Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations

TL;DR

This paper investigates the long-term behavior of solutions to a monostable reaction-diffusion equation, demonstrating convergence to a pushed front profile and linking the front's position to mean curvature flow with drift.

Contribution

It establishes the convergence of solutions to a pushed front profile in higher dimensions and connects the front's evolution to mean curvature flow with a drift term.

Findings

Solutions approach a pushed front profile over time.

The front's position is approximated by mean curvature flow with drift.

Conditions on initial data ensure convergence to the front.

Abstract

We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=\Delta_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), with the unstable equilibrium point $0$ and the stable equilibrium point $1$. Under the condition that the corresponding one-dimensional equation has a pushed front $\Phi_{c^*}(z)$ with $\Phi_{c^*}(-\infty)=1$, $\Phi_{c^*}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $\Phi_{c^*}(y-\gamma(x,t))$ for some $\gamma(x,t)$ as $t \to \infty$, if initially $u(x,y,0)$ decays sufficiently fast as $y \to \infty$ and is bounded below by some positive constant near $y=-\infty$. It is also shown that $\gamma(x,t)$ is approximated by the mean curvature flow with a drift term.