Extinction Versus Persistence in Strong Oscillating Flows

In this paper, we give some conditions for finite-time extinction or persistence of the solutions of diffusion–advection equations in strong and oscillating flows under Dirichlet boundary conditions. The enhancement of the diffusion rate depends on the interplay between strong advection and time-homogenization, and in particular on the ratio between the strength of the flow and its frequency parameter. Quantitative estimates of this ratio, which depend on the geometry of the domain, are provided in the case of a uniform flow. In the general time–space dependent case, the finite-time behavior of the solutions is related to the existence of first integrals of the flow.