Interface Dynamics in Evolution Equations – IDEE

In some classes of reaction diffusion equations, solutions may develop sharp internal layers, or interfaces, that separate the spatial domain into different phase regions. This happens, in particular, when the reaction term is very large compared with the diffusion term.This project is concerned with the singular limit of reaction diffusion equations, as a parameter related to the thickness of a diffuse interface tends to zero. We shall be concerned with both Fisher-KPP (monostable) nonlinearities and Allen-Cahn (bistable) nonlinearities.

1.Fisher-KPP:

Reaction diffusion equations with logistic nonlinearity were introduced in the pioneer works of Fisher or Kolmogorov, Petrovsky and Piskunov. They are widely used in the literature to model phenomena arising in population genetics, or in biological invasions. The main property of such equations is to admit (biologically relevant) travelling wave solutions with some semi-infinite interval of admissible wave speed.

Is is known that, under some assumptions on the initial data, the singular limit of the rescaled Fisher-KPP equation is an interface moving by constant speed which turns out to be the minimal speed of monotone related one-dimensional travelling waves.

In this part of the project, we would like to address the following issues:

a. What happens if one introduces a delay effect?
b. What happens if one introduces a non-local effect?
c. What is the real link between the convergence of the problem and the stability of the associated travelling waves?

2. Allen-Cahn:

Allen and Cahn have introduced a microscopic diffusional theory for the motion of a curved antiphase boundary. The interfacial velocity is found to be linearly proportional to the mean curvature of the boundary and the constant of proportionality does not include the specific
surface free energy. This new theory is confirmed by experimental measurements of domain coarsening kinetics in Fe---Al alloys.

For even not well-prepared initial data, precise studies of both the generation and the motion of interface have shown that the rescaled Allen-Cahn equation converges to motion by mean curvature (MMC in short). In the classical MMC framework (i.e. for small times), fine estimates of the thickness of the transition layers are also known. In the viscosity framework, the convergence for all times to the generalized MMC defined by Evans-Spruck and Chen-Giga-Goto is proved.

As far as Allen-Cahn equations are concerned, we would like to address the following issues.

a. Can we estimate the thickness of the transition layers past the onset of singularities?
b. What happens if we consider density-dependent diffusion?
c. What happens if we replace the classical Laplace operator with the fractional Laplace operator?
d. What happens when the comparison principle does not hold?