Asymptotic analysis of localized patterns in PDEs (CAIMS)
Org: David Iron and Theodore Kolokolnikov (Dalhousie) [PDF]
 GUILLEMETTE CHAPUISAT, ENS Cachan, 61, av. du Président Wilson, 94235 Cachan Cedex, France
Model of spreading depression and existence of travelling front
[PDF] 
Spreading depression is a transient depolarization of neurons that spreads slowly through a part of the brain during stroke, epilepsy or migraine with aura. They have been observed and studied in most animal species for more than 50 years, but their existence in the human brain is still discussed. Mathematical models of spreading depressions have been established; they are linked to a reactiondiffusion mechanism.
After some numerical experiments, we have made the following
hypothesis: The mechanisms that trigger spreading depressions are the same in the human brain as in the rodent brain, but the morphology of the human brain could explain the nonobservation of these waves.
Hence I have studied the following equation:
¶_{t} u  \triangle u = lu (uq)(1u) 1_{z < R}  au 1_{z ³ R} 

where (x,z) Î R^{N} is the space variable.
I have proved that if R is small enough, there is no travelling front solution of these equation. And if R is large enough, there exists a travelling front in the xdirection. This result is obtained by studying the energy of a solution with special initial conditions in several travelling referentials.
 DAVID IRON, Dalhousie University, Department of Mathematics and Statistics, Halifax, NS B3H 3J5
Stability of curved interfaces to the two dimensional perturbed AllenCahn equations
[PDF] 
We consider equilibrium solutions to a perturbed AllenCahn model in bounded 2dimensional domains that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the associated twodimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.
 THEODORE KOLOKOLNIKOV, Dalhousie
Ring solutions in R^{N} and smokering (vortex) solutions in R^{3} for GiererMeinhardt Model
[PDF] 
We consider the classical GiererMeinhardt Model in N dimensions,
e^{2} Duu+ 
u^{p}
v^{q}

= 0, Dvv + 
u^{m}
v^{s}

= 0 

where e is assumed to be small.
A ringtype solution in R^{N} is a solution that concentrates on the surface of an Nsphere as e® 0. On the other hand, a smokering or vortex solution in R^{3} is a solution that concentrates on the perimeter of a twodimensional circle.
For ring solutions, assume
0 < 
p1
q

< a_{¥} if N = 2, and 0 < 
p1
q

< 1 if N ³ 3 

where a_{¥} > 1 whose numerical value is a_{¥}=1.06119. We prove that there exists a unique R_{a} > 0 such that for R Î (R_{a},+¥], there is a ringtype solution inside the ball of radius R (R=+¥ corresponds to R^{N} case), that concentrates on the surface of a ball of radius 0 < r_{0} < R. Moreover depending on parameter values, there are either exactly one or two choices for r_{0}.
For smokering solutions, we study the case when the domain is all of R^{3}. We then show that a smokering solution concentrates on a circle whose radius is precisely r_{0} = 0.43385.
The analysis of ring solutions relies heavily on manipulation of Bessel functions. The analysis for smokering solutions involves a deep expansion of a certain singular integral.
This is a joint work with Juncheng Wei (rings) and with Xiaofeng Ren (smokerings).
 MICHAL KOWALCZYK, Universidad de Chile
Stationary spot solutions in an activatorinhibitor system
[PDF] 
In this talk I will discuss the existence of spot solutions in a FitzHughNagumo type system. These solutions turn out to be stable thanks to a delicate balance between the tendency of the system to locally minimize its energy and the curve shortening effect.
This is a joint work with Xinfu Chen.
 YASUMASA NISHIURA, RIES, Hokkaido University, N12, W6, Kitaku, Sapporo, Japan
Dynamics of Particle Patterns in Dissipative Systems
[PDF] 
Particle patterns mean any spatially localized structures sustained by the balance between inflow and outflow of energy/material which arise in the form of chemical blob, discharge pattern, morphological spot, and binary convection cell. These are modeled by typically threecomponent reaction diffusion systems or a couple of complex GL equations with concentration field. They collide with each other, interact with defects and experience large deformation and/or basinswitching of dynamics in the form of merging, annihilation, rebound, and pinning to the defect created by heterogeneities. A new viewpoint based on a network of hidden saddles is presented to reveal the skeletal structure of those complex transient dynamics.
 XIAOFENG REN, George Washington University
Ideal and Defective Solutions to a Free Boundary Problem from Block Copolymer Morphology
[PDF] 
The OhtaKawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. In a proper parameter range an equilibrium pattern of many droplets is proved to exist in a general planar domain. A subrange is identified where the multiple droplet pattern is stable. A defective ring pattern solution is also found. The importance of the the resonance condition is carefully studied.
 MICHAEL WARD, Dept. of Mathematics, UBC, Vancouver, V6T 1Z2
SelfReplicating Spots for ReactionDiffusion Models in Two Space Dimensions
[PDF] 
We analyze the dynamical behavior of multispot solutions in a twodimensional domain W for certain twocomponent reactiondiffusion models, including the GrayScott model, in the singularly perturbed limit of small diffusivity e for one of the two components. A formal asymptotic analysis, which has the effect of summing infinite logarithimic series in powers of 1/loge, is used to derive an differential algebraic system of ODE's characterizing the slow dynamics of the spot locations. By numerically examining the stability thresholds for a single spot solution, a specific and simple criterion is formulated to theoretically predict the initiation spotreplication events. The analytical theory is compared with full numerical results.
 REBECCA WHITE, Dalhousie

