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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
##### Keywords:
heterogeneous media; periodic perimeter; traveling speed
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##### References:
  Ambrosio L., Corso Introduttivo Alla Teoria Geometrica Della Misura Ed Alle Superfici Minime (1997)  Ambrosio L., Functions of Bounded Variation and Free Discontinuity Problems (2000) · Zbl 0957.49001  DOI: 10.1007/978-3-0348-7928-6  DOI: 10.1007/s00526-002-0186-5 · Zbl 1036.35001  DOI: 10.1137/100800014 · Zbl 1228.35026  DOI: 10.1016/j.matpur.2010.03.006 · Zbl 1209.37069  DOI: 10.1137/S0036141000369344 · Zbl 0986.35047  Brezis H., Operateurs Maximaux Monotones (1973)  DOI: 10.1016/j.matpur.2009.01.014 · Zbl 1180.35070  DOI: 10.3934/nhm.2009.4.127 · Zbl 1186.35013  DOI: 10.1016/j.jde.2009.07.019 · Zbl 1182.35073  DOI: 10.1007/PL00009915  DOI: 10.1016/j.jmaa.2007.06.052 · Zbl 1136.35011  DOI: 10.1017/S095679250800764X · Zbl 1185.53076  DOI: 10.2307/1971452 · Zbl 0696.53036  DOI: 10.1007/978-3-642-61798-0 · Zbl 0361.35003  DOI: 10.1007/BF01393250 · Zbl 0381.35035  Giusti E., Minimal Surfaces and Functions of Bounded Variation 80 (1984) · Zbl 0545.49018  DOI: 10.1016/j.anihpc.2004.10.009 · Zbl 1135.35092  DOI: 10.3934/dcds.2009.25.231 · Zbl 1179.35091  DOI: 10.1007/BF00250439 · Zbl 0305.49047  Monneau R., Ann. Sci. Ec. Norm. Supér 46 pp 215– (2013)  DOI: 10.3934/dcdsb.2004.4.867 · Zbl 1069.35031  DOI: 10.1007/s00526-007-0125-6 · Zbl 1166.35387  DOI: 10.1007/s000300050029 · Zbl 0887.35070  Ninomiya H., Methods Appl. Anal. 8 pp 429– (2001)  DOI: 10.1515/crll.1982.334.27 · Zbl 0479.49028
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