34,395 research outputs found
Pattern formation driven by cross--diffusion in a 2D domain
In this work we investigate the process of pattern formation in a two
dimensional domain for a reaction-diffusion system with nonlinear diffusion
terms and the competitive Lotka-Volterra kinetics. The linear stability
analysis shows that cross-diffusion, through Turing bifurcation, is the key
mechanism for the formation of spatial patterns. We show that the bifurcation
can be regular, degenerate non-resonant and resonant. We use multiple scales
expansions to derive the amplitude equations appropriate for each case and show
that the system supports patterns like rolls, squares, mixed-mode patterns,
supersquares, hexagonal patterns
Global existence and asymptotic behavior for a nonlinear degenerate SIS model
In this paper we investigate the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in the modeling and the spatial spread of an epidemic disease
On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities
We study a non-local variant of a diffuse interface model proposed by
Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical
species acting as nutrient. The system consists of a Cahn--Hilliard equation
coupled to a reaction-diffusion equation. For non-degenerate mobilities and
smooth potentials, we derive well-posedness results, which are the non-local
analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015).
Furthermore, we establish existence of weak solutions for the case of
degenerate mobilities and singular potentials, which serves to confine the
order parameter to its physically relevant interval. Due to the non-local
nature of the equations, under additional assumptions continuous dependence on
initial data can also be shown.Comment: 28 page
Blow-up properties for a degenerate parabolic system with nonlinear localized sources
AbstractThis paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction–diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304–320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball
A coupled system of PDEs and ODEs arising in electrocardiograms modelling
We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure
Perturbation Analysis of the Kuramoto Phase Diffusion Equation Subject to Quenched Frequency Disorder
The Kuramoto phase diffusion equation is a nonlinear partial differential
equation which describes the spatio-temporal evolution of a phase variable in
an oscillatory reaction diffusion system. Synchronization manifests itself in a
stationary phase gradient where all phases throughout a system evolve with the
same velocity, the synchronization frequency. The formation of concentric waves
can be explained by local impurities of higher frequency which can entrain
their surroundings. Concentric waves in synchronization also occur in
heterogeneous systems, where the local frequencies are distributed randomly. We
present a perturbation analysis of the synchronization frequency where the
perturbation is given by the heterogeneity of natural frequencies in the
system. The nonlinearity in form of dispersion, leads to an overall
acceleration of the oscillation for which the expected value can be calculated
from the second order perturbation terms. We apply the theory to simple
topologies, like a line or the sphere, and deduce the dependence of the
synchronization frequency on the size and the dimension of the oscillatory
medium. We show that our theory can be extended to include rotating waves in a
medium with periodic boundary conditions. By changing a system parameter the
synchronized state may become quasi degenerate. We demonstrate how perturbation
theory fails at such a critical point.Comment: 22 pages, 5 figure
Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth
The global existence of bounded weak solutions to a diffusion system modeling
biofilm growth is proven. The equations consist of a reaction-diffusion
equation for the substrate concentration and a fourth-order Cahn-Hilliard-type
equation for the volume fraction of the biomass, considered in a bounded domain
with no-flux boundary conditions. The main difficulties are coming from the
degenerate diffusivity and mobility, the singular potential arising from a
logarithmic free energy, and the nonlinear reaction rates. These issues are
overcome by a truncation technique and a Browder-Minty trick to identify the
weak limits of the reaction terms. The qualitative behavior of the solutions is
illustrated by numerical experiments in one space dimension, using a BDF2
(second-order backward Differentiation Formula) finite-volume scheme
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