19 research outputs found
Multiplicity of supercritical fronts for reaction-diffusion equations in cylinders
We study multiplicity of the supercritical traveling front solutions for
scalar reaction-diffusion equations in infinite cylinders which invade a
linearly unstable equilibrium. These equations are known to possess traveling
wave solutions connecting an unstable equilibrium to the closest stable
equilibrium for all speeds exceeding a critical value. We show that these are,
in fact, the only traveling front solutions in the considered problems for
sufficiently large speeds. In addition, we show that other traveling fronts
connecting to the unstable equilibrium may exist in a certain range of the wave
speed. These results are obtained with the help of a variational
characterization of such solutions
Striped periodic minimizers of a two-dimensional model for martensitic phase transitions
In this paper we consider a simplified two-dimensional scalar model for the
formation of mesoscopic domain patterns in martensitic shape-memory alloys at
the interface between a region occupied by the parent (austenite) phase and a
region occupied by the product (martensite) phase, which can occur in two
variants (twins). The model, first proposed by Kohn and Mueller, is defined by
the following functional: where
is periodic in and almost everywhere.
Conti proved that if then the minimal specific
energy scales like ,
as . In the regime , we improve Conti's results, by computing exactly the
minimal energy and by proving that minimizers are periodic one-dimensional
sawtooth functions.Comment: 29 pages, 3 figure
Domain structure of bulk ferromagnetic crystals in applied fields near saturation
We investigate the ground state of a uniaxial ferromagnetic plate with
perpendicular easy axis and subject to an applied magnetic field normal to the
plate. Our interest is the asymptotic behavior of the energy in macroscopically
large samples near the saturation field. We establish the scaling of the
critical value of the applied field strength below saturation at which the
ground state changes from the uniform to a branched domain magnetization
pattern and the leading order scaling behavior of the minimal energy.
Furthermore, we derive a reduced sharp-interface energy giving the precise
asymptotic behavior of the minimal energy in macroscopically large plates under
a physically reasonable assumption of small deviations of the magnetization
from the easy axis away from domain walls. On the basis of the reduced energy,
and by a formal asymptotic analysis near the transition, we derive the precise
asymptotic values of the critical field strength at which non-trivial
minimizers (either local or global) emerge. The non-trivial minimal energy
scaling is achieved by magnetization patterns consisting of long slender
needle-like domains of magnetization opposing the applied fieldComment: 38 pages, 7 figures, submitted to J. Nonlin. Sci
Phase Dynamics of Nearly Stationary Patterns in Activator-Inhibitor Systems
The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model
are studied using a phase dynamics approach. A Cross-Newell phase equation
describing slow and weak modulations of periodic stationary solutions is
derived. The derivation applies to the bistable, excitable, and the Turing
unstable regimes. In the bistable case stability thresholds are obtained for
the Eckhaus and the zigzag instabilities and for the transition to traveling
waves. Neutral stability curves demonstrate the destabilization of stationary
planar patterns at low wavenumbers to zigzag and traveling modes. Numerical
solutions of the model system support the theoretical findings
Propagation and blocking in periodically hostile environments
We study the persistence and propagation (or blocking) phenomena for a
species in periodically hostile environments. The problem is described by a
reaction-diffusion equation with zero Dirichlet boundary condition. We first
derive the existence of a minimal nonnegative nontrivial stationary solution
and study the large-time behavior of the solution of the initial boundary value
problem. To the main goal, we then study a sequence of approximated problems in
the whole space with reaction terms which are with very negative growth rates
outside the domain under investigation. Finally, for a given unit vector, by
using the information of the minimal speeds of approximated problems, we
provide a simple geometric condition for the blocking of propagation and we
derive the asymptotic behavior of the approximated pulsating travelling fronts.
Moreover, for the case of constant diffusion matrix, we provide two conditions
for which the limit of approximated minimal speeds is positive
Spike autosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model
PACS. 47.54.+rPattern selection; pattern formation – 82.20.-wChemical kinetics and dynamics – 05.45.-a,
Layer solutions for a one-dimensional nonlocal model of Ginzburg–Landau type
We study a nonlocal model of Ginzburg–Landau type that gives rise to an equation involving a mixture of the Laplacian and half-Laplacian. Our focus is on one-dimensional transition layer profiles that connect the two distinct homogeneous phases. We first introduce a renormalized one-dimensional energy that is free from a logarithmic divergence due to the failure of the Gagliardo norm to be finite on smooth profiles that asymptote to different limits at infinity. We then prove existence, uniqueness, monotonicity and regularity of minimizers in a suitable class. Lastly, we consider the singular limit in which the coefficient in front of the Laplacian vanishes and prove convergence of the obtained minimizer to the solutions of the fractional Allen–Cahn equation