Elaine Crooks

On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions

We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions.

Travelling waves for reaction-diffusion-convection systems

(1) ut = Auxx + f(u), u ∈ R , x ∈ R, t ∈ [0,∞), where A is a real, positive-definite, N × N matrix and f : R → R is a continuously differentiable nonlinear function. The vector u may represent, for example, the concentrations of chemicals or the population densities of interacting species, the interactions between components of u being modelled by the reaction term f(u) and their diffusion by Auxx. Travelling waves are solutions u of (1) in the form

A Spatially Extended Model of Kinase-Receptor Interaction

We perform a mathematical analysis of a spatially extended model describing mutual phosphorylation of cytosolic kinases and membrane receptors. The analyzed regulatory system is a part of signal transduction mechanisms, which enables communication of the cell with its extracellular environment or other cells. The mutual receptor-kinase interaction is characteristic for immune receptors and Src family kinases. From the mathematical viewpoint, the considered system is interesting because it couples differential equations defined in a domain

Compensated Convexity, Multiscale Medial Axis Maps and Sharp Regularity of the Squared-Distance Function

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Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations

and on its boundary

Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds

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Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions

via nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which the Robin boundary condition on the external boundary of the spherical shell

Front-like entire solutions for equations with convection

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Local minimizers and planar interfaces in a phase-transition model with interfacial energy

is replaced by a uniform Dirichlet boundary condition. This method allows us to find the stationary spherically symmetric solutions, both stable and unstable. Interestingly, numerical computations sug...

On the Vol'pert theory of travelling-wave solutions for parabolic systems

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale