In this talk I will discuss the front propagation for a certain class of nonlinear diffusion equations on hyperbolic space . More specifically we consider solutions whose intital data are nonnegative and compactly supported, and study how their fronts (i.e. the level surfaces near the transition layer) spread over the space. Much attention will be directed to the similarities as well as the differences between the case of and that of the Euclidean space .
Among other things we show that the global shape of the expanding fronts will remain close to an expanding geodesic sphere, while the local profile of the solution near the front area converges to what we call a “horospheric wave”.
This is joint work with Fabio Punzo and Alberto Tesei.