Evolution of Interfaces for the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations

TL;DR

This paper classifies the short-time behavior of interfaces in nonlinear p-Laplacian reaction-diffusion equations, providing explicit formulas and analyzing how diffusion and reaction terms influence interface dynamics.

Contribution

It offers a complete classification of interface evolution for the nonlinear p-Laplacian reaction-diffusion equations, including explicit formulas for solutions and interfaces.

Findings

Interface can expand, shrink, or stay stationary depending on parameters.

Explicit formulas for interface and solutions are derived.

Methods involve nonlinear scaling laws and barrier techniques.

Abstract

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration \[ u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0 \] The interface may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters $p,\beta, sign~b$, and asymptotics of the initial function near its support. In all cases, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. The methods of the proof are based on nonlinear scaling laws, and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves.