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

We study the reaction diffusion equation ut − duxx = 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.