Convergence to a receding wave in a monostable free boundary problem

TL;DR

This work analyzes a monostable reaction-diffusion equation on a semi-infinite habitat with a moving free boundary driven by the edge density $\delta$, i.e. $u_t= d u_{xx}+f(u)$ for $x>g(t)$ and $g'(t)= -\frac{d}{\delta}u_x(t,g(t))$ with $u(t,g(t))=\delta$. In the high-density regime $\delta>1$, the authors prove global existence of solutions and show that the left boundary $g(t)$ retreats to infinity with a strictly positive asymptotic speed $c(\delta)$, while the shifted density $u(t,x+g(t))$ converges uniformly to a retreating semi-wave profile $q_{c(\delta)}(x)$. A novel touching method is developed, constructing parametric families of semi-waves via auxiliary perturbed problems to create upper and lower barriers that touch the solution at the moving boundary, enabling sharp control of $g'(t)$ and $u(t,x)$. The semi-wave analysis, including existence, uniqueness, monotonicity, and perturbation stability, underpins the asymptotic description and offers a technique potentially applicable to other free-boundary problems with retreating fronts.

Abstract

We study a monostable reaction-diffusion equation of the form $u_t=du_{xx}+f(u)$ over a semi-infinite spatial domain $[g(t),\infty)$, with $x=g(t)$ 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 $[g(t),\infty)$ maintains a certain density $δ$ at the habitat edge $x=g(t)$. In the ``high-density'' regime, where $δ$ exceeds the carrying capacity of the favourable environment represented by a monostable function $f(u)$, it is known (see \cite{DLNS} 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 $[g(t),\infty)$ allows us to prove that, as time $t$ converges to infinity, the free boundary $x=g(t)$ converges to $\infty$ with a constant asymptotic speed $c(δ)>0$ determined by an associated semi-wave problem, and the population density $u(t,x)$ has the property that $u(t,x+g(t))$ converges uniformly to $q_{c(δ)}(x)$, the semi-wave profile function associated with the speed $c(δ)$. It turns out that in the retreating situation considered here, some key techniques developed for advancing fronts in related free boundary models do not work anymore. This difficulty is overcome here by a ``touching method", which uses a family of lower and upper solutions constructed from semi-waves of some carefully designed auxiliary problems to touch the solution $u(t,x)$ at the moving boundary $x=g(t)$, thereby generating a setting where the comparison principle can be used to obtain the desired estimates for $g'(t)$ and $u(t,x)$. We believe this method will find applications elsewhere.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Lemma 2.2: A priori bounds
• proof
• proof : Proof of Theorem \ref{['th1.1']}
• Lemma 3.1
• proof
• Theorem 3.2