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Interface Development for the Nonlinear Degenerate Multidimensional Reaction-Diffusion Equations

This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE \[ u_t-Δu^m +b u^β=0, \ x\in \mathbb{R}^N, 0<t<T \] with nonnegative initial function $u_0$ such that \[ supp~u_0 = \{|x|<R\}, \ u_0 \sim C(R-|x|)^α, \quad{as} \ |x|\to R-0, \] where $m>1, C,α, β>0, b \in \mathbb{R}$. Interface surface $t=η(x)$ may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters $m,β, α, sign\ b$ and $C$. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface.