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Long-time behavior of the mean curvature flow with periodic forcing. (English) Zbl 1288.35071
The existence and the asymptotic behavior as $$t \to \infty$$ of the traveling wave solutions of the following equation are studied: $\begin{cases} u_t = \sqrt{1 + |Du|^2}\mathrm{div}({\frac{Du}{\sqrt{1 + |Du|^2}}) +g \sqrt{1+|Du|^2}} \text{ in } (0, +\infty) \times\mathbb R^n,\\ u(0, \cdot) = u_{0}.\end{cases}$ Assume that there exists $$A \subseteq (0, 1)^n$$, such that $\int_A g(y)dy > P er(A, T^n ),$ where $$P er(A,\mathbb T^n )$$ is the periodic perimeter of $$A$$, it is proved that one may define a unique traveling speed c, such that the equation: $-\mathrm{div}(\frac{D \psi}{\sqrt{1 + |D\psi |^2}}) = g(y) - \frac{c}{\sqrt{1 + |D\psi |^2}}$ has a $$\mathbb Z^n$$-periodic solution $$\psi$$. Under the assumption that there exists a global traveling wave solution $$u(t, y)$$ it is shown that $$u(t, y) - ct \rightarrow \psi(y)$$ in $$C^{1+\alpha} (\mathbb R^n )$$.

MSC:
 35B40 Asymptotic behavior of solutions to PDEs 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 35C07 Traveling wave solutions 35K59 Quasilinear parabolic equations 35K15 Initial value problems for second-order parabolic equations
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