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Front propagation for nonlinear diffusion equations on the hyperbolic space. (English) Zbl 1322.35065
The paper addresses nonlinear parabolic equations of the Kolmogorov-Petrovskii-Piskunov (KPP) and Allen-Cahn (AC) types, which, unlike their well-studied counterparts in the Euclidean space, are considered in a hyperbolic space of dimension $$n$$. While the usual KPP and AC equations in the multidimensional Euclidean space are actually applied for plane waves, the counterparts of these equations apply here to the so-called horospheric waves. The main result is that the well-known asymptotic velocity $$c$$ of the KPP equation is shifted to $$c - (n-1)$$. As a result, the KPP equation in hyperbolic space may give rise to both propagation and extinction (decay of the field), the latter effect occurring when the shifted velocity becomes formally negative. The AC equation in hyperbolic space may also give rise to either propagation, with the shifted asymptotic velocity, or extinction.

##### MSC:
 35K55 Nonlinear parabolic equations 35K57 Reaction-diffusion equations
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