# Interface Development for the Nonlinear Degenerate Multidimensional Reaction-Diffusion Equations

## TL;DR

This paper classifies the short-time behavior of interfaces in nonlinear degenerate reaction-diffusion equations, providing explicit formulas for interface asymptotics and local solutions based on parameters.

## Contribution

It offers a complete classification of interface behaviors and explicit asymptotic formulas for a class of nonlinear degenerate PDEs, extending understanding of interface dynamics.

## Key findings

- Interface can shrink, expand, or stay stationary.
- Explicit formulas for interface asymptotics are derived.
- Local solutions near the interface are characterized.

## Abstract

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.