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

We study the behavior of solutions of a monostable reaction-diffusion equation $u_t={\mathrm{\Delta }}_{x}u+u_{yy}+f(u)$ ($x\in {\mathbb{R}}^{n-1}$, $y\in \mathbb{R}$, $t>0$), 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 ${\mathrm{\Phi }}_{c^{\ast }}(z)$ with ${\mathrm{\Phi }}_{c^{\ast }}(-\mathrm{\infty })=1$, ${\mathrm{\Phi }}_{c^{\ast }}(\mathrm{\infty })=0$, we show that the solution $u(x,y,t)$ approaches ${\mathrm{\Phi }}_{c^{\ast }}(y-\gamma (x,t))$ for some $\gamma (x,t)$ as $t\to \mathrm{\infty }$, if initially $u(x,y,0)$ decays sufficiently fast as $y\to \mathrm{\infty }$ and is bounded below by some positive constant near $y=-\mathrm{\infty }$. It is also shown that $\gamma (x,t)$ is approximated by the mean curvature flow with a drift term.