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The continuum reaction-diffusion limit of a stochastic cellular growth model

Stephan Luckhaus, Livio Triolo (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered....

The CUDA implementation of the method of lines for the curvature dependent flows

Tomáš Oberhuber, Atsushi Suzuki, Vítězslav Žabka (2011)

Kybernetika

We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the...

The defocusing energy-critical Klein-Gordon-Hartree equation

Qianyun Miao, Jiqiang Zheng (2015)

Colloquium Mathematicae

We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution u t t - Δ u + u + ( | x | - 4 ∗ | u | ² ) u = 0 in spatial dimension d ≥ 5. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the momentum vector...

The Dirichlet problem in weighted spaces on a dihedral domain

Adam Kubica (2009)

Banach Center Publications

We examine the Dirichlet problem for the Poisson equation and the heat equation in weighted spaces of Kondrat'ev's type on a dihedral domain. The weight is a power of the distance from a distinguished axis and it depends on the order of the derivative. We also prove a priori estimates.

The dynamics of a levitated cylindrical permanent magnet above a superconductor.

Michael Schreiner (2003)

Revista Matemática Complutense

When a permanent magnet is released above a superconductor, it is levitated. This is due to the Meissner-effect, i.e. the repulsion of external magnetic fields within the superconductor. In experiments, an interesting behavior of the levitated magnet can be observed: it might start to oscillate with increasing amplitude and some magnets even reach a continuous rotation. In this paper we develop a mathematical model for this effect and identify by analytical methods as well with finite element simulations...

The dynamics of weakly interacting fronts in an adsorbate-induced phase transition model

Shin-Ichiro Ei, Tohru Tsujikawa (2009)

Kybernetika

Hildebrand et al. (1999) proposed an adsorbate-induced phase transition model. For this model, Takei et al. (2005) found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in ℝ 2 and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates...

The Effect of Bacteria on Epidermal Wound Healing

E. Agyingi, S. Maggelakis, D. Ross (2010)

Mathematical Modelling of Natural Phenomena

Epidermal wound healing is a complex process that repairs injured tissue. The complexity of this process increases when bacteria are present in a wound; the bacteria interaction determines whether infection sets in. Because of underlying physiological problems infected wounds do not follow the normal healing pattern. In this paper we present a mathematical model of the healing of both infected and uninfected wounds. At the core of our model is an...

The existence of a periodic solution of a parabolic equation with the Bessel operator

Dana Lauerová (1984)

Aplikace matematiky

In this paper, the existence of an ω -periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions f ( t , r ) , h ( t ) , a ( t ) are assumed to be ω -periodic in t , f ∈ L 2 ( S , H ) , a , h such that a ' ∈ L ∞ ( R ) , h ' ∈ L ∞ ( R ) and they fulfil (3). The solution u belongs to the space L 2 ( S , V ) ∩ L ∞ ( S , H ) , has the derivative u ' ∈ L 2 ( S , H ) and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

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