A free boundary problem with nonlinear advection and Dirichlet boundary condition

Abstract

We study a free boundary problem for Fisher–KPP equation with nonlinear advection ut=uxx−uux+f(u) on [0,h(t)], which can model the spreading of chemical substances or biological species in the moving region. In this model, the free boundary h(t) indicates the spreading front of the species. Due to some factors (such as the migration of species), the advection is affected by population density. This paper mainly studies the asymptotic behavior of solutions. We prove that, the solution is either spreading (the survival area [0,h(t)] tends to [0,+∞), the solution converges to a stationary solution defined on the half-line), or converging to small steady state ([0,h(t)] goes to a finite interval and the solution converges to a small stationary solution with compacted support), or converging to big steady state ([0,h(t)] tends to a bigger finite interval, the solution converges to a large stationary solution with compacted support). Besides this, we also prove that, when the input of the species is a critical value, the solution is either spreading or in converging to medium steady state. Additionally, we also have two different spreading results. Finally, using traveling semi-wave, we give the spreading speed when spreading happens.