APDE seminar @BCAM: Traveling wave behavior for KPP equation on the hyperbolic space

Abstract

In this talk we present the Cauchy problem on hyperbolic space for the heat equation with a KPP type forcing term. Our goal is to understand how hyperbolic geometry affects the dynamics of solutions. We address the question of propagation versus extinction, including the critical case. In the case of propagation, we show that if the initial datum has a certain symmetry, the solution converges asymptotically to a traveling wave of minimal speed in a moving frame. The choice of this moving frame  depends on the symmetries of the initial datum, which, in turn, is closely related to the three types of isometries in hyperbolic space: elliptic, hyperbolic, and parabolic.