## Optimal rate of convergence to the motion by mean curvature with a driving force.(English)Zbl 1171.35009

The author studies the rate of convergence of hypersurfaces associated with solutions of a singularly perturbed parabolic equation with double well potential $$\Psi(r)=(1-r^2)^2/4$$ and forcing term $$g$$, which approximates the mean curvature flow with driving force: $\begin{cases} u_t^\epsilon -\Delta u^\epsilon + \frac{1}{\epsilon^2}\psi(u^\epsilon)=\frac{c_0}{\epsilon}g, \;\;\mathrm{in} \;(0,T)\times \mathbb{R^N}, \cr u^\epsilon (0,x)=u_0^\epsilon(x), \;\;x \in \mathbb{R^N}.\end{cases}$ Here $$\epsilon$$ is a parameter, $$\psi(r)=\Psi'(r)$$, $$c_0$$ = constant, and $$u_0^\epsilon$$ is a bounded, continuous function.
It has been shown that the interface $$\Gamma^{\epsilon}(t)=\{ x\in {\mathbb R}^n: u^{\epsilon}(t,x)=0 \}, 0\leq t \leq T,$$ converges to a hypersurface $$\Gamma(t)_{0\leq t \leq T}$$ moving by the mean curvature flow as $$\epsilon \downarrow 0$$. The author investigates the optimal rate of convergence before singularities form. He shows that the Hausdorff distance satisfies the estimate
$sup_{t\in[0,T]} d_H (\Gamma(t),\Gamma^\epsilon(t))\leq L(T) \epsilon^2,$
where $$L$$ is a constant depending on $$T$$. This improves current estimates and is optimal in the case of radial symmetry. Techniques in the proof include the construction of sub and super-solutions of the equation, suggested by formal asymptotics.

### MSC:

 35B25 Singular perturbations in context of PDEs 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations