Some nonlocal diffusivities governing biological processes

Abstract: It is well known that the Laplacian operator reproduces a great deal of diffusive processes, due mainly to its capture of aleatory movements through its mean value property, and also thanks to other features such as regularizing effect, infinite speed of propagation, etc. 

However, there are not few situations in which particles or individuals do not move randomly in their media, but according to certain conditions, such as convenient habitats, different cellular interactions, etc. From the middle fifties with the works of Fife, among others, a new line of research was born, that focuses on defining appropriate nonlocal diffusive operators to describe  specific biological processes. 

We have some experience with nonlocal operators, resulting from both, a convolution with a regular kernel, as well as with operators of fractional type. Working in Fractional Orlicz Sobolev spaces,  we have prescribed different behaviour depending on the position of the particle, via the Orlicz function of the fractional incremental  quotient. 

In addition, if we consider a regular kernel, we are reproducing real (and not differential) jumps, which leads to surprising facts. Even when with a symmetric kernel we also obtain geometry properties of solutions such as the median property, in general the results completely differ from the Laplacian or p-Laplacian operator (large solutions, blow-up conditions, irregular aggregation of particles etc.)