How free boundaries couple Salmonella with hysteresis

In this talk we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes "with memory" in which various substances interact according to hysteresis law. In particular, the growth of a colony of Salmonella typhimurium on a Petri dish is modelled with this parabolic problem. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called "strong solutions" belonging to the anisotropic Sobolev class $W^{2,1}_q$ with sufficiently large q. Several open problems in this direction are proposed as well.