Study On The Reaction-diffusion Equation With Double Free Boundaries

This paper researches the reaction-diffusion equations with advection term and double free boundary conditions ut=uxx-β(t)ux+f(u),where uxx is diffusion term,ux is advection term,β(t)is the coefficient of advection term and f(u)is the reaction term.This equation can be used to describe the dynamic behavior of biological population in advection environment.We first consider the advection-reaction-diffusion equation,where β is a constant and f(u)=αum(1-un)(α>0,m,n ∈ N*).Through the phase diagram analysis,the influence of the parity difference of m and n on the heteroclinic orbits is studied,and the exact heteroclinic orbit of the system with corresponding value is solved by using the logarithmic sequence scaling method.By studying the influence of advection coefficient and initial conditions on the long-time behavior of the solutions,we obtain a spreading-transition-vanishing trichotomy whenβis small,a virtual spreading-transition-vanishing trichotomy when β is a medium-size constant,and all vanishing happens when β is large.Numerical analysis is given to show the theoretical results more intuitively.Furthermore,a time aperiodic advection-reaction-diffusion equation with free boundaries and mth-order Fisher-KPP nonlinearity(f(u)=αum(1-u)(α>0,m ∈ N*))is proposed.By considering the long-time behavior of the solutions,we obtain a spreading-vanishing dichotomy when β(t)is a small function,a spreadingtransition-vanishing trichotomy whenβ(t)is a medium-sized function,and all vanishing happens when β(t)is a large function.The partition of β(t)relies on the mean value(?)dt and the shape(?)We choose appropriate parameters in the simulation to intuitively show the theoretical results.In addition,the wave-spreading and wave-vanishing of the solutions are observed in numerical analysis.