Interface Development for the Nonlinear Degenerate Multidimensional Reaction-Diffusion Equations

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 \[ u_t-Δu^m +b u^β=0, \ x\in \mathbb{R}^N, 0<t<T \] 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 $m>1,C,\alpha ,\beta >0,b\in \mathbb{R}$. 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 $m,\beta ,\alpha ,signb$ and $C$. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface.

arXiv comment17 pages, 1 figure

References in corpus (3)

• Evolution of Interfaces for the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations
• Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption
• Evolution of interfaces for the non-linear parabolic p-Laplacian type reaction-diffusion equations. II. Fast diffusion vs. absorption