Head and tail speeds of mean curvature flow with forcing. (English) Zbl 1430.35125
Summary: In this paper, we investigate the large time behavior of interfaces moving with motion law \(V = -\,\kappa + g(x)\), where \(g\) is positive, Lipschitz and \({\mathbb{Z}}^n\)-periodic. We show that the behavior of the interface can be characterized by its head and tail speeds \({\bar{s}}\) and \(\underline{s} \), which only depend on its overall direction of propagation \(\nu \). We discuss the large time behavior of the moving interface in terms of \({\bar{s}}\) and \(\underline{s} \), which is shown to vary continuously in \(\nu \). In the laminar setting we show that when \({\bar{s}}>\underline{s}\) there exists an unbounded stationary solution as well as localized traveling waves with different speeds.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 53E10 | Flows related to mean curvature |
| 35R35 | Free boundary problems for PDEs |
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