R. Marangell and D. E. Pelinovsky
Selection of the ground state on a compact metric graph
Abstract:
We show that the ground state in the Fisher–KPP model on a compact metric
graph with Dirichlet conditions on boundary vertices is either trivial (zero) or nontrivial
and strictly positive. For positive initial data, we prove that the trivial ground
state is globally asymptotically stable if the edges of the metric graph are uniformly
small and the nontrivial ground state is globally asymptotically stable if the edges are
uniformly large. For the intermediate case, we find a sharp criterion for the existence,
uniqueness and global asymptotic stability of the trivial versus nontrivial ground state.
Besides standard methods based on the comparison theory, energy minimizers, and
the lowest eigenvalue of the graph Laplacian, we develop a novel method based on the
period function for differential equations to characterize the nontrivial ground state
in the particular case of flower graphs.
Keywords:
Fisher-KPP equation; flower graphs, tadpole graphs,
ground state, period function.