Singular Limit of a Damped Wave Equation with a Bistable Nonlinearity
Dorothea Hilhorst, Mitsunori Nara · SIAM Journal on Mathematical Analysis · 2014
In this paper, we study the singular limit of the damped wave equation $\varepsilon^2 u_{tt}+\gamma u_t=\Delta u+\varepsilon^{-2}f(u)$ on $\mathbb{R}^n$, where $f$ is of bistable type and $n=2$ or $3$. In order to understand interfacial phenomena, we derive estimates for the generation and the motion of interfaces. We prove that steep interfaces are generated in a short time of order $O(\varepsilon^2|\ln\varepsilon|)$, and that their motion is governed by mean curvature flow in the limit $\varepsilon\to 0$ under the assumption that the damping is sufficiently strong. To that purpose, we prove a comparison principle for the damped wave equation and construct suitable subsolutions and supersolutions.