Quasi-optimal error estimates for the mean curvature flow with a forcing term

We study a singularly perturbed reaction-diffusion equation with a small parameter $\epsilon>0$. This problem is viewed as an approximation of the evolution of an interface by its mean curvature with a forcing term. We derive a quasi-optimal error estimate of order $\O(\epsilon^2|\log\epsilon|^2)$ for the interfaces, which is valid before the onset of singularities, by constructing suitable sub and super solutions. The proof relies on the behaviour at infinity of functions appearing in the truncated asymptotic expansion, and by using a modified distance function combined with a vertical shift.