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

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.