artificial advection-diffusion problem

Pierre Cornilleau, Sergio Guerrero (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the null-controllability of an artificial advection-diffusion system in dimension n. Using a spectral method, we prove that the control cost goes to zero exponentially when the viscosity vanishes and the control time is large enough. On the other hand, we prove that the control cost tends to infinity exponentially when the viscosity vanishes and the control time is small enough.

On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

J.A. Sherratt (2010)

Mathematical Modelling of Natural Phenomena

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front...

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