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Volume Filling Effect in Modelling Chemotaxis

D. Wrzosek (2010)

Mathematical Modelling of Natural Phenomena

The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized models have been proposed to describe the aggregation phase of this process. Most of...

Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation

Sylvia Serfaty (2007)

Journal of the European Mathematical Society

We study vortices for solutions of the perturbed Ginzburg–Landau equations Δ u + ( u / ε 2 ) ( 1 | u | 2 ) = f ε where f ε is estimated in L 2 . We prove upper bounds for the Ginzburg–Landau energy in terms of f ε L 2 , and obtain lower bounds for f ε L 2 in terms of the vortices when these form “unbalanced clusters” where i d i 2 ( i d i ) 2 . These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including...

Wave Equation with Slowly Decaying Potential: asymptotics of Solution and Wave Operators

S. A. Denisov (2010)

Mathematical Modelling of Natural Phenomena

In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.

Weak- L p solutions for a model of self-gravitating particles with an external potential

Andrzej Raczyński (2007)

Studia Mathematica

The existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential is studied in weak- L p spaces (i.e. Markiewicz spaces). The main goal is to prove the existence of global solutions and to study their large time behaviour.

Weak Solutions for a Fourth Order Degenerate Parabolic Equation

Changchun Liu, Jinyong Guo (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We consider an initial-boundary value problem for a fourth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the discrete-time method. The asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: d d t u + i - Δ + V ( x ) u = ν Δ u - γ R | u | 2 p u - i γ I | u | 2 q u + ν η ( t , x ) . ( * ) The force η is white in time and smooth in x ; the potential V ( x ) is typical. We are concerned with the limiting, as ν 0 , behaviour of solutions on long time-intervals 0 t ν - 1 T , and with behaviour of these solutions under the double limit t and ν 0 . We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for(...

Worst scenario method in homogenization. Linear case

Luděk Nechvátal (2006)

Applications of Mathematics

The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized...

Σ -convergence of nonlinear monotone operators in perforated domains with holes of small size

Jean Louis Woukeng (2009)

Applications of Mathematics

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in N with isolated holes of size ε 2 ( ε > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator...

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