Cross-diffusion models in biomathematics are of strong interest, e.g. in ecology.
A well known example in microbiology is
the by now classical Keller-Segel model
for chemotaxis. The original system of
four PDEs can be reduced to two PDEs:
a diffusion equation with strong nonlinear drift for chemotactically moving cells and a reaction-diffusion equation for the attractive chemical agent. In a further reduction this model relates to classical models for self-gravitational collapse. A change of sign for the nonlinear drift relates to semi-conductor equations.
Interestingly, the occuring blowup of solutions relates to the biological phenomena of self-organisation.
In two spatial dimensions a crucial dichotomy was proved in the 90's, namely blowup of solutions vs. existence of global solutions in dependence of a critical parameter, which relates to the strength of the nonlinear drift or to a critical mass.
Proofs depend, e.g. on the Moser-Trudinger inequality and non-trivial stationary states relate to a certain extent to the Gauss-Bonnet formula.
Solving the stationary reaction-diffusion equation for the chemical agent, and plugging it into the diffusion-drift equation, a non-local equation with Newtonian or Bessel potential results.
Generalizing these potentials relates to the analysis of by now so-called aggregation equations.
In this talk we present qualitative results on pattern formation within this class of nonlinear equations, including the development of singularities.