Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition

We study the reaction diffusion equation $ u_t-du_{xx} = f(u) $ with a monostable nonlinear function $ f(u) $ over a changing interval $ [g(t), h(t)] $, viewed as a model for the spreading of a species with population range $ [g (t), h(t)] $ and density $ u(t,x) $. The free boundaries $ x = g(t) $ and $ x = h(t) $ are not governed by the same Stefan condition as in Du and Lin [20] and other previous works; instead, they satisfy a related but different set of equations obtained from a 'preferred population density' assumption at the range boundary, which allows the population range to shrink. We obtain a rather complete understanding of the longtime dynamics of the model, which exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for $ x $ over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution $ u $ of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem.