Front propagation for nonlinear diffusion equations on the hyperbolic space

We study the Cauchy problem in the hyperbolic space ℍⁿ (n ≥ 2) for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space ℝⁿ new phenomena arise, which depend on the properties of the diffusion process in ℍⁿ. We also investigate a family of travelling wave solutions, named horospheric waves, which have properties similar to those of plane waves in ℝⁿ.