Jack Carr

Applications of Centre Manifold Theory.

0.- 4.5. The Case ?1 < 0.- 4.6. More Scaling.- 4.7. Completion of the Phase Portraits.- 4.8. Remarks and Exercises.- 4.9. Quadratic Nonlinearities.- 5. Application to a Panel Flutter Problem.- 5.1. Introduction.- 5.2. Reduction to a Second Order Equation.- 5.3. Calculation of Linear Terms.- 5.4. Calculation of the Nonlinear Terms.- 6. Infinite Dimensional Problems.- 6.1. Introduction.- 6.2. Semigroup Theory.- 6.3. Centre Manifolds.- 6.4. Examples.- References.

Uniqueness of travelling waves for nonlocal monostable equations

We consider a nonlocal analogue of the Fisher-KPP equation u t = J * u - u + f(u), x ∈ R, f(0) = f(1) = 0, f > 0 on (0,1), and its discrete counterpart u n = (J * u) n - u n + f(u n ), n E Z, and show that travelling wave solutions of these equations that are bounded between 0 and 1 are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikeharas Theorem (which is a Tauberian theorem for Laplace transforms).

The Discrete Coagulation-Fragmentation Equations: Existence, Uniqueness, and Density Conservation

The discrete coagulation-fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. These models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys. Our results relate to existence, uniqueness, density conservation and continuous dependence and they generalise the corresponding results in [ref. 2] for the Becker-Doring equations for which the processes are restricted to clusters gaining or shedding one particle. Examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation-fragmentation equations.

The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density ϱ is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions byc(ϱ), 0 ≦ ϱ ≦ ϱs, the principal result is that if the initial density ϱ0 ≦ ϱs then the solution converges strongly toc(ϱo), while if ϱ0 > ϱs the solution converges weak* toc(ϱs). In the latter case the excess density ϱ0–ϱs corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions.

Abelian integrals and bifurcation theory

Abstract Conditions are given for uniqueness of limit cycles for autonomous equations in the plane. The results are applicable to codimension two bifurcations near equilibrium points for vector fields.

Invariant manifolds for metastable patterns in ut=ε2uxx-f(u)

We consider the above equation on the interval 0 ≦ x ≦ 1 subject to Neumann boundary conditions with f ( u ) = F ′( u ) where F is a double well energy density function with equal minima. Our previous work [3] proved the existence and persistence of very slowly evolving patterns (metastable states) in solutions with two-phase initial data. Here we characterise these metastable states in terms of the global unstable manifolds of equilibria, as conjectured by Fusco and Hale [6].

Instantaneous gelation in coagulation dynamics

The coagulation equations are a model for the dynamics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters. For a certain class of rate coefficients we prove that the density is not conserved on any time interval.

Asymptotic behavior of solutions to the Coagulation-Fragmentation Equations. II. Weak Fragmentation

The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.

Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system

Travelling waves are natural phenomena ubiquitously for reaction-diffusion systems in many scientific areas, such as in biophysics, population genetics, mathematical ecology, chemistry, chemical physics, and so on. It is pretty well understood for a diffusing Lotka-Volterra system that there exist travelling wave solutions which propagate from an equilibrium point to another one. In this paper, we prove there exists, at least, a wave front--the monotone travelling wave-- with its minimal speed.

The application of centre manifolds to amplitude expansions. II: Infinite dimensional problems

Abstract The theory of centre manifolds for infinite dimensional systems is described, with emphasis on the practical computational aspects of applying the theory to near-critical problems, and in particular to computation of the centre manifold. The calculations are illustrated by detailed analyses of two specific problems.