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Abstract: In this paper we study the long time-long range behavior of reactiondiffusion equations with negative square root -type reaction terms. Inparticular we investigate the exponential behavior of the solutions after astandard hyperbolic scaling. This leads to a Hamilton-Jacobi variationalinequality with an obstacle that depends on the solution itself and defines theopen set where the limiting solution does not vanish. Counter-examples show anontrivial lack of uniqueness for the variational inequality depending on theconditions imposed on the boundary of this open set. Both Dirichlet and stateconstraints boundary conditions play a role. When the competition term does notchange sign, we can identify the limit, while, in general, we find lower andupper bounds for the limit. Although models of this type are rather old andextinction phenomena are as important as blow-up, our motivation comes from theso-called -tail problem- in population biology. One way to avoid meaninglessexponential tails, is to impose extra-mortality below a given survivalthreshold. Our study shows that the precise form of this extra-mortality termis asymptotically irrelevant and that, in the survival zone, the populationprofile is impacted by the survival threshold except in the very particularcase when the competition term is non-positive.

Author: Sepideh Mirrahimi LJLL, Guy Barles LMPT, FRDP, Benoit Perthame LJLL, INRIA Rocquencourt, Panagiotis E. Souganidis


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