Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
1999-08-11
Nonlinear Sciences
Pattern Formation and Solitons
26 pages, Latex, 13 ps figures
Scientific paper
10.1016/S0167-2789(99)00209-2
In the focusing problem we study a solution of the porous medium equation $u_t=\Delta (u^m)$ whose initial distribution is positive in the exterior of a closed non-circular two dimensional region, and zero inside. We implement a numerical scheme that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition is a function with circular level sets distorted with a small sinusoidal perturbation of wave number $k\geq 3$. We find that for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a self-similar regime, characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational symmetry number k. For m greater than the critical value, the final form of the interface is circular.
Angenent Sigurd B.
Aronson D. G.
Betelu S. I.
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