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 $$u_t-\Delta u^m +b u^\beta=0, \ x\in \mathbb{R}^N, 0<t<T$$ with nonnegative initial function $u_0$ such that $$supp~u_0 = \{|x|<R\}, \ u_0 \sim C(R-|x|)^\alpha, \quad{as} \ |x|\to R-0,$$ where $m>1, C,\alpha, \beta >0, b \in \mathbb{R}$. Interface surface $t=\eta(x)$ 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 $m,\beta, \alpha, sign\ b$ and $C$. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface.