We study the Cauchy problem in the hyperbolic space 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 new phenomena arise, which depend on the properties of the diffusion process in . We also investigate a family of travelling wave solutions, named horospheric waves, which have properties similar to those of plane waves in .
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Hiroshi Matano, Fabio Punzo, Alberto Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space. J. Eur. Math. Soc. 17 (2015), no. 5, pp. 1199–1227DOI 10.4171/JEMS/529