Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations
Ryo Kiyono
TL;DR
The paper analyzes the large-time behavior of nonnegative solutions to a monostable reaction-diffusion equation in $\mathbb{R}^{n-1}\times\mathbb{R}$ when the one-dimensional problem has a pushed front with speed $c^*>2\sqrt{f'(0)}$. It proves that, for suitably decaying initial data, solutions converge to a translated pushed-front profile $Φ_{c^*}(y-γ(x,t))$, where the moving interface $γ(x,t)$ is asymptotically governed by mean curvature flow with a drift, i.e., the interface evolves as a geometric flow plus a constant drift. The authors construct supersolutions and subsolutions and analyze $ω$-limit points and level-set geometry to link the interface to a semilinear approximation of the mean curvature flow. This extends known bistable-type results to pushed fronts in monostable equations and clarifies how non-compact initial data selects a curved interface evolving by a geometric flow, with implications for pattern formation and front propagation in higher dimensions.
Abstract
We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=Δ_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), 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 $Φ_{c^*}(z)$ with $Φ_{c^*}(-\infty)=1$, $Φ_{c^*}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $Φ_{c^*}(y-γ(x,t))$ for some $γ(x,t)$ 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 $γ(x,t)$ is approximated by the mean curvature flow with a drift term.