| Contents |
| We study a parabolic problem of the form
$$
u_t = ({\mathcal L}(u_x))_x\quad \hbox{in } (0,b) \times (0,T).
$$
The characteristic trait of function ${\mathcal L}$, which we consider is that it is increasing and it has jump(s). A jump of ${\mathcal L}$ at $p_0$ leads to creation of facets with slope $p_0$.
We concentrate on two instances of ${\mathcal L}$:
1) ${\mathcal L}(p) = \hbox{sgn}\,(p+1) + \hbox{sgn}\,(p-1)$;
2) ${\mathcal L}(p) = \hbox{sgn}\,p + \epsilon p$.
The first case appears to admit infinite oscillations. We address this issue, showing that we have always only a finite number of facets with non-zero curvature. Moreover, with the help of the comparison principle for viscosity solutions we estimate the extinction time.
The second case is interesting because it is an instant of two types of competing diffusion. Counting facets becomes a more subtle job. We also show that their number is decreasing. |
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