A finite difference scheme for some nonlinear diffusion equations in an absorbing medium: Support splitting phenomena

摘要

In the porous media equation v(t) = (v(m))(xx) (m > 1), it is well known that there appears a finite propagation of the initial support and that, if the initial support is connected, so is supp v(t, .) for t > 0. When the effect of the absorption is considered as the additional lower order term -cv(p) (c > 0, p > 0), the possibility that the support will split is expected. Rosenau and Kamin [Phys. D, 8 (1983), pp. 273-283] tried the numerical computations and suggested the support splitting phenomena. But the theoretical justification is not discussed. In this paper, such phenomena are investigated by use of finite difference schemes, and the sufficient conditions under which the support begins to split are obtained in the specific case where m + p = 2 and 0 < p < 1. [References: 25]