The Trichotomy Of Solutions And Sharp Transitions For R-A-D Equations In Cylinders

In this paper,the initial-boundary value problem of the reaction-advection-diffusion equations of the form ut-Δu+α(y)us=f(u)with bistable nonlinearity in cylinders is studied.We prove the trichotomy of the solutions of the initial boundary value problem by using a comparison argument and giving certain growth conditions to the one-parameter family of initial value,so as to obtain the existence of the threshold solutions.And then by constructing appropriate auxiliary functions,we prove that any threshold solution is monotonic with respect to x when |x| is large enough,and in turn is applied to the spatial exponential decay of the threshold solutions.On this basis,we show that the transition from extinction to propagation is sharp.The proof rely on the method of dimension reduction and the results on exponential separations and principle Floquet bundles for the linear parabolic equations on R.