Convergence to a receding wave in a monostable free boundary problem

Abstract We study a monostable reaction-diffusion equation of the form $$u_t=du_{xx}+f(u)$$ u t = d u xx + f ( u ) over a semi-infinite spatial domain $$[g(t),\infty )$$ [ g ( t ) , ∞ ) , with $$x=g(t)$$ x = g ( t ) the free boundary whose evolution is governed by equations derived from a “preferred population density” principle, which postulates that the species with population density u ( t , x ) and population range $$[g(t),\infty )$$ [ g ( t ) , ∞ ) maintains a certain density $$\delta $$ δ at the habitat edge $$x=g(t)$$ x = g ( t ) . In the “high-density” regime, where $$\delta $$ δ exceeds the carrying capacity of the favourable environment represented by a monostable function f ( u ), it is known; see [6] for the case of a bounded population range [g(t), h(t)], that for large time, the front retreats as time advances. In this work, the unboundedness of the population range $$[g(t),\infty )$$ [ g ( t ) , ∞ ) allows us to prove that, as time t converges to infinity, the free boundary $$x=g(t)$$ x = g ( t ) converges to $$\infty $$ ∞ with a constant asymptotic speed $$c(\delta )>0$$ c ( δ ) > 0 determined by an associated semi-wave problem, and the population density u ( t , x ) has the property that $$u(t,x+g(t))$$ u (